continue from PAP_JYOT

These will not only make the total correct, but also bring about agreement with the [Siddhantas], which all generally agree with the actual as given by modern astronomy.

Degrees moving

behind -15o -60o -60o -90o -50o -70o -15o -360o

`Days' given 56 188 108 73 68 220 56 769

Actual 'days' 54 188 106 72 75 220 54 769

The great difference in 'days' between the 68 given and 75 actual must be explained by their following next to be retrograde period, where even a large number of days can produce a very small difference in degrees. So, correction to whole degrees can produce this difference in days.

Note 2. Verse 67 means that 6329 'days' after epoch, there is conjunction, which repeats after each synodic cycle. The cycle for Mars is 768 3/4 'days'. So instead of dividing by 768 3/4, we are asked to multiply by 4 and divide by 3075. To get back the true remainder, the remainder here is divided 4.

Note 3. (a) In the case of all star-planets the total 'days' should be equal to the days of the respective cycle.

(b) In the case of the superior planets, (Mars, Jupiter and Saturn), the degrees are all negative and add upto --360o. When the given degree is numerically greater than the corresponding 'days', (for eg.,-90o for 78 'days' here)the planet is retrograde.

(c) The heliacal setting and rising are at teh beginning and end of the cycle for all. But for the two inferior planets, Mercury and Venus, there is another setting and rising at inferior conjunction, when the two are retrograde.

(d) The rising and setting are given by observation at different regions and different conditions of the atmosphere, and therefore vary among the [Siddhantas].

(e) In the case of the inferior planets the degrees should add upto zero. When the degrees are positive and greater than the days, the planet is gaining upon the sun, and the total gain is its elongation. when the

degrees are less than the 'days', the planet is lagging behind. When the degrees are negative and numerically greater than the 'days', the planet is retrograde and comes at the middle of the cycle, if the cycle begins and ends at superior conjunction.

Example. Compute Mars at 1,20,553 days from epoch.

The solar days are, 1,20,553X360./.365-15-30=118,817.5, and the sun is 17o.5, taking the sun at epoch as zero, which it nearly is, as already shown.

118,817.5-6329=112,488.5

112,488.5X4./.3075 leaves the remainder 1004.

1004 ./.4 =251, real remainder of 'days'.

During this period we get the movement :-15o in 56'days', -60o in 188 'days', and -4o for the 7 'days' remining, total -79o. Adding -79o to the sun, 17o.5, True Mars =298o.5

Verses 70-72 deal with Mercury.

Ë´É (Ê´É) ¶ÉʶɴɺÉÖ®úºÉäxuù(xuäù)

xÉ´ÉxÉ´É (ªÉ¨É) MÉÖÊhÉiÉä %EÇò®úÉ [¨É]MÉÖhɦÉHäò*

MÉÖhÉEòÉ®úRóiÉä ±É¤vÉÉ-

xªÉ½þÉÊxÉ ¶ÉÒiÉÉÆ ¶ÉÖ{ÉÖjɺªÉ **70**

nù¶ÉʦÉnÚù (uùÉÇ)nù¶É½þÒxÉÉ& (xÉ)

|ÉÉMÉÖÊnùiÉÉä ¨ÉÊxÉ (xÉÖ) ʦɰüxɦɷÉÉƶÉÉ&(xÉxnùÉƶÉÉ&)*

vÉÞÊiÉʦÉ& ZÉ(ºÉ) xÉ´ÉÉä%ºiÉÉǨÉiÉ&

ËjɶÉÎfÂø¨É°ünäùÊkÉ (ÊiÉ) ºÉ®úºÉÉ·É& (¶É®úÉÊ·É&)**71**

+¹]õÉnù [¶É]ʦÉ&[ºÉ]xÉ´É&

¹ÉÉäb÷[¶É] ʦɷÉɹ]õ (EÇò)´ÉÔVÉiÉÉä %ºiÉʨÉiÉ&*

{É·ÉÉnÚùºÉÖʦÉxÉÇ´É´É-

ÊVÉiÉÉä ÊxÉ®ÆúºÉ (¶ÉÆ) ¤ÉÖvÉÉä%Ê{É (delete)ªÉÉÊiÉ**72**

70. Subtract 14,681 from the 'days'. Multiply the remainder by 29, and divide out by 3312. Take the remainder and divide by 29. The 'days' for Mercury in the cycle is got.

71. In 10 'days' Mercury falls back by 12o, and rises in the east. In 14'days' more he lags by 9o. Then in 18 `days' he gains 9o. Then he sets, and in 30 `days' gains 25o and rises heliacally.

72. Then in 18 `days' he gains 9o. In 16 `days' he lags 12o, and sets in the west. Then in 8 'days' he lags 9o, and gets into conjunction.

Note1. In verses 71 and 72 my corrections are based on the need to conform as nearly as possible to reality, and they are also, as far as possible, kept close to the text. ['sarasvih], may also be [jala'svih], i.e. 24. The values are :--

-------------------------------------------------------------------------------- 1 2 3 4 5 6 7 Total

--------------------------------------------------------------------------------Given -12o -9o +9o +25o +9o -12o -9o 0o

degrees [vakrasta][vakra] [or+24] [vakra][vakrasta]

[asta]

Given 10 14 18 30 18 16 8 114

'days'

Correct 8 16 18 30(?28)18 18 6(?8) 114

-------------------------------------------------------------------------------- In columns 6 and 7 it should be -9o and -12o, or at least -10o -11o though the text etters are unmistakable.

Note 2. The constant for subtraction, 14,681 shows that the planet is in superior conjunction, since the epoch position of the planet must agree with that given by modern astronomy and other [Siddhantas], at least with in a few degrees. (See table appended). If so, the Table of cycle motions given should begin and close with superior conjunction. BUt in the Table given, the cycle begins and closes with the inferior conj. as can be seen from the retrograde motion with which the Table begins and ends, and the most rapid motion (aticara) coming at the middle.

The astronomer of very inferior calibre, who has made this interpolation, has been misled by the two sets of heliacal rising and setting in the case fo the inferior planets, Mercury and Venus. He has wanted to begin the motions with the rising in the east and setting in the west, to fall in line with others not realising that this occurs during it sretrogradee motion which falls at the inferior conjunction coming in the middle. This is another proof that VM cannot be the author of this set of dealing with teh star-planets. (He has committed the same mistake int he similar case of Venus, where the mistake can be seen glaringly when the true Venus got is compared with that of the other[siddhantas] or modern astronomy). If he

does want to begin with rising in the east and end with the inferior conj. and to do this he must add to the days to be subtracted half the cycle days, equal to 573/29 days. (The cycle days=3312./.29=1146/29).

Next follows Jupiter, in three verses:

®úʽþiÉä¹]õ (delete)ÊuùªÉ¨É¶É®úÉι]õʦÉ& (¹]õlÉÉÆ)

xÉÉMÉ (xÉMÉÉ) ½þiÉä ÊuùʴɹɪɷÉ(º´É) ®úÉʷɽþiÉä*

ºÉ{iɽþiÉä näù´ÉMÉÖ®úÉè (®úÉä&)

¦É´ÉÉxiÉ Ênù´ÉºÉÉ ÊiÉ®úÉƺÉÆMɨªÉÉ& (ÊxÉ®Æú¶ÉºªÉ)** 73**

ºÉ´Éè%EòÉ (EòÉÇ) iºÉƶÉÉävªÉÉ&

¹ÉÉäb÷¶ÉǦÉuùÉÇnù¶ÉÉäÊnùiÉ& |ÉÉEÚò (SªÉɨÉ)*

EÞòiÉʴɹɪÉè& EÞòiÉ´ÉänùÉ&

ºÉ{iÉiªÉÉ ºÉÉhÉÇ´ÉÉ& ¹Éι]õ&**74**

xÉ´É ÊnùÎM¦É& ¶ÉÚxªÉÉEòÉÇ&

[ºÉÉ]hÉÞ¶ÉxªÉÉ ®úºÉº´É®úÉtÉʦÉ& (¶SÉè´É)*

¶ÉÚxªÉEÞòÊiÉ (iÉè uùÉÇÊjÉ®úÉiÉÂ

iÉÉäiÉ֨ɺiÉMÉÉ (´ÉiÉÉä%º´ÉMÉ&) ¹ÉÉäb÷¶ÉʦɮúEÇòiÉ (EòÉÇ)**75**

73. Deduct 16,522 from the 'days'. Multiply the remainder by 7 and divide by 2752. Divide the remainder here by 7. The 'days' from conjunction are got.

74-75. All degrees given are to be subtracted from the sun. In 16 'days' he moves 12o and rises in the east. Then in 54, 70, 49, 88, 40, days he moves 44o, 64o, 120o, 76o, 32o. Then he sets in the west, moves 12o in 16 days and joins the sun.

Note 1. The first foot of verse 73 is faulty containg 3 [matras] extra, and corrupt. So it has been corrected. The rest of verses 73 and 74 are with TS's emendations. In 75, all emendations are TS'S, excepting those for grammar.

Note 2. The cycle is 2752 ./.7=393 1/7 days. The days and degrees are :

1 2 3 4 5 6 7 Total

Degrees -12o -44o -64o -120o -76o -32o -12o -360o

Given days 16 54 70 109 88 40 16 393

Near correct days16 54 70 109 88 40 16 393

Venus is dealt with in 3 verses:

xɪÉxÉÉEÇòʨÉÊiÉnÖù (vÉÞiÉÒxuù)xÉä

ÊuùMÉÖhÉä °ü (°ü){ÉäÎxuùªÉè& º´É (ªÉä·É)®èú¦ÉÇHäò*

¶Éä¹É& (¹ÉÆ)ªÉkÉnù (où)ʱÉiÉÆ

¦ÉÞMÉÖiÉxɪÉÊxÉ®Æú¶ÉÊnù´ÉºÉÉ&ºªÉÖ&**76*.

ʴɹɪÉèxÉÇ´ÉEòʴɽþÒxÉ&

|ÉÉMÉÖÊnùiÉκ´ÉÊlÉ[ʦÉ]®äúEò{É (ªÉ) ¨É½þÒxÉ&*

´ÉºÉÖEÞòiªÉÉ ÊiÉlªÉÖxÉ (lªÉÚxÉ&)

EÞòiÉÉι]õʦÉ& ºÉ[{É\SÉEòÉϺjɶÉiÉÂ]**77**

¹É¹]õÉ ({É\SÉÉ) [¹]õ] EäòxÉ ºÉnù¶É (¶É&)

ÊxÉ®ÆúºÉiÉÉä (¶ÉiÉÉä) iÉÉä ʴɱÉÉä¨ÉMÉ& ¡ò·ÉÉiÉÂ*

=nùªÉ (nèù) ÊiÉ ÊlÉ% (ÊxÉ) ®Æú¶ÉÉEòɱÉÉä (±Éä)

xÉªÉ (|ɪÉÉ) ÊiÉ ´ÉÉ (SÉÉ)ºiÉÆ Ê´ÉxÉÉlÉ (±ÉÉä¨É) MÉÊiÉ&**78**

76. Dedcut 1,18,122 from the 'days'. Multiply by 2 and divide by 1151. Take the remainder and divide by 2. We have the 'days' from the conjunction of Venus.

77-78. In 5 'days' Venus lags by 9o, and rises in the east. In 15 days he lags 21o. In 64 days days he lags 15o. In 164 days he gains 10o, and joins the sun. Then, moving in accordance with teh reversed order of the days for cycles given, he rises after the days given from setting to conjunction (i.e, 40 days) in the west, and moves till he reaches the setting in retrograde (and getting into the inferior conjunction)section.

NOte 1. I have corrected [mitindu] into [dhrtindu] for agreement with the sun in superior conjunction which alone fits. TS's correction [mahindu] does not bring agreement with the sun either at superior conj. or inferior conj. There can be another possible correction [matindu] [mati is 8.]. In the 64 days, flanking the retrograde, the days may be a little more or less, since a small error of observation can produce a difference of a large number of days. The lacunae is filled by me with [sa-'pancakas-trim'sat], meaning 35o, to fit the number of degrees wanted to make up the total zero, and fitting the number of days given. TS's emendation, [krtastabhih] will be far from fitting the total. Moreover, their filling the lacuna by [sesuh] meaning 5o is quite inadequate to make up zero. I have emended [sastastakena] into [pancastakena], meaning 5X8=40, which will fit the number of days. Also, 10o synodic motion there requires 40 days and it is also the period from setting to going into superior conjunction. [sasta] is patently wrong

spelling, and [sastastakena] is meaningless. But TS keep it, which is wrong. That this is the segment of heliacal setting to conj. cna be inferred from [udayati] given for the next segment, and 10o for heliacal setting and rising at superior conj. is given by many [siddhantas]. The other minor emendations are TS's.

Note 2. 118,122 seems to be a very large subtactive constant, equal to more than 300 years, while all others are very near VM's time. But I cannot think of any other number to fit.

Note 3. The maximum elongation is seen to be 45o, correctly, (cf.Table).

Degrees -9o -21o 15o +35o +10o +10o +35o 15o -2 -9o 0

[vak- [vakra] asta asta asta [vak-

rasta] rasta]

Given 5 5 64 164 40 40 164 64 15 5 576

days

Correct 5 15 64 164 40 40 164 64 15 5 576

days

Note 4. The remark about Mercury, that the cycles begin and end with the superior conjnction according to tghe subtractive constant given, but the motions in the cycles begins and end with the inferior conjunction, holds in the case of Venus also, showing thereby that the author is an ignorant imposter, and cannot be VM. To correct the fault, 287 3/4 'days' should be added or subtracted from the subtractive constant.

Example. Compute Venus at 1,20,553 days from epoch:

If the subtractive constant given in the text is used, 1,18,817.5 (already found in the example in Mars)-1,18,122=695.5.

This X 2 ./. 1151 leaves the reaminder 240.

This divided by 2 gives 120 'days' gone in the cycle. We have for the first 5 days-9o, and the next 15 days-21o and the next 64 days-15o and the remaining 36 days, 36X35./.164=7o 40', totally-37o 20'. Adding the sun 17o.5 already found in the example for Mars, the true longitude of Venus is -20o, i.e. 340o. (The example in the [Saurasiddhanta] for the same date has given 46). The error in Venus, in using this method here, is

66o. On the other hand, let us use the cycle order re-arranged to begin from superior conjunction. It is 10o for 40 days, 35o for 164 days etc. We have 10o for 40 days, and the remaining 80 days, 80X35o./.164=17o, total 27o. Adding the sun, 17o.5, we have, true Venus, 44o.5. This is close to the correct 46o. This expose the ignorance of the impostor.

Saturn follows in the next three verses:

Ê´ÉvÉÞÊiÉ¶É ®úºÉ¹É]ÂõEò´ÉEÇò (delete)¶É¶ÉÉRÂóEäò

jÉ´vÉä vÉÞÊiÉ°üpù¦É ÊVÉiÉä %OÉRóiÉä*

ºÉÉè®úº´É ºªÉ) vÉÞÊiÉ [¦É]¹]õÉʦɮú(ι]õ&)

ºÉÉvÉÌEò (SÉ) ½þ ÊxÉ°üÊnùiÉ& |ÉÉEÂò**79**

+¹]õxÉ´ÉÊiÉVªÉÉÇ (iªÉÉ) xÉ´ÉÊiÉ-

nÇù±ÉÆ SÉ ¨ÉxÉÖ ¦É ºjɪÉÉänù¶ÉʴɽþÒxÉÉ&*

MÉÖhÉ°üuèùUô ¶ÉÚxªÉÉEòÉÇ&

nÚùlÉÚxÉäxÉ ¶ÉiÉäxÉ ¶É ʶÉxÉ´ÉEò¨É **80**

+ÊiÉVÉMÉ... ®úEòÉÇ[VÉMÉiªÉÉ ºÉÉvÉÉÇEòÉÇ]-

®ú(xÉ) ºiɨÉäiªÉÆ (iªÉ) iÉÉä xÉ´ÉÊiÉ(EÖò)ʦÉÌ´É (ÌxÉ) ®Æú¶É¨É *

¹ÉÉäb÷¶É ºÉÉvÉÉÇiºÉÉè (xÉ ºÉÉè)®ú-

·É®úÊiÉ ®ú´Éä ººÉ´ÉÇnùÉ ½þÒxÉ& **81*.

70. Subtract 16,518 from the 'days', multiply by 3 and divide by 1118. Take the remainder here and divide by 3. The days left over in the cycle are got. In 18 days Saturn lags behind by 16 1/2, and rises in the east.

80-81. In 98, 14, 113, 98, and 13'days' he falls behind 90 1/2o, 13o, 120o, 91o,and 12 1/2o respectively. Then Saturn sets in the west, and joins the sun passing 16 1/2o in 19 days.

Note 1. In verse 79, ¹É]ÅõEò´ÉEÇò is patently extra, forming syllables not required for the foot, and has been deleted by me as also by TS. +¹]õ ʦÉ& is corrected into +ι]õ& ´É¤É ±ÉÉ, as also by TS to conform to grammar and facts.

In the rest of the minor corrections there is no difference between our corrections.

In verse 80, I have retained the Êuù in uùªÉÚ´ÉäxÉ, while TS have made it tÖ, meaning one day, which is not necessary, and which leads into trouble later, needing further correction. The minor corrections are common to both. In 81, I have filled up the lacuna by (iªÉÉ ºÉÉvÉÇ) while TS have made it

(ÊiÉʦɮúvÉÇ).The word is +ÊiÉVÉMÉiÉÒ and not +ÊiÉVÉMÉÊiÉ&, which alone can justify TS's... ÊiÉʦÉ& Also only ºÉÉPÉÉÇEÇò can mean 12 1/2, but their +PÉÉÇEÇò can mean only 6. I have corrected xÉ´ÉÊiÉ into xÉ´ÉEÖò keeping the. But TS have corrected it into +ÊiÉvÉÞÊiÉ, making unnecessary changes in the lettering, though both of us mean the same. The rest of the corrections are minor, and common to bot of us.

Note3. The text ends abruptly without the usual verses giving details about the author, his parentage, date fo writing etc.

Note 4. The colophon is simply "The star-planets of the [Pauli'sa siddhanta] ends". But after this is found details about the scribe, his lineage, his time of writings, viz. 1673 Vikrama Samvat, and 1538 ['Saka], equal to 1616 AD, and the purpose of his copying the work, ("for his own reading and helping others" i.e., other astronomers).

THE 'SAKA ERA Of VARAHAMIHIRA ['SALIVAHANA 'SAKA]*

Introduction

With reference to chronology the word 'Saka is used in two senses: (1) As a common noun meaning any era (as for e.g., in the terms [Yudhisthira 'Saka], Vikrama 'Saka, Malava 'Saka, 'Salivahana 'Saka etc). and , (2) As a proper noun to mean a particular era called the 'Saka Era is the same as what later is generally referred to as the ['Salivahana 'Saka] which commenced with the month of [Caitra] occurring in 78 A.D., i.e., at the end of 3179 years of the Kali Era, for it can be shown that all astronomical works and commentaries thereon, wherever they mention a 'Saka ERa, mean only the 'Salivahana Era, starting, as mentioned above, from 3179 Kali elapsed. But some like the late T.S.Narayana Sastri, Gulshan Rai, Kota Venkatachelam, and V.Thiruvenkatacharya (VT) take the word to mean a certain Cyrus Era or Andhra Era, Which they say, started from 550 B.C. Kane mentions two others of the group: Jagannatha Rao, Age of Mahabhara war (1931), C.V.Vaidya starting the 'Saka-Kala from Buddha's [nirvana]. We noe find that T.S.N is the source for all these people, and almost every argument used by them is his. In his Age of 'Sankara he has used a [Yudhisthira Era] of 3140 39 B.C., and a 'Saka Era of 576 BC, which he later shifted to 550 B.C. Still another view is expressed by K.Rangarajan, who takes it to mean an era which

commenced from 523/22 B.C. with the first Viceroy fo India appointed by the Persian Emperor. They also try to show that it never means the ['Salivahana 'Saka]. What astounds us is that even where there is clear evidence that 'Salivahana 'Saka is to be taken, (in the shape of statements that 3179 is to be added to the years gone in the 'Saka Era to get the years gone in Kali) these scholars ignore it implicitly as in teh case of the 'Saka Kala mentioned by Brahmagupta and [Bhaskara] II. When this is the fate of such clear evidence, we need not be surprised if they identify with their alleged Cyrus or Andhra Era, the 'Saka Era mentioned in giving the epochs of Karanas (astronomical manuals) as in the case fo the [Pancasiddhantika](PS), the [Khandakhadyaka] or the [Laghumanasa], or in giving the date of a work given by the author, as for instance by [Bhattotpala] at the end of his commentary on the [Brhajjataka] or in inscriptions like the [Aihole] Inscription, or in sundry other places as in the [Brhatsamhita] I.13, in all of which cases the identification has got to be made by examining the months and [tithis] and [Ksepas] mentioned therewith.

The reason why they want to identify the ['Sakakala] with the so-called Cyrus or Andhra Era is this: They believe that there was a "plot hatched by European Indologists" to post-date by several centuries the ancient events of Indian history, and that most Indian Indologists have become unconscious victims of that plot. They try to show that the [Yudhisthira] and [Saptarsi Eras] are every-where identical, and were actually started 25 years after the beginning of the Kali Era. Using this they try to show that it is [Samudra Gupta] of the [Gupta] dynasty that is to be identified with the Sandracottus of the Greks, and not Candragupta Maurya, which latter identification has been taken by the European Indologists as the sheet-anchor of Indian chronology, and the chronology of the dynaties before and after that time is established thereform. Now, the identification of 'Sakakala] with 'Salivahana 'Saka stands in their way. Hence their attempt to identify it with the so-called Cyrus or Andhra Era whose very existence is a matter of dispute, there being no evidence for it.

Most historians have not taken these people seriously, thinking that the extravagance of their claims would be a deterrent to the acceptance of their views. But attempts have been made by Professors Gulshan Rai and

VT to give astronomical and mathematical proofs to show that [Varahamihira (VM) belongs to 123 B.C. and not to 505 A.D., (as he is generally believed to be ), and thereby that the 'Sakakala mentioned by VM is the Cyrus or Andhra Era. They also attempt to show that the 'Sakakala mentioned by Bhattotpala as stated above is the Cyrus or Andhra ERa, and therefore the 'Saka year 888 given him corresponds to 338 or339 A.D., which would mean that Brahmagupta, [aryabhata] [Bhaskara] I etc. must precede this date. The present article is intended to expose the hollowness of the above theory and to show that the astronomical arguments addueced in support of it (which to the lay reader may look formidable) are erroneous, and thus knock the bottom out of the claims of this set of writers.

[Varahamihira] uses the word ['Sakakala] in a few places in his works.

(1) In the [Brhatsamhita] he says:

[asan maghasu munayah sasati prthvim yudhisthire

nrpatau/

[sad-dvika-panca-dvi-yutah 'Sakakalas tasya

rajna's ca//XIII.3

"The Sapta-rsis were in the asterismal segment [Magha] when [Yudhisthira] was ruling over the earth. Any date by the 'Saka Era plus2526 gives the time from that king, i.e., the date in the [Yudhisthira Era]."

(2) In his [Pancasiddhantika](PS) the following occurs:

sapta-a'svi-veda-sankhyam 'Sakakalam apasya

caitra-'sukladu/

ardhastamite bhanau yavanapure

somadivasadye.//I.8//

"Deducting 427 of the 'Saka Era, (from the years in the Era) at the beginning of the light half of Caitra, which falls near sunset at [Yavanapura], beginning a Monday..."

(3) Br.Sam. VIII 20-21. This will be discussed, later.

(4) In [Pancasiddhantika], XII.2. but it is not used by these scholars.

In (1), a synchronism is found between the 'Saka Era and the [Yudhisthira Era. We shall not discuss this synchronism here but rest content with saying that whatever be the 'Sakakala mentioned in (2), it is

highly probable that the same is mentioned by (1). In (2), it is clear, the epoch of the [Pancasiddhantika] is given as 427 'Saka elapsed, which means the date of the work must be c.427 'Saka, and thus VM's time can be fixed. If as VT and others say the 'Sakakala meant here is the Cyrus or Andhra Era of 550 B.C., then the date of VM must be 427 years after 550 B.C., i.e., 123 B.C., which Gulshan Rai and VT have tried to establish by their special arguments. If it is same as the 'Salivahana 'Saka, then VM's date must be 427 years after 78 A.D., i.e., 505 A.D.

Here we do not propose to go into the question whether there was a 'Saka Era beginning from 550 B.C. or whether it is necessary to postulate such an era in view of the reference in the [Brhatsamhita 'sloka] quoted above which is discussed in the next paper 'The untenability of the 'Postulated 'Saka era of 550 B.C.' We shall confine ourselves to showing that the 'Sakakala of VM's PS is the 'Saligahana 'Saka, and therefore 427 'Saka (elapsed) corresponds to 505 A.D. As we have stated before, we shall also show that the special arguments to the contrary advocated by Gulshan Rai and VT and their conclusion that VM'S date is 123 B.C. cannot stand.

Internal Evidence for 'Salivahana 'Saka

There is plenty of internal evidence to show that the date meant by VM is 505 A.D. and not 123 B.C. It consists of the many [Ksepas] ( i.e. values of the Mean longitudes etc. at Epoch) found in the work, and the names of certain authors which it mentions . We shall take the [Ksepas] first.

In PS I.14 VM gives a [Saura] period of 1,80,000 years or revolutions of the Sun, in which there are 66,389 intercalary months and 10,45,095 suppressed [tithis]. From this we can get that there are in this period 2,406,389 revolutions of the Moon and 65,746,575 civil days. Comparing this with the Yuga-elements derivable from the Khandakhadyaka of Brahmagupta (which follows the [Ardharatrika] system of [Aryabhata] and whose elements are identical with those of a [Pauli'sa Siddhanta] wuoted by [Bhattotpala] in his commentary of the [Brhat Samhita], -not the [Pauli'sa] of the PS- We find that this is only a sub-yuga forming a twentyfourth part of the yuga given by them, and this suggests that the Yuga-elements of the

original [Saura Siddhanta], of which the [Saura] of the PS is a compendium, are identical with those of the [Khandakhadyaka] etc. mentioned above, these elements, therfore, may also be called hereafter, the [Saura] elements. Now, all these systems have arrived at 0o Mean longitude for the Sun, Moon, Mars etc., 3 [ra'sis] for the Moon's [ucca] (Apogee), and 6 [rasis] for [Rahu's] head viz., midnight at Ujjain, Thursday/Friday, 17/18, Febrauary, 3102 B.C. Taking that 'Saka 427 mentioned in PS I.8, refers to 'Salivahana Saka 427, (equivalent to 3606 Mean Solar years after the beginning of Kali), we have 1,317,123 days, 3 [nadis],9 [vinadis], after the midnight at Ujjain, Sunday/Monday, 20/21 March 505 A.D. The [Saura] of the [PS] takes this midnight as the Epoch for the compuation of its Star-planets (Tara-grahas), viz., Mars etc. If we compute the Mean Mars etc. for this epoch, using the Saura elements, the results agree with the respective Ksepas given in the PS to the second in the case of Jupiter and Saturn, with in 4" in the case of Mars and Venus, and 7" in the case of Mercury. Even this small difference is due to VM having arrived at the Ksepas using the shortcut given by him in the [karana] and the number of days gone in Kali as the [Ahargana] (days from epoch). If we also do the same there is complete agreement in teh case of Venus also, and the difference is reduced to 2" in the case of Mars. In the case of Mercury there is difference of a few seconds still, which may be due either to VM desiring to give its (Ksepa) correct to the minute only, or to sone defect in the manuscript reading which has omitted the seconds; and one of the manuscripts has actually a reading '[vilipti]' here. For the Mean Sun and Moon, and the Moon's Ucca and [Rahu], the epoch taken is the Midday at Ujjain just preceding the epoch of the Star-planets, i.e., the midday of Sunday. Here too, checking the Ksepas in the manner given for the Star-planets, we find perfect agreement with in 4" in the case of the [Ucca]. In the case [Rahu] the available manuscripts are so vitiated that Thibaut and Sudhakara Dvivedi (T-S) have failed to give the Ksepa fully. Using the letters available in the manuscripts, the relevant verse may be read as:

[trighana'sataghne navakaikapaksaramendu-

dahanasat-sahite/

svarayamavasubhutarnavagunadhrti-bhakte

Kramad rahaoh//IX.6//]

The Ksepa for [Rahu] enunciated in this verse as reconstructed above, agrees within 1" with its value according to the [khandakhadyaka]elements.

This perfect agreement is the reason why S.B. Dikshit has retained the date March 505 A.D. in spite of the difficulties he encountered in interpreting PS I.8 with reference to the [Saura]. For, no date, within many thousand years before or after 505 A.D. will agree with teh Ksepas in the manner shown above, not to speak of 123 B.C. When such is the case, VT quoting from Dikshit, a passage, which to those that have not read Dikshit's article in full will appear to involve an irremediable contradiction, says that 505 A.D. should be abandoned in favour or 123 B.C. on account of this. As the manner of VT's quoting from the article may create an impression in the reader's minds which Dikshit did not intend, and as VT himself concludes from the quotation that the agreement in the Ksepas discovered by Dikshit is null and void, and as he does not realise (as seen from his remarks under the quotation) that if he gives 123 B.c. for VM, he still has the responsibility to point out that the [Ksepas] agree with his date, we intend making the discussion a little elaborate so that we may give [Dikshit's] views in full with some pertinent observations on them.

The [Saura] Epoch occurs before the True [Vai'sakha 'Sukla] Pratipad, ending on Tuesday. Dikshit wants to reconcile this with the statement in PS I. 8, [caitra-'sukladau]. He considers the point that according to Mean reckoning, it is 'Adhika'-Caitra 'Sukla, but dismisses it, giving two objections: i. Why does not VM use the Epoch? Dikshit concludes by saying that '[caitra-'sukladau]' might stand, because [Amanta-Vai'sakha-'Sukla] is [Caitra-'Sukla] according to [Purnimanta] reckoning. so there is no troble at all for Dikshit as far as this goes. Therefore therw is no need for VT to abandon 505 A.D., go to 123 B.C., and show that the Caitra 'Sukla Pratipad of this year occurs on Wednesday, which weekday also is admissible according to one manuscript reading.(Cf. the readings given above). It should be remembered that VT can score a point only if the weekday, viz., Wednesday of the Caitra 'Sukla Pratipad of 123 B.C. alone can effect the reconciliation, and not the Tuesday of the Caitra 'Sukla Pratipad of 505 A.D.

But really speaking, there is no need to reconcile the [Saura] with any

part of PS I. 8, because it has reference only to the Romaka and the [Pauli'sa]. (If it can be applied to the Saura also, as indeed it can, it is good, but we have no right to demand it as Dikshit does) PS I. 8-10 give the computation of [Ahargana] according to the Romaka; and I.8 and 11 (and perhaps also 12 and 13) according to the Pauli'sa. I.8 gives the Epoch, which is thus the same for both. The Epoch is the beginning of Caitra 'Sukla which ends 427 'Saka, and the exact time is sunset at Yavanapura beginning Monday, i.e., 7 [nadis], 20 [vinadis] after sunset at Ujjain. This is equivalent to 37 [nadis] 20 [vindis] after Ujjain sunrise on Sunday, 20 th March 505 A.D. The [Ahrgana] with which to compute the Mean Sun etc. must be reckoned from this point for Romaka and the Pauli'sa, and their Ksepas are for this point. The expression Caitra-'sukladau is an indication that the months gone are to be counted from Caitra in computing the [Ahrgana], and the words "beginning Monday" is a check for the [Ahargana], Monday being stated to be the first day of the [Ahargana]. For, the [Ahargana] got by computation may be a day more or less than the correct one (a fact well known to astronomers) because the 'varying' True Tithis has got to be used in the formula; and checking by Monday beginning from the Epoch, viz., 0 Ahargana, it may happen that one day has got to be added or subtracted. This can be made clear by an example.

Problem: What is the Ahargana for Saturday beginning, next to the Epoch?

By counting we see we must get 5 days for [Ahargana]. Let us noe compute it. By the Romaka or Pauli'sa (or even Saura) almanac, the tithi gone at Saturday beginning is Caitra 'Sukla Caturthi. Using it in the above formulae enunciated in PS I. 8-11, 4 is got as [Ahargana]. But counting from Monday, 4 will give only Friday beginning. So we must add 1, and give 5 as the correct [Ahargana] if it should agree with Saturday beginning. We see here the use of the check. This is the purpose for which the weekday beginning the Epoch is given, and it si not merely to satisfy the curiosity of the reader. From this we can see that'Monday' is necessary, and 'Tuesday' or 'Wednesday' willbe wrong. So in PS I.8, 'soma-divasadye' or 'soma-divasadyah' must have been VM's original

reading. 'Bhauma' must have been a scribal error, or the correction of some reviosor who did not understand what was necessary, but thought thatthe weekday of the True 'Suklapratipada gone must be given here, and this must have given rise to 'saumya', a mixing of the two. Here T-S have rightly given the emendation 'soma-divasadya'. We may venture to give another suggestion, even if it may not appear very convincing to some. The emendation of T-S is not really essential and we can adopt the manuscipt reading 'saumya' as such and take it in the [yaugika] (derivative ) sense, meaning 'day pertaining to the Moon', i.e. Monday. Though there is the dictum 'Rudhir yogam apaharati' ('the meaning obtained by usage is stronger than that got by derivation'), still at such an ancient period as VM's, when the weekdays must have come into use very recently, the word [Saumya-divasa] might not have become [rudha] in [budha-vara] as it is noe. Also when other things require the derivative sense, we are permitted to abandon the [rudha] sense.

The above discussion has been necessaitated here by a desire to remove any doubt created in the readers' minds by Dikshit's dissatisfaction, which may be interpreted as going against the case for 505 A.D.

We may now proceed to show that the Ksepas of the Romaka and the Pauli'sa also as well as the [adhimasa] and [avama'sesa] of their rules for [ahargana], agree with 505 A.D. and not with 123 B.C. We have seen that the epoch for the Romaka and Pauli'sa is 37 [nadis], 20 [vinadis] from sunrise at Ujjain on Sunday, 20 March 505 A.D. (It must be noted that Dikshit does not question this,) The Romaka Mean Sun at Epoch can be seen to be 359o 34 1/4 by taking the [ahargana] as zero in PS VIII. 1, and working with the Ksepa left. This means that 26 [nadis] after Epoch, the Mean Solar month Mesa begins. In the same way we get the Mean Moon at Epoch from PS VIII.4, to be 356o 12' using the emended reading [Krtastanacakhaika]'; if the reading '[Krtastanavakaika]' found in the manuscript and followed by TS and Dikshit is used, it is 359o 19' at Epoch for TS, and 2o24' for Dikshit who taken that the Moon is given for sunset at Ujjain. From this we see that the Mean New Moon according to out interpretation will take place at 16 [nadis], 36[vinadis] after Epoch, i.e., 9 [nadis], 24 [vinadis] before the Mean Sun comes to Mesa. According to TS's value for the Moon, it is 24 [nadis], 42 [vinadis] before the Mean Sun

at Mesa; and according to Dikshit, 32 [nadis. It must be noted that accoriding to all the three interpretations, the Mean New Moon end begins the Mean Caitra and is very near the Epoch. The corresponding Ksepas of the Pauli'sa also will be found to give the same result. Thus the word Caitra in PS i. 8 presents no difficuylty, as it is mentioned only in relation to the Romaka and the Pauli'sa. Also, the [avama] and the [adhimasa 'sesas] fo the Romaka and the Pauli'sa found in PS I.9-11, agree, within the limits of accuracy, with the time fo the day when the respective New Moon occurs, and its distance from the beginning of the Mean Solar year as found from PS VIII and III. From the fore-going facts we see that the beginning of Caitra should fall very near the beginning of the Mean Solar year, which it does if we take 505 A.D. If 123 B.C. is taken, it is about 20 days away, and so there is disagreement with the Ksepas of PS I.8-11, and those of the Sun and the Moon in PS III, VIII and IX.

In the case of the [Saura], an examination of the Sun's and Moon's Ksepas given in PS IX. 1-2 will show that the Mean Solar year ends at 3 [nadis], 9 [vinadis] after midnight, Sunday/Monday, 20/21 March, (as we have already shown), and the Mean New Moon falls about 12 [nadis], 30[vinadis] after the Mean Solar year. Thus there is a Mean [Adhimasa] following the Epoch of the Star planets, i.e., midnight, which can be called Caitra, as Dikshit himself has accepted. As it is very close to (i.e. only 12 [nadis], 30 [vinadis] from) the beginning of the Mean Solar year, and as in computing the [Ahargana] the practice is to see whether an [Adhimasa] ahs taken place or not in the months gone used for reckoning, and adjust the number of [adhimasas] got by adding one or reducing the adhimasas by one, by treating a large [adhimasa-sesa] as unity or not counting one just got by computation, no harm will ensure if this [Adhika-Caitra] is treated as regular Caitra, taking the previous regular Caitra as [Adhikaphalguna]. And there is the advantage of dispensing with a Ksepa for months gone at Epoch. So even if PS I.8 applies to teh [Saura] as [Dikshit] thinks, the objection which he has to using the term '[caitradau]' for this vanishes, and there is no need to explain it in the manner he has done. Thus all the difficulties raised by Dikshit are answered, and not a trace of any objection for 505 A.D. is left.

We may now proceed to give another piece of evidence to show that the date cannot be 123 B.C. In PS XV we find the following 'sloka:

[Lankardharatrasamaye dinapravrttim jagada

caryabhatah//

bhuyas sa eva suryodayat praha Lakayam//20//

Here is a reference made by VM to [Aryabhata] and his two works, the well-known [Aryabhatiya], and his less known work referred to by later authors as his [Ardharatrika] System, manuscripts of which are yet to be discovered, but whose nature is fully given by [Bhaskara]I (6th-7th cent). in his [Mahabhaskariya]. In the [Kalakriyapada] of his [Aryabhatiya], [Aryabhata] says:

[sastyabdanam sastir yada vyatitas traya's ca

yugapadah/

tryadhika vim'satir abdas tadeha mama janmano

'titah//10//

This says that at 3600 Kali (expired) [Aryabhata] had completed twenty-three years of age, and 3600 Kali is 499 A.D. VM's reference is certainly to this [Arybhata] as can be gathered from the mention of him as the author of bothy the [Ardharatrika] and the [Audayika] systems. It follows from this that VM must be later or at least a contemporary of this [Aryabhata]. So VM can belong to 505 A.D. and not to 123 B.C. Thus all internal evidence and we have seen plentlyof it -points to 505 A.D. as the time of VM.

The [Ayanamsa argument examined

Now Profs. Rai and VT have advanced an argument based on [Ayanam'sa] to show that VM must be as early as 123 B.C. Being interspersed with mathematics, this argument may seem unassailable to some, unless its hollowness is exposed.

Being more full, we may discuss VT first. What VT says may be put succinctly as follows: (i) At the time of VM the Summer Solstice was at the end of the asterismal segment [Punarvasu], (or what comes to the same thing, the Vernal Equinox had a longitude of 3o 20' reckoned from the zero point of the Ecliptic), as gathered from VM's own statements in the PS and the [Brhat Samhita]. (ii) Taking the [Ayanam'sa] (i.e. the total precession) to

be zero at VM's time, there is an [Ayanam'sa]of 28o 15' in April 1909 A.D. (It comes to this : The Vernal Equinox has receded 28o 15' from the original position of 3o 20', and its position in 1909 is 335o5' from the zero point of the Ecliptic). (iii) Using the correct rate of precession (ayana-calana) per annum, 50".2585-nX0".000225, where n is the number of years before 1909, for a precession of 28o 15' to take place, n must be 2031 years. (iv) This means 2031 years before 1909, i.e. in 123 B.C., the [Ayanam'sa] was zero, and therefore 123 B.C. is the date of PS.

We admit that if (i)and (ii) are correct, (iii) and (iv) follow automatically. But (i) and (ii) are not correct, as we shall show. Relating (i) there are the following three 'slokas of VM, which are quoted by VT also :

[a'slesardhad daksinam uttaram ayanam raver

dhanisthadyam/

nunam kadacid asid yenoktam purvasastresu//

sampratam ayanam savituh karkatakadyam

mrgadita's canyat/

uktabhavo vikrtih pratyaksapariksanair vyaktih//

Br. Sam. III. 1-2//

a'slesardhad asid yada nivrttih kilonakiranasya/

yuktam ayanam tada'sit, sampratam ayanam

Punarvasutah//PSIII.21//

"Certainly at one time the turning of the Sun towards the south was from the middle of the [A'slesa] segment, and the turning north was from the beginning of the [Dhanistha] segment, because this is mentioned in ancient works.

"But now the turnings are from the beginning of the [Karkataka] and Makara rasi] segments, respectively. If this does not happen (in future, on account of precession), the amount of deviation is to be determined by observation." (Br. Sam. III.1-2).

"When the Sun turned away south from the middle of [Aslesa], it was proper for that time. But now the turning away is from Punarvasu." (PS III.21).

Now in the 'sloka from the PS, "from the middle of [A'slesa]" corresponds to the same phrase in the quotation from the Br.Sa, III.1; and "from Punarvasu" corresponds to " the beginning of Karkataka" in

Br.Sam.III.2 above, the same phenomenon of precession being described in both. So "from Punarvasu" must be taken to mean a point three quarters from the beginning of the segment, for that is the point corresponding to the beginning of Karkataka. But VT who wants the end of [Punarvasu] to be the turning point, wants us to shut our eyes to the specific reference to the "somewhere" in Karkataka, giving the reason that the word is found in a mere [Samhita] and not in a [Karana] like the PS. It seems he has not taken note of the many passages in the PS itself that specifies the 'beginning' of Karkataka as the point. For instance, in the 'sloka next but one, i.e., PS III.23, we find "[mesa-tuladau visuvad]", "at teh beginning of Mesa and Tula are the Equinoxes". One 'sloka later we have again:

[udagayanam makaradau rtavah 'si'siradaya's ca

suryava'sat/

dvibhavanakalasamanam, daksinam ayanam ca

karkatakat //PS III.25//

In XIII.10, we have, "At the end of [Mithuna] the Sun revolves at an altitude of 24o at teh N. pole".

Also VT says taht [Puunarvasutah] can mean only from the "end of Punarvasu". This interpretation is wrong. It only means "from Punarvasu", and can mean any point in it. [Gramatah pattanam pratisthate] does not only mean 'he starts from the border of the village'. It can mean any point in the village.

Further, the context in which [aslesardhat] etc, is found,itself specifies a point 1 3/4 segments from the middle of [A'slesa] and this point is three quarters of Punarvasu. In the immediately preceding 'sloka, VM states that [Vyatipata-punyakala] occurs when Sun plus Moon equals 17 asterismal spaces, i.e., 17X13o 20', or 226o 40', as opposed to out expectation taht it should occur at teh middle of the 14th (i.e.,at 180o) according to teh definition given in the 'Sastras. There is a difference of 46o 40' or 3 1/2 spaces that has to be explained. As yoga is obtained from the combined longitudes of the Sun and the Moon, a change of 1 3/4 spaces that has to be explained. As yoga is obtained of each, caused by the shifting of the origin of refernce will expalin the difference of the 3 1/2 spaces. This shifting of the origin, by the precession of the equinoxes, is mentioned in

[a'slesardhat] etc., and this must be 1 3/4 segments as required, and the point at 3/4 Punarvasu follows, for it is this point that is 1 3/4 segments behind the middle of [A'slesa].

Still another proof can be adduced to show that 3/4 [Punarvasu] is to be considered as the point in question. If it is the end of Punarvasu, the Vernal Equinox will be, as we have already stated, at +3o 20' from the zero point from which the longitudes of the Sun, the Moon etc., to compute the daylight, the shadow and other things, in short, for all work usually given in the [Tripra'snadhikara] of a [siddhanta], we must be instructed to deduct 3o 20' from the longitudes got by computation, and use this for the calculation, as the longitudes from the Vernal Equinox are to be used here. In a much as such an instruction has not been given anywhere in the text, we must take it that the zero point and the Vernal Equinox were coincident, which mean s that the Summer Solstice was at 3/4 Punarvasu. Now in (ii), VT has budgetted for an cut of 3o 20', 15, But the above fact will result in a cut of 3o 20', and VM will be lifted 240 years from the intended 123 B.C. towards the true place, 505 A.D.

Now we may pass on to consider (ii), viz., VT's statement that in April 1909 A.D. there is an [Ayanam'sa] of 28o 15', taking it to be zero at VM's time, when according to VT the VErnal Equinox was +3o 20' from the zero point, i.e.there is a total [ayancalana] of 28o 15', from VM'stime to 1909 A.D. VT makes up the 28o 15' necessary for him, by piecing together four different quantities: 9a) the distance between the Vernal Equinox and the zero point, both referring to VM's time, equal to 3o 20'; (b) the late L.D.Swamikannu Pillai's (LDS) calculation of the [Ayanan'sa] in 1909 to be 22o 25' which is the equivalent in degrees of the time from the Sun at the Vernal Equinox of 1909 to its entering the Sign Mesa in the same year according to [Surya Siddhanta]; (c) what VT calls a [Bija] (i.e. correction) of 2.18 days, equivalent to 2o9'; and (d)an error or observation equal to 16'. Of these four quantities, we have already seen that VT cannot have (a), by the fact that the summer Solstice was at 3/4 Punarvasu and not at the end of Punarvasu in VM's time. So 3o 20' is cut off from the 28o 15'. We shall not discuss (d), for we except to point out below what mischief even this can do. That leaves us (b) and (c) to deal with.

We shall take (b) first. VT uses the [Ayanam'sa] 22o 25' calculated by

LDS in a manner not intended by him. To understand how it is so, it is necessary to make clear the principle involved in the calculation. LDS found from the [Nautical Almanac] that at 0.2143-day on the 21st March 1909, the True Sun was the Vernal Equinox. He found that according to the [Surya Siddhanta] (Modern, not the Saura of PS), the True Sun reached the First Point of Mesa at 0.9492 day on 12th April 1909. From the difference between the two moments, equal to 22.7349 days, using the rate of motion of the sun at that internval, he calculated the [Ayanam'sa] to be 22o 25'. Suppose LDS had used the time of the True Sun at Mesa (Mesa Sankramana as it is called) fo some other Almanac like the Drk Almanac or VVakya Almanac, he would have got different [Ayanam'sas], for it is a well-known fact that Almanacs vary in their times of [Sankramana]. Which [ayanam'sa] are we to adopt? which is the 'correct', [Ayanam'sa]? By 'correct' [Ayanam'sa] is meant the total precession in degrees, fo the Vernal Equinox, from a specific point on the Ecliptic, which we call the zero point, during the interval 1909 and the time when we take the Vernal Equinox to coincide with thezero point, in out case the time of VM. According to this criterion none of the present-day Almanacs gives the 'correct' [Ayanam'sa]. The following is the reason: If the length of the year adopted by an almanac is the correct sidereal year, viz., 365 days 15 [nadis], 22.9 [vindas], so that at the end of every year the Sun returns to the specified zero point, then this way of finding the [Ayanam'sa] will yield the 'correct' [Ayanam'sa]. But the old system Indian Almanacs use, instead of the above correct Sidereal year, the Sidereal year of the [Aryabhatiya] or of the new [Surya Siddhanta] (365-15-31-31), adopted generally by LDS in his Ephemeris) and the like, which though called Sidereal, are very nearly Anomalistic, being about 8-5 or 8-6 [vinadis] longer than the correct Sidereal year. As a result, the First point of Mesa moves forward leaving the zero point behind at the rate of 8.5"per annum. So if we adopt LDS's methods of using the time of the True Sun at teh First Point of Mesa according to a particualr Almanac to get the [Ayanam'sa], we must deduct from the gross [Ayansm'sa] got, the accumulated interval between the zerp point and the First point of Mesa of that Almanac, to get the correct [Ayanam'sa]. (This accumulated interval may be called the 'Procession' of the First Point of Mesa for that Almanac). It is this correct [Ayanam'sa] that should be divided by the correct rate fo

precession of 50" 2585 etc. to get the year when the Vernal Equinox was at the zero point. If, on the other hand, we use the gross rate of precession (which is the correct rate of precession 'plus' about 8 5"), to get the year, for the gross [Ayanam'sa] increases not by the correct rate of precession but by the gross rate of precession, viz. 50" 2585 etc. etc. increased by about 8.5". This is the reason why most Indian systems give nearly 1'as the rate of precession. The reader will find out statement corroborated by sections 64 and 277 of LDS's Indian Chronology (Madras, 1911).

This is the reason why LDS divided his gross [Ayanam'sa] by 58" 78 and got 536 A.D. for VM as a first approximation. The gorss rate of precession 58"78 is got from the rate of Procession (viz.8"52) plus the rate of correct precession 50"26, for it is at this combined rate of 58" 78 that the Vernal Equinox recedes with reference to the First Point of Mesa, per annum. From this we see that it is wrong to use for this purpose the actual rate of precession given, even by the system, if any, as for eg. 54" annum in the case of the new [Surya Siddhanta] or 1' per annum in the case of certain other [Siddhantas], and so on; for these [Siddhantas] have found the rates or precession by actual observation of the Sun at the Vernal Equinox, and there is likely to be an error in the observation. According to the error the rates may vary. The nearer their rates are to 58".78, the better are their observation.

It is incumbent on our part, in the present context, to answer certain remarks made by VT on the above procedure of LDS. VT remarks: "There are the following drawbacks in the whole argument (of LDS):

"(a) It was considered that Dakshinayana began when the Sun reached the beginning of Karkataka instead of the end of Punarvasu.

"(b) The fact that the modern tropical year goes on decreasing at the rate of 0.53 seconds per century was not taken into consideration.

"(c) At least at teh time of Varahamihira, the Indian Siderial year-so designated at present-was really a tropical year and the value for the presession of the equinoxes must be taken as 50".2585 -nX.000225" and not as 54".7505 as assumed by Swamikannu Pillai."

Of these (a) has already been answered. AS for (b), int he 14 centuries considered by LDS, the time neglected by him is about 56 seconds, equivalent in 2" of the Sun's motion. Is this not negligible in the context?

As for (c), this is against the internal evidence of the PS. Excepting [Vasistha] and the Romaka, all the other [Siddhantas] in it give Sidereal years. The Pauli'sa gives 365 days, 15 [nadikas], and 30 [vinadikas], the Saura, 365-15-31-30, and the [Paitamaha 366 days. By what stretch of imagination can these be called Tropical years, these years that are so far greater than the correct Sidereal year that they border on the Anomalistic? As for LDS not taking 50".2585 for division, we have answered it by saying that this would be proper only if the correct Sidereal year, 365-15-22.9. had been used throughout the period of which we are considering the [Ayanam'sa]. Secondly, where has LDS assumed a rate of precession of 54".7505, and in which context?

It should not be thought that because the modern Drk Almanacs use the correct Sidereal year equal to 365-15-22.9, the time of the True Sun at their First point of Mesa will give the correct [ayanam'sa]. These almanacs were started recently and they arbitrarily fixed for themselves such [ayanam'sas] as would keep their [sankramanas] within reasonable distance from those of the old almanacs. The very fact that the [sankramanas] vary only within a matter of [nadis], shows this, for considering the difference between the correct Sidereal year and the so-called Sidereal years of our [siddhantas], even within a period of 420 years there will be a difference of one day, and for the period we are considering, viz. 1400 years or more, there should be a difference of more than three days, the Drk [Sankramanas] occurring earlier. To avoid the hue and cry that would be rasied if the [Sankramanas] in their almanacs are foundto occur thus, more than three days earlier, the Drk Almanac makers fixed for themselves [aynam'sas] that would keep their [sankramanas] near enoguh to those of the old system almanacs. The [Caitra or the Dhanistha Paksa] has come in handy for them to fix their [ayanam'sa] in this manner, but these [Paksa] are contradicory to all schools of traditional astronomy which have adopted the [Raivata-paksa] alone.

To continue the main argument. As, in the manner already stated, the number of years got to be deducted from 1909 to find the zero point should be the same, whether we divide the gross [ayanam'sa] by the gross rate of precession, or the correct[aynam'sa] by the correct rate of precession, and as the rates in the ration 7:6 approximately, the gross

[ayanam'sa] found by LDS (by using the time of [sankiramana] of the new [Surya Siddhanta] should be reduced by one seventh of itself to get the equivalent correct [ayanam'sa], which we find to be 19o 12'. So VT can have only 19o 12' and not 22o 23' by (b).

Now, we pass on to (c). (This is VT's special.) What VT says amounts to this : Kali began at midnight0.579 days after sunrise on 15th February as the Epoch of [Kaliyuga]. So there is a difference of 2.18 (?2.17) days which must be a [bija] correction. So we must add 2.18 days to the time of the True Sun reaching the First Point of Mesa in (b), viz. 0.9492 on 12th April. Thus the interval in days is increased by 2.18 days; which means 2o 9' more in [ayanm'sa], which will make up the 28o 15' required.

Now, what VT thinks to be a [bija] is really the interval between the times of the True and the Mean Suns reaching the First Point of Mesa. According to Indian astronomy the Sun's Equation of the Centre is about 2o9' at the time of Mesa [Sankiramana]. So the True Sun is in advance of the Mean Sun by 2o 9' and reaches the First Point of Mesa earlier by about 2.18 days. As the Apogee of the Sun has an extremely slow motion according to Hindu astronomy, the 2 18 days practically continue throguh the ages to be the same. In (b), LDS took for calculation the interval between the True Sun at the VErnal Equinox and the True Sun at the First Point of Mesa which is quite proper. If he had taken the interval between the times of the Mean Sun at Vernal Equinox and the True Sun at the First Point, then indeed we shall be justified in adding 2.18 days; for then the interval first got would have been less by 2.18 days, on account of the Mean Sun reaching the Veranl Equinox later by 2.18 days than the True Sun. If we add 2.18 days, as we ought to now, we get the same interval of 22.735 days. Thus VT cannot have (c).

The error of observation, (d), is possible and may be allowed if required; but it must be remembered that it is arbitrary, indefinite and may be plus or minus. VT has taken (d) as error or observation, not from [apriori] considerations, but [aposteriori], because this alone will give him, when added to the other quantities and divided by 50".2585 etc., 2031 years to be deducted from 1909 and get 123 B.C. So the reader is warned against getting predisposed in facour of 123 B.C.,simply because 1909

A.D. [minus] 2031 is exactly equal to 123 B.C., for this particular amount of error of observation has been arbitrarily presumed to get this very result.

In conclusion, we find that in VT's [ayanam'sa] of 28o 15'. (a) is cut off,(c) is cut off, (d) may be ignored, and (b) is reduced to about 19o 12'. If we divide this by the correct rate of precession, 50".2585 etc., we get c. 534 A.D. as VM's time. It may be noted how far away this is from 123B.C., and how near it is to 505 A.D. (epoch).

The [ayanam'sa] argument of Prof. Rai (Jpuhs I.124-27) is the same as VT's (a) and (b), with the difference that he takes (b) for 1931 instrtead of 1909. This amounts to 26o 3' 40" according to him, and committing the same mistake as VT of dividing this gross [ayanam'sa] by the actual rate of precession, he says VM lived 1866 years before 1931, i.e., in 65 A.D. Since this does not take him to the desired 123 B.C. Prof. RAi thinks that this descrepancy may be overcome by assuming an appropriate error or observation, which in this case has to be as large as 3 degrees or so!

With showing that 427 'Saka of VM is 505 A.D. and that the [ayanam'sa] argument is fallacious, the main object of this paper is over. It is unnecessary to discuss the 'slokas from [jyotirvidabharana] quoted by these scholars enumerating the nine gems of [Vikramaditya's] court (dhanvantari-ksapanaka etc.) and the year given therein; for in the light of the foregoing discussion these must be taken as part of a romance, or an attempt at imposture by the author of the work. Nobody will take seriously this 'sloka jumbling men of different ages together, as no one will take seriously the other romance, the [Bhoja Prabahdha], for matters of history.

The Date of Bhattotpala

Both Rai and VT seek additional evidence for VM's earlier date by making his commnetator Bhattotpala himslef earlier than 505 A.D. We shall examine this now. Bhattotpala says at the end of his commentary on VM's Brhajjataka that he finished writing it in 'Saka 888 (elapsed) on Caitra 'Sukla Pancami, which was a Thursday:

[caitramasasya pancamyam sitayam guruvasare/

vasvastastamite 'sake krteyam vivrtir maya//

Here VT says that "the weekday does not come out correctly if we take either the ['Salivahana 'Saka] or the Vikrama 'Saka. So the 'Saka mentioned by...Bhattotpala refers only to the 'Saka with 550 B.C. as

epoch. This means that if Bhattotpala's 'Saka is taken as given in the 'Saka of 550 B.C., the weekday agrees; and so the date referred to is 888 years after 550 B.C., i.e., 339 A.D., (but in his 'Andhra Saka' he gives 340 A.D., cf. fn. 11 above) and so VM must be earlier still. But we have made the calculations, and we find that it is 339 A.D. that does not give the agreement ; in that year the Caitra Pancami falls on Friday, ending at about 35 [nadis]. In his Popular [Astronmy}, pp. 136-37, VT has chaged the date of Bhattotpala to 338 A.D., in accordance with his changing the 'Saka Epcoh to 551 B.C. Strangely enough, here too VT asserts that he finds agreement with the weekday, i.e. three days off, on this date. On the other hand there is perfect agreement with 'Salivahana 'Saka 888 (corresponding to 966 A.D.) if Caitra is in the Purnaimanta reckoning which was prevalent in Bhattotpala's time and place. If 'Saka 888 is elapsed year, [Caitra'suddha-pancami] falls on Thursay, at 25 [nadis], Februaru 28, 966 A.D. So we get the time of Bhattotpala's finsihing the work correctly as we expressed. Because we took the current 'Saka instead of the elapsed (elapsed is the more usual practice of the Hindus), we had recourse [Purnimanta] reckoning, where too it is the previous [Phalguna] that agrees.

And there is also positive evidence to show that Bhattotpala has meant only the ['Salivahana 'Saka]. He has commneted on the [Khandakhadyaka] of Brahmagupta, who says that it is a re-presentation of the (Ardharatrika) system of [Aryabhata. This means that Brahmagupta is later than [Aryabhata] (3600 Kali, corresponding to 499 A.D.) and that Bhattotpala must be alter stll. It is not possible to drag down, as VT and others do, both [Aryabhata] and [Brahmagupta] together into the earlier centuries, for the following reasonds: The date of [Aryabhata] is definitely 3600 Kali, as already shown. Brahmagupta gives 587 'Sakakalaas the wpoch of his [Khandakhadyaka (I.3). Brahmagupta elsewhere states that 3179 is to be added to the 'Saka year to get the corresponding Kali year (Cf. Brahmasphutasiddhanta, I.26). [Amaraja] commenting on the above verse of [Khandakhadyaka] (I.3) gives the Kali year corresponding to the epoch of the work ('Saka 587) to be 3766 by adding 3179 to 587; and also calculates and verifies the Ksepas and the weekday do the epoch taking the Kali year 3766, which is A.D. 665, which therefore must be the time of Brahmagupta. Further, Brahmagupta is linked to [Bhaskara]Ii (who VT at

least admits wrote his Siddhanta-'siromani in 1150 A.D.) by an observed [ayanam'sa] of about 11o. Bhaskara Ii also says that in Brahmagupta's time the [ayanam'sa] was so little that it was "unobservable even to that expert astronomer". So Brahmagupta cannot be dragged too far away from [Bhaskara], and this condition will be fulfilled only if his epoch is int he ['Salivahana 'Saka]. (TSN says in this connection that Brahmagupta, whom Bhaskara II eulogises as his learned ancient teacher, could not detect an observational error of 5'!!) And so his commentator, Utpala's date, 'Saka 888, has also to be in the 'Salivahana 'Saka.

Prof. Rai advances another argument, which is his own and not given by anybody else. It is this : [Brhat Samhita], VIII. 20-21, gives a rule from which, by using the ['Saka] year, the corresponding [Jovian] year in the 60 year cycle [Prabhava] etc., can be got. He works it out for 1932 using the 'Saka starting from 550 B.C., and gets 52 years gone in the [Prabhava]series. Using the years gone in the ['Salivahana 'Saka] of 78 A.D., he gets the 18th year in the series, viz. [Tarana]. He finds this [Tarana] given in North Indian almanacs. But he says this proves nothing beyond showing that the North Indian almanac-makers have adopted the 'Salivahana 'Saka for this rule. But the point at issue is which is the correct 'Saka to take. This can be found by working out the year from the Kali years gone till 1932, and seeing which of the two (52 or 18) it agrees with. Prof. Rai works out the Jovian years gone from the beginning of Kali, using the elements of the [Surya Siddhanta], and dividing the result by 60 gets the remainder 52. Lo! this is the same as the remainder got by using the 'Saka of 550 B.C. So, that is the 'Saka intended by Vm in his [Brhat Samhita], he says. The argument seems to be perfect.

But this is the fallacy in it: If the Jovian year is to be worked out a [priori] using the [Kali] years gone, the years should be counted from [Vijaya] and not from [Prabhava]. This condition is specified in the very [Surya Siddhanta] whose elements Prof.Rai uses for computation, and this has been missed by him. Now counting 52 from Vijaya, we get only the 18th year of the [Prabhava] series, viz. [Tarana], and this agrees with 'Saka of 78 A.D. and not the 'Saka of 550 B.C. Thus Prof. Rai's argument fails. In the result, it is only a proof for taking VM's 'Sakakala to be the 'Salivahana 'Saka, and discarding the 'Saka of 550 B.C.

Prof. Rai seeks further support to his theory by stating (ib., p.71) that, "Albiruni writing in 1030 A.D. not only talks of Bhaskaracharya, but also mentions his book 'Karana Kutuhala'", that the date oc composition of Karanakutuhala given in the work itself, viz. 'Saka 1105, if taken in the 'Salivahana 'Saka would be 1183 A.D., i.e. 150 years after Albiruni, which is patently impossible, that "Weber in his book on Sanskrit Literature (p.262) notices this anomaly, but is unable to offer any expalantion" (Weber in his Book on Sanskrit Literature, English Translation, London, 1914), and that "if we take this 'Saka commencing from 550 B.C., the riddle is solved", for this would take [Bhaskara] to the 6th cent. A.D., long prior to Albiruni.

The answer to Prof. Rai is given by [Bhaskara] himself who indicates thathe uses only the 'Salivahana 'Saka, for he says that 3179 is to be added to the 'Saka year to get the Kali years gone (Cf. Siddh. 'Siromani, Ganita, Madhyama, [Kalamana,28). Moreover Albiruni's words in the context do not warrant the name [Bhaskara] at all, nor does he mention anywhere a work [Karana-'Kutuhala]' (the work named being a Karana-'Sara). It has also to be added that Prof. Rai is not speaking the facts when he says that Weber offers on the very page that Rai refers to (page 262) several explanations: Weber says that "we have scarcely any alternative save to separate Albiruni's 'Bashkar' son of 'Mahdeb', and the author of "Karanasara' from the [Bhaskara], son of [Mahadeva], and author of [Karanakutuhala]". (Note that none of the three names, neither that of the author, nor of his father, which is really Mahe'svara and which Weber himself draws attention to in a footnote, nor of the work, tallies). Weber again suggests that his translation of the Arabic words of Albiruni might be wrong, for "Alhiruni usually represents the Indian bh by b-h, and for the most part faithfully preserves the length of the vowels, neither of these is here done in the case of Bashkar, where, moreover, the s is changed into sh", and adds in a footnote that in the passage under discussion "there lurks not a [Bhaskara] at all, but perhaps, Pushkara". Even if the passage refers to a [Bhaskara], Weber suggests that "we may have to think of that elder [Bhaskara], 'who was at the head of the commentatots of [Aryabhata, and is repreatedly cited by [Prthudakasvamin], who was himself anterior to the author of the 'Siromani". It is in the face of these facts that Prof. Rai

coolly asserts that Weber "is unable to offer any explanation!" (Here Rai only follows T.S.N's remarks.)

We may add here that the epoch of [Karanasara] which is mistaken for [Karanakutuhala], is given by Albiruni as 'Saka 821 (A.D. 899) (Alberuni's India, Sachau, I.392), and obviously Prof. Rai's [Bhaskara] of the 6th cent. cannot write a work 300 years later! So Prof. Rai's argument only goes against his own theory.

Thus nothing can shake the evidence showing that the 'Saka mentioned by VM is the 'Salivahana 'Saka and that the date 'Saka 427 given by him in his PS is 505 A.D. Incidentally it has also been shown that the 'Saka era used by Brahmagupta, Bhattotpala and [Bhaskara] II is the 'Salivahana 'Saka of 78 A.D.

We propose to show in a subsequent article the untenability of certain other claims of these scholars referred to in the Introduction and that everywhere when the word 'Saka occurs as the name of an era, it is only the 'Salivahana 'Saka that is meant, and therefore or otherwise there is no case for postulating a Cyrus or Andhra Era of 550 B.C.

THE UNTENABLILITY OF

THE POSTULATED 'SAKA OF 550 B.C.*

1. Introduction

It has been shown in the preceding study that the 'Saka ERa used or alluded to by astronomers like [Varahamihira (VM), Brahmagupta, Bhattotpala, 'Sripati, Bhaskaras I and II, etc. is the era starting from 78 A.D., later known as the 'Salivahana 'Saka, and not the era of 550 B.C. postulated by the late T.S. Narayana Sastri (TSN) or V. Thiruvenkatacharya (VT) and called by them the Cyrus Era or the Andhra ERa, respectively. Incidentally we have shown to be untenable their statements that [Aryabhata] belonged near to 2742 B.C., VM to 123 B.C., Brahmagupta to 36 A.D., Bhattotpala to 339 A.D. and Bhaskara II to 522 A.D., and thereby we have proved that VM belongs to 505 A.D. and Bhattotpala to 966 A.D. and indicated that the real date of [Aryabhata] is 499 A.D. and of Brahmagupta 654 A.D.

In the same way it can be shown that wherever other astronomers or writers like Kalhana and Albiruni mention a 'Saka Era, it is this 'Saka of 78

A.D. they mean. The tradition of alamanac-makers also supports this, for they all give in their alamanacs only this 'Saka Era and not the alleged other one. In inscriptions and documents also, in short, in every case where a date in 'Saka Era is given, it is this 'Saka alone, though this is disputed by TSN and (till recently) by Sri Kota venkatachelam (KV) in the case of the Aihole Inscription (to which we shall revert later).

Ii. VM's Brhat-Samhita XIII. 3 considered

We shall not take up for discussion [Brhat-Samhita] of VM (Br.Sam.) XIII. 3 referred to by us in the previous paper, which TSN and others consider as their stronghold, and which we left over for detailed consideration later:

[asan magha su munayah 'sasati prthvim yudhisthire

nrpatau/

sad-dvika-panca -dvi-yutah 'Sakakalas tasya

rajna's ca//

This stanza occurs in the context of the 'Saptarsi-cara or the alleged 'Motion of the Seven Sages', (i.e., the group of stars Ursa Major or the Great Bear), among the twenty-seven asterimss, given for use in astrological prediction. To gind the position of the group at any time, three things are necessary: (i) its postion at a given time; (ii) the time elapsed from the given time to the time for which the postion is required,and (iii) its rate of motion. The above stanza gives (i) and (ii),viz., that at the time fo [Yudhisthira's] rule the Sages were at teh asterismal segment [Magha], and the time elapsed from this time to any year in the 'Saka Era is the number of the year in the 'Saka Era plus 2526. (Requirement iii is given in the next stanza, XIII.4, as one asterismal segment for 100 years).

Now TSN and KV argue thus: (a) This stanza is a quotation from Vrddha Garga (VG), and so VG knows a 'Saka ERa which he mentions here. It is accepted by all that VG lived long prior to 78 A.D., the starting point of the 'Salivahana 'Saka. So this 'Saka must be an earlier 'Saka, viz., that of 550 B.C. postulated by them. (b) The first half of this stanza says the Sages were in [Magha] during [Yudhisthira's] time. The [Puranas] and VG etc. say that at the junction of [Dvapara] and Kali yugas, the Sages were at Magha and Yudhisthira was ruling. 25 years after the advent of Kali, the Sages moved to the next asterism from [Magha] and in that year

Yudhisthira left this world for heaven. The second half of the stanza states that the 'Sakakala mentioned therein started 2526 years after [Yudhisthirakala]. If we take [Yudhisthirakala] to have started from the time he went he heaven, i.e. after 25 Kali equivalent to 3076 B.C., this 'Saka must have started 2526 years after this, i.e. from 550 B.C., and is evidently quite different from the 'Saka starting from 78 A.D.

It is in the light of this conclusion, and in support of it, that TSN etc. say (as we have discussed already in the previous paper), that the 'Sakakala mentioned by VM in pther places in his works, and also by other astronomers like [Brahmagupta], is this 'Saka of 550 B.C. But we have proved conclusively in the previous study that in those places it is the 'Saka of 78 A.D. that is referred to. Therefore the conclusion here arrived at by TSN etc. must stand on it sown legs. We sahll proceed to examine this now. Even at teh outset we can say that it is extremely unlikely that VM means here alone a 'Saka different from what he means by the same word elsewhere in his works; and therefore he must mean the 'Saka of 78 A.D. here also. All the same we shall examine their arguments.

(a) The alleged quotation of Vrddha Garga

The reasoning (a) is based on the assumption that the stanza is a quotation from VG, which it is not. The actual words of VG are quotod by the commentator Bhattotpala in his commentary on thsi stanza: cf. [tatha ca Vrddha-Gargah]: "Kali-Dvaparayoh sandhau sthitas te pitrdaivatam" (At the junction of Kali and Dvapara, they -the sages-were at Magha). It is to be noted that thus would be redundant if the staza in question also were VG's, both giving the same idea, viz. the situation of the stages. It may also be noted that this is in a different metre. What VM means by his statement in the introductory stanza, [Kathayisye Vrddha-Garga-matat (Br.Sam.XIII.2) is only that he is giving the astrological predictions due to the motion of the Sages as based on the work of VG, as indicated by the word [matat]('opinion') used here. Also in all the other [caras] given in the other cahpters of Br.Sam, like [Adityacara, Candracara, Rahucara] etc., what VM means by [cara] is the prediction based upon the motion and not the actual motion, and so must it be here also, (the actual motions being given in a ganita work like the Pancasiddhantika). If in the case of the Sages the motion also is given, it is because it is simple, has not been

given elsewhere, and is necessary for the main purpose, viz., the prediction that Vm says he gives according to VG. So this stanza which serves to find the position of the Sages need not necessarily be, and as we have shown, is not, VG's. This being the case, it cannot be argued that Garga who came long prior to 78 A.D. knows a 'Sakakala and therefore this 'Sakakala must be the earlier postulated one of 550 B.C.

(b) The Time of Yudhisthira

We now pass on to consider (b), the second and more important reasoning of TSN etc., viz., that VM in this verse refers to Yudhisthira who lived at the beginning of Kali and rose to heaven 25 years after Kali set in (i.e. in 3076 B.C.) and so the 'Saka Era beginning 2526 years after that must be the postualted 'Saka of 550 B.C. But we answer, there is nothing in this verse to show that in VM's opinion Yudhisthira lived at the beginning of Kali. On the other hand, it can be shown that VM might have meant a time about 650 years after Kali, or even that he did mean this later period for the time of [Yudhisthira], and therfore the 'Saka Era following 2526 years after, cannot be the postualted 'Saka of 550 B.C., but can only be the well-known 'Saka of 78 A.D. It is a fact well known to scholars (inclusive of TSN etc.) that the junction of Dvapara and Kali (3102 B.C.) is not the only period with which Yudhisthira is associated. This is according to one school; but there is at least one other school (e.g. that of the Jain Buddhist writers) who take it that Yudhisthira lived about 500 years later. They use a [Yudhisthira Era] which began in 468 Kali (corresponding to 2634 B.C.). EVen of the first school mentioned not all associate the same event of Yudhisthira's life with the beginning of Kali, 3102 B.C. There are four sub-schools here (Fleet says three, but mentions all four, JRAS (1911)676-78, 'The Kaliyuga Era of B.C. 3102'). One sub-school believes that the first coronation of Yudhisthira at Indraprastha was the beginning of Kali and the commencement of the [Yudhisthira Era]. Another makes the [Bharata] war and the beginning of Kali synchronous. A third says that Kali began at the death of Krsna and his ascent to heaven. The fourth sub-school says that Yudhisthira's adbication and starting on the [Mahaprasthana] was at the beginning of Kali. The reason why there are so many views must be explained by the fact that the traditional idea of the ages like [Krta], [Treta], [Dvapara] and [Kali] with their specific

characteristics, was earlier that the integration of the beginning of the traditional Kali with that of the astronomical Kali answering to 3102 B.C., which was computed later by astronomers like [Aryabhata] so as to form a convenient point of reference for the Mean Planets. Thus the Kali Era, said to begin with 3102 B.C., is an extrapolated era, and in examining any date mentioned in this Kali Era, this fact should be borne in mind.

Now, in this multiplicity of schools on this point, which is a fact accepted by all, resulting from the integration of the traditional Kali with the astronomical Kali, there is the possibility of VM's statement representing one other school or at least a variant of the Jain school, differing as it does, from it only by about two centuries. Kalhana, the Kashmirian chronicler of the 12th cent. A.D., is one of those that subscribe to thsi school; for not only does he quote in his [Rajatarangini] this verse of VM. but also expresses his own concurrence with it in so many words:

[Bharatam Dvaparante bhud vartayeti vimohitah/

Kecid evam mrsa tesam kalasankhyam pracakrire//

I.49//

* * * * * *

['satesu satsu sardhesu tryadhikesu ca bhutale

Kaler gatesu varsanam abhuvan Kuru-Pandavah//

I. 51//

* * * * * *

sat-dvika-panca-dvi-yutah 'Sakakalas tasy rajana's ca//

I.56b//

"Some people have been misled by the statement that the Bharata (War) was at the end of [Dvapara], and have given a wrong chronology to the kings (the Pandavas, Gonanda etc.)... The Kurus and the Pandavas came when 653 years had passed in Kali...... The time in the 'Saka ERa plus 2526 is the time of his rule, i.e. the time in the epoch beginning from his (Yudhisthira's) rule."

It may noted that 653 plus 2526 (the numbers here given) equal 3179, the well-known converter of 'Saka into Kali and [vice versa]. Not only is Kalhana a believer in this school, but he is also certain that VM belongs to this school, as seen by his statement 'Samhitakaraih' (Rajatarangini,I.55) and his quotation of VM following immediately (I.56). Cunningham also

thinks the same as seen from his statement, "As VM places the GReat War 653 years after the beginning of the Kali AGE....." (op.cit.p.11). Again, Prof. P.C. Sengupta, who in his Ancient Indian Chronology (Univ. of Calcutta, 1947) in seeking to determine the date of the Bharata War astronomically (see chs.I-III) favours this school, and comes to the conclusion that: "The date of the Bharata Battle is thus astronomically established as the year 2449 B.C. (Kali 653), which is supported by the VRiddha Garga tradition recorded by Varaha Mihira., (see p. 19). Now it must be noted that the mere possibility of following this school is sufficient for our purpose, as we have stated above,

Nor, can the objection be raised that VG and the [Puranas] associate the Sages with Magha at the beginning of Kali, and that in this verse too, as the Sages are declared to be at [Magha] in [Yudhisthira's] reign, the time here should be taken as the beginning of Kali, and so the time given for [Yudhisthira's] rule must be the beginning of Kali, and not 653 Kali. For, the beginning of Kali is not associated with [Maghra] alone. The [Matsya Purana] says (ch.271, st.41) that according to the 'Srutarsis the Sages were at [Krittika] at the beginning of Kali, and TSN and KV are aware of it (TSN quotes it and explains it, see The Age of Sankara, Madras, 1916 ff., App.,pp.166-67; so also KV, plot in Indian Chronology, 34-36). They themselves say that in VM's opinion also the sages were at [Krittika] at the beginning of Kali (TSN. ib., p.171; KV, ib. p. 36, and 'Indian Eras', JAHRS XX.77). According to [Aryabhata II and [Para'sara] too (for details see below), the Sages were at [Krttika] at the beginning of Kali. They were at 'Sravana according to 'Sakalya and Muni'svara, and at Rohini according to Lalla (for details, see below). So the objection rasied above does not stand. Now, according to [Aryabhata II] and [Para'sara], who give [Krttika], it is easily seen that the Sages will be in [Magha] in the 7th cent. Kali, for [Magha] is the seventh asterism from [Krttika] and the motion is about one century perasterism. Thus, there can be no objection to the Sages being in [Magha] in the 7th cent. Kali. It is only if the motion of the Sages is taken to be retrograde (as TSN and certain others think) according to [Aryabhata II], [Para'sara] and VM, but would be far away from it. But it is not retrograde according to [Aryabhata II], [Para'sara] and VM, as also according to other astronomers who give rules for the motion, which we

shall show.

III. (a) 'Motion of the Sages"-Direct, not Retrograde

This requires a knowledge of the motion of the Seven Sages which we shall give in some detail because there is a lot of misconception among scholars (inclusing TSN, KV and VT) about this, which in turn vitiates the results of their researches. It was believed by the authors of the ancient [Jyotisa Samhita], like the planets, the rate of motion being given as 100, or nearly 100, years per asterism (13o 20'). It may be said, even at the outset, that there is no such motion as claimed to exist by the authors of these [Samhitas] and followed up the [Puranas] and some of the later astronomers; that the Sages are always to be associated only with the [Phalgunis], [Hasta] and [Citra] asterisms (see fn.10); and that the theory of their motion is wrong, howsoever it might have originated. That is why many standard astronomers and astronomical works like [Aryabhata] I, Brahmagupta, [Sripati], Bhaskaras I and II, the [Suryasiddhanta], etc., do not deal with the subject at all as being outside the pale of real astronomy. That is also why [Kamalakara] is constrained to say in his [Siddhanta Tattvaviveka](Banaras, 1880-85), [Bhagrahayutyadhikara]:

['Sakalyasamjna-munina kathitas sabanah

saptarsitarakabhava dhruvaka's cala's ca/25a/

* * * * * *

[Yair golatattvam vivrtam hi tai's ca

suryadibhir naiva vi'sesa esah/

[Proktas sva'sastre 'sti gatir muninam

ato na yukta divi golaritya]//30//

* * * * * *

[adyapi kair api narair gatir aryavaryaih

drsta na yatra kathita kila 'Samhitasu]/32a/

* * * * * *

[Prayo 'tha te ca munayah kila devatam'sa

drggocara nahi nrnam iha satphalaptyai]//36a//

"Sage 'Sakalya has given the motion of the Sages with their poitions in his time ...[Surya] and others who explain the nature of the celestial sphere in their works do not give it and therefore the theory cannot be

sustanied astronomically....Even today this motion mentioned in the [Samhitas] is not observed by knowing astronomers... Therefore the seven real Sages who are (only) the Presiding deities (of these stars) are to be considered to be moving, unobserved by men, for the prediction of the fruits thereof."

But the motion has been accepted as a fact by the common people and the authors of the [Puranas], and an era called [Laukika Era] (by the people belonging to the Kashmir region) and the Saptarsi ERa (by the Puranas), has been founded upon this theory. As already mentioned, there are also astronomers like VM, [Aryabhata II] (cf. his Aryasiddhanata or Mahasiddhanta, Madhyamadhyaya, Ii), [Para'sara] (cf. Aryasiddhanta, Para'saramatadhyaya,9), Lalla (Quoted by Muni'svara in his commentary on the Siddhanta 'Siromani), 'Sakalya (quoted by Munisvara, ib., and by Kamalakara in his Siddhanta Tattvaviveka, Bhagrahayutyadhikara 25 and under), [Vate'svara], and [Muni'svara] (see his Siddhanta Sarvabhauma), who have accepted the motion as real on teh authority of the ancients and given rules for the motion, which necessarily must agree with their own observation, or else they would be meaningless even for them. This means that whatever be the rule, if applied to the time of the author, the position of the Sages must be got as between [Magha and Svati]. In giving the rules, the authors all consider the motion as direct and never as retrograde as fancied by some scholars like TSN, KV, VT etc, for which fancy there is no support anywhere. Let us take the rules one by one and examine them for the facts mentioned.

VM's rule is as follows (Br.Sam., XIII.3-4): The number of years gone from the time of Yudhisthira is to be found by adding 2526 to the 'Saka years gone at the time for which the position of the Sages is wanted. This is to be divided by 100, which gives the number of asterismal segments gone, and these are to be counted from [Magha] to get the position. In the context there is no mention of any retrograde motion, not is it mentioned that the number got is to counted backwards as in the case fo the [patas] like [Rahu]. In the absence of any such specific indication we do what is normally doen, i.e., count the segments forward, as in the case of all other [grahas] like the Sun etc. Working for VM's time, i.e., 427 'Saka, we get the middle of [Uttara Phalguni] as the position of the Sages, which is well

within the limit for agreement with observation. If we count backwards taking the motion as retrograde, we get the middle of [Pusya], which is far outside the limit, and this also shows that the rule implies only forward motion, as we have already determined.

[Arybhabhata II gives the rule in the form of cycles per Kalpa, even as the [Siddhantas] do for the planets. He says there are 15,99,998 cycles in the Kalpa and the cycles commence 30,24,000 years from the beginning of the Kalpa. Here too there is no indication of retrograde motion. Calculating from the above date we find that at the beginning of the present Kali the Sages are at 2.38 segments (counting from A'svini), i.e., they have passed [Bharani] and been in Krttika for 38 years. It is easily seen that at 662 Kali the Sages will cross to [Magha] according to this [Siddhanta]. Compare this with VM's rule that would give the crossing to [Magha] in the 7th century Kali (exactly speaking 653 Kali; for going back 2526 years from zero 'Saka, equal to 3179 Kali, we arrive at this date).

We now pass on the [Para'sara]. His rule is the same as that of [Aryabhata] II, with the difference that in [Para'sara] case the cycles begin at the commencement of the Kalpa itself. This would give for the commencement of Kali the position 2.34 [naksatras], counting, of course, from [A'svini], i.e., after crossing [Bharani], the [Sages] have been in [Krttika] for 34 years, and at 666 Kali the Sages pass on to [Magha]. See how close this is to 662 and 653, the dates according to [Aryabhata] II and VM, derived above.

Now for Lalla's rule: As said before, the rule is quoted by [Muni'svara] in his commentary on the [Siddhanta 'Siromani]. It is this : Deduct 14 from the Kali years gone and divide the remainder by 100. Asterisms are got, which are to be counted from [Rohini]. Here too it is to be noted that no backward counting is enjoined, and the rule must normally mean forward counting as in all rules where nothing is said. It is to be noted that according to thsi rule, the Sages pass to [Rohini] from [Krttika] 14 years after Kali set in, i.e., they have been in [Krttika] for 86 years before the beginning of Kali, (agreeing with VM with in half a segment). Taking the date of Lalla to be 650 A.D., the Sages would be at [Citra] in his time, According to 'Samhita, "Kratu, one of the Sages, enters 'Sravana at the commencement of Kali and the Sages have direct motion every year at the

rate of 8' (Which rate is equivalent to one asterism per century);" cf.

[yugadau visnitarayah kratur bhadye vyavasthitah/

pratyabdam "Praggatis" tesam astau lipta Muni'svara]//

(Quoted in Kamalakara's Com. on his own Siddh.

Tat. Viv., Bhagrahayutyadhikara, under stanza 25)

According to Vate'svara, Madh.15,

Eò¨É±Éʴɹ]õ®ú´ÉCjɺɮúÉä°ü½þ º¡Öò]õÊMÉ®úÉ%ʦÉʽþiÉÉ ¨ÉÖÊxÉ{ɪÉæªÉÉ&*

ªÉ <½þ iÉÉxÉÊ{É ´ÉÎS¨É ªÉÖMÉänÂù¨É ´ÉÉxÉ tÖSÉ®ú±É¤nù´É®úÉä ¨ÉÖVÉMÉÉä¹]õªÉ&** 1692**

So the rule would be to multiply the Kali years by 8 and divide by 800, to get the asterisms. These are to be counted forward from 'Sravana. It may be noted that here eastward motion, i.e., direct motion, is specifically stated. [Kamalakara], too, expalining the motion as really that of the presiding sges, says in the same context that the motion is eastward, i.e., cf., [sa praggatir munivarair bhagata muninam] (ib., 36). According to this rule, after 1100 A.D. the Sages would move to [Magha], and we can place this work at the earliest in c.1100 A.D.

Lastly, for the rule of [Muni'svara] given in his [Siddhanta Sarvabhauma]. Deduct 600 from the Kali years, Double the remainder and divide by 15. The position of of the Sages in degrees in got. This divided by 30 gives the position in the [rasis]. This rule again clearly takes the motion as direct. According to [Muni'svara] the Sages cross to A'svini at 600 years Kali (which is equivalent to the statement of 'Sakalya, for accoding to 'Sakalya's rule too the Sages enter A'svini at 600 Kali). At teh time of [Muni'svara], according to hi sown rule, the Sages would have crossed from [Citra] to [Svati] which is just outside the limit, and which position [Muni'svara] should have accepted as agreeing with with observation because the difference was not much. (It would have satisfied him better if some astronomer had said or if he could obtain, by quibbling, the Sages were at 'Sravistha or 'Satabhisak at teh commencement of Kali. No such thing was available, and the best he could have was 'Sakalya, and he had to be satisfied with that).

Thus all authorities either state or imply only direct motion, and there is no authority for retrograde motion. That is why scholars like [Cunningham] (as already mentioned), "Sriyut 'Sris' Chandraa Vidyarnava, Dr. Jayaswal and many others" (in the words of Dr. Triveda,JIH XIX, ii) have

considered the motion famous astrologers of Banares who informed Col. Wilford", (cf. Triveda, ib., p. 10), and the author of [Kaliyugarajavrttanta] who believe the motion to be retrograde. But in the light of what we have said, they must be wrong (see Fn. 15).

III (b). The Puranas on the motion of the Sages

Also, the [Puranas] do not say whether the motion is direct or retograde. We cannot get any indication regarding the direction of the motion of the Sages from the Puranas themselves, as they are vitiated by emendations and interpolations, made to affect the very point which we are trying to decide. Still, some scholars resort to them for support, and it is not surprising that they fail to establish anything. Dr. Triveda is one such: he not only does not prove his point, but proves the contrary of what he desires to prove, as also the lack of clearness in his mind. For e.g.: He says: (i) "But in fact their (the Sages') motion is retrograde as from the word Precession, pre=purva or east, and cession from Fr. cedare=go" (ib., P.11). (ii) "Kamalakara Bhatta also says in his [Siddhanta Tattvaviveka], 'Pratyabdam Praggatis tesam'; that is, in every year their motion is from West to East." (ib.p.11). (iii) "C.A. Young in his A Text Book of General Astronomy published in 1904 says on page 141, 'The Equinox moves westward on the ecliptic, as if it advanced to meet the Sun on each annual return'. So it is certain that their motion is contrary to that of the SWun, and it is retrograde." (ib.,p.11)

Let us examine his statements here. (i) If Precession is retrograde, why should the motion of the Sages be retrograde also? Are they the same phenomenon? If he means that the motion is derivable from precession, he has not shown it, and cannot show it, because it is not so (see Fn.11 (i) above). Even if some connection be established between the two, in the period we are considering the simulated motion would be directed only opposite to Precession (see Fn.11(i). He is not aware that the derivation he gives for the word 'Precession would mean direct motion even for Precession, and not retrograde motion, for 'going east' means 'direct motion'. (ii) Dr. Triveda's quotation from Kamalakara is plainly against himself, for 'from west to east' is 'direct motion', and not otherwise as Dr. Triveda thinks. (iii) Young rightly says that the Equinox moves westward, i.e. it has retrograde motion. But what has that to do with the Sages? Triveda

does not perceive that from here it can be understood that it is westward motion that is retrograde, and not eastward motion, as he thinks.

Under the delusion that he has proved the motion of the Sages to be retrograde, Dr. Triveda proceeds to apply this to the following statement in the [Puranas] in order to establish his thesis that the interval between [Pariksit] and Nanda is 1500 years (given by one reading ) and not 1015 or 1050 years (given in certain other readings)(cf. Triveda, ib., pp.1-3, 12-15). We shall briefly examine this in order to expose the errors in his reasoning, for if he establishes his reasoning, for if he establishes his point by using his theory of retrograde motion, that might be taken by some as a point in favour of the theory of retrograde motion itself, even after all that has been said by us to establish that the motion is direct.

The Puranic stament is as follows:

[Mahapadmabhisekat tu yavaj janma Pariksitah/

(oR yavat Pariksito janma yavan Nandabhisecanam.)

evam varsasahasram tu jneyam panca'saduttaram]

(1050)//

The last foot has the variants : 'satam panca da'sottaram (i.e., 1510) or [pancada'sottaram] (i.e., 1015) or [panca'satottaram (i.e., 1500) (Visnu Purnana, IV.xxiv.104; [Bhagavata], XII.ii 26; etc.). Triveda's thesis is to establish the interval to be 1500 or 1510 (according to two readings given) using the Saptarsi Era given in the following Puranic statement:

[Prayasyanti yada caite purvasadham maharsayah]/

(OR yada maghbhyo yasyanti purvasadham maharsayah)/

[tada Nandat Prabhrtyesa Kalir vrddhim gamisyati]//

(Visnu Purana, IV. xxiv.112; [Bhagavata],XII.ii.32; etc.)

It is said here that when the Sages pass from [Magha], (their position at the beginning of Kali when [Pariksit] was ruling), to [Purvasadha] at the time of Nanda, the Kali will worsen. From [Magha] to [Purvasadha] the Sages pas 10 asterisms in their course, taking the motion to be direct, (as we have established), i.e., about 1000 years from [Pariksit] to Nanda (or 3700 years if one cycle has gone), and this is supported by two readings. But Triveda suggests 1500 years for this interval, supported by the other two readings. Counting backwards from [Magha] to [Purvasadha] (in accordance with his theory of retrograde motion0 he should get at least 16,

not counting both. Thus he should get at least 1600 years as the interval. But this will not suit his theory, and so he omits to count 'Sravana, and gets the 15 asterisms he wants, to give him the required interval of 1500 years! (see ib., p. 12, lines 10-11). This is proof that the author of the Puranas, who employed the Saptarsi Era for purposes of chronology,has taken the motion only to be direct and used the Era: and not retrograde, for if taken as such, at least 1600 years will be got as the interval, which is not supported by any reading of the text.

One thing clearly emerges from this discussion, viz. that the motion of the Sages as given by the astronomers and the Puranas is direct and not retrograde. So VM can be right in saying that the Sages were in [Magha] in the 7th century Kali and in this he is supported by [Aryabhata]II, and [Para'sara], as well as the 'Srutarsis. Therefore Yudhisthira's reign associated with the Sages at [Magha] can well be in the 7th cent. Kali, also supported as it is by a whole school of chronologists. As the 'Saka Era mentioned is to come 2526 years after this period, it is the 'Saka of 78 A.D. that must have been meant by VM in the 'sloka, Br.Sam.XIII.3, and not the one postulated by TSN etc., concurring with what we have established already in the previous study from an astronomical point of view.

IV. The Aihole Inscription

Now we shall take up the Aihole Inscription and show that the 'Saka used in it is only that of 78 A.D. and not the other one alleged by TSN and echoed by certain other scholars. Discussing the age fo VM in his Age of Sankara, Pt.I-D, pp. 224ff., TSN takes up the Aihole Inscription for consideration, and tries to show that the 'Saka Era mentioned therein is his own 'Saka of 550 B.C. from the synchronism found in it between the 'Saka Era and the Bharata War. The portion of the inscription relevant to our discussion is the following:

[trim'satsu trisahasresu bharatad ahavaditah/

saptabda'satayuktesu 'sa (?ga) tesvabdesu pancasu/

panca'satsu Kalau Kale satsu panca'satasu ca/

samasu samatitasu 'Sakanam api bhubhujam]//

In trying to interpret this passage, Dr. Fleet at first (Indian Antiquary, V(1876)67-73) made the mistake of thinking that the time of the inscription

is given in three eras, viz. [Bharata War, the Kali and the 'Saka. Perhaps he was led into this mistake by the word 'sata' occuring thrice (sapataba'satayuktesu, 'satesvabdesu, and panca'satasu) and the statement in the Puranas that the Kali epoch is different from the Bharata War. But subsequently, in IA VIII (1879) 240-41, Dr. Fleet acknowledged his mistake and gave the correct reading by emedning 'satesu into gatesu (for, in the Kanarese-Telugu script in which the inscription is engraved on roch, ga, with a horizontal stroke across would become 'sa and the engraver might have been misled into adding the stroke here by the large number of 'sa letters occurring the context; or it might have been caused by weathering) and interpreting the passage as 3735 years from the Kali epoch, after the Bharata War, and 556 years 'Saka kings, i.e. 556 years in ('Salivahana) 'Saka Era. This interpretation is accepted by all scholars (see for instance, Kielhorn, Ep. Ind., VI (1900-01) 1-12), except TSN and KV. But the emendation of 'satesu into gatesu is accepted by TSN. He also accepts the fact that only two dates are given, of which one is 'Saka Era. This necessitates the two expressions 'after the [Bharata War]' and 'from the Kali epoch' to be taken together, as giving one date. If the Kali epoch is meant as important and the [Bharata War] is mentioned here simply to described it, without any more troble we get the interpretation, '3735 years from the Kali epoch', which beautifully synchronises with the 'Salivahan 'Saka year 556 given, (about this number there is no dispute), for if we deduct from 3735 the wellknown converter 3179 we get 556, which itself proves that this must be the 'Salivahana 'Saka of 78 A.D. If, on the other hand, the [Bharata War] is taken as important, and also that the War was fought 36 years earlier (TSN makes it 38 to suit his calculations) according to one sub-school taken advantage of by TSN, then there is trouble, for the War took place in 3140 B.C. according to TSN. 3735 years from this date there is no 'Saka epoch to synchronise with. But TSN sorely wants it to synchronise with the 'Saka of 550 B.C. postulated by him. He clutches at an error committed in a collection of old records publihed for literary study, the [Pracinalekhamala, (N.S. Press, Bombay, Kavyamala Series 16), thinking that it will help him. In the [Pracinalekhamala, saptabda'sata] is printed as [sahabda'sata]. Whether this is a misprint or an intended emendation, we do not know. But this much we can say, that the letter is

certainly pata and not ha, as anyone can verify from the photo-print of the inscription reproduced in IA V (1876) op.p.69,ib.VIII (1879) op.p.241, Ep.Ind.VI (1900-01) op. p.7, etc.) and comparing the letters. Not only this; the word [saha] will be a repetition, because there is the word [yukta] giving the same meaning; also [saha] requires an instrumental to govern, which is not available in the verse. In spite of all this, TSN takes this [saha] instead of [sapta] and gets the number 31355, of course, as we have pointed out, with a duplicate [saha] serving no purpose in the interpretation ) and begins to effect the sybchronisation thus (see p.189, plot in Indian Chronology);The Aihole Inscription is 3135 years frm the War, viz. 3140 B.C. So the date of the inscription is 5 B.C. And then the inscription is 556 years from the 'Saka epoch (of TSN), viz. 550 B.C. 556 years from 550 B.C. is 6 B.C. (so says TSN), for he wants it, and wish is father to thought). 6 B.C. is only one year off 5 B.C. (obtained above), which can be easily accounted for, and the synchronism established; which shows that the 'Saka mentioned in the inscription is his 'Saka of 550 B.C. But TSN and KV who quotes him seem to be unaware of the blunder in the calculation, and that 556 years from 550 B.C., is not 6 B.C., but 7 A.D.; and this date is 11 years off 5 B.C., and no amount of jugglery can spirit this period of 11 years off and the synchronism is far from being establised. Waht is more, having failed to prove the 550 B.C. 'Saka, but thinking that it has been proved, TSN indulges in a tirade against Orientalists and their ways (see p.190, ibid.), unconscious all the while, that it all applies to TSN himself!: "Alas ! it is a great pity that these Orientalists should at first conceive a theory of their own, and then actively set themselves to work out the same by hook or by crook, by changing every authority to suit their own favourite hypotheses, and by hoisting up the fabricated text as the only true version, while they perfectly know all the while in their own heart of hearts that they have been able to achieve their objects only by fabricated evidence and meddling with the original authorities..... The Orientalists simply beg the question, and beat about the bush in discussing such matters (here, explanation of the word 'Saka), blowing hot and cold at the same time, misjudging themselves, and misleading others, and thereby keeping back the Truth as far away as possible from the ken of ordinary public." How aptly these words apply to TSN himself!

V. The Evidence of the Jyotirvidabharana

Even though we have stated that the evidence of the [Jyotirvidabharana] does not merit any consideration (see previous paper), still because it is made much of by TSN, KV and VT (see for e.g.,KV : JAAHRS XXI (1950-52) 28-32, Chronology of Nepal History, Vijayawada, 1953, pp. 14-19; VT: JIH XXVIII (91950) 107-08), we shall consider that too. Their contention is that the author of the [Jyotirvidabharana] is the famous Kalidasa himself as claimed by the work in Kali 3068 (34 B.C.), and that therefore VM cannot belong to 427 in the 'Saka of 78 A.D. (corresponding to 505 A.D.), but only in the postulated Cyrus (or Andhra) 'Saka or 550 B.C. (corresponding to 123 B.C.), and that thus the Cyrus (or Andhra) 'Saka is proved. But the work could not have been written before 78 A.D., (though it says it was written in 34 B.C.), for in that work 'Salivahana is mentioned as a 'saka-kara (founder of an era), and that he founded the 'Saka Era 135 years after Vikrama founded his own 'Saka 3044 years after Kali (i.e. 57 B.C.). How could [Kalidasa], the alleged author of the work, be the contemporary (however junior it might be ) of VM (said to have lived in 123 B.C.) and at the same time know the starting of the ['Salivahana 'Saka in 78 A.D.?

The late date of the [Jyotirvidabharana] can be established also by other internal evidence in that work. Thus in giving the rule for the calculation of [ayanam'sa], it is stated that 445 is to be decucted from the years in the 'Saka Era and the remainder divided by 60. Cf.

['Sakah 'sarambhodhiyugo (445) nito hrto

manam khatarkair (60) ayanam'sakas syuh]/ (1.18a)

This means that in 445 'Saka the [ayanam'sa] is zero. This can be only the ['Salivahana 'Saka], for Indian astronomical works give zero [ayanam'sa] for c. 421 ['Sali]. ['Saka (Kali 3600, (some give 444). It cannot be argued that the author means the postulated Cyrus Era here, because firstly among the six ['saka] at all, and secondly nodody gives zero [ayanam'sa] for 123 B.C. (though he takes it as the starting point for calculation) (see Jih XXVIII. 106) and our discussion on it in the previous paper). Thus, having seen that he is alter than 445 of this 'Saka, (523 A.D.), for this rule can be applied only later than 445 'Saka, no instruction begin given as to what to do if the time taken is before 445 'Saka.

Again, the rule given in the [Jyotirvidabharana] for finding the year in the 60-year cycle of Jupiter corroborates this. If it is applied to the current year, 1881 'Saka (1959-60), we get the year [Virodhikrt], which we also get if we work it out according to the methods in the [Siddhantas]. If the year reckoned in the Cyrus Era or if the Vikrama Era is used in the rule, there is disagreement. So it is the ['Salivahana 'Saka] that is required to be used in this rule and not the Cyrus 'Saka not the Vikrama 'Saka which reigned in his time (for he says he is a contemporary of Vikrama). Thus, again, the conclusion that the author is later than the starting of the ['Salivahana 'Saka] follows, and his use of that 'saka.

In answer to these objections KV seems to have argued that Kalidasa could actually have lived earlier than the ['Salivahana 'Saka] epoch and have mentioned that epoch as a future historical event ont he basis of the ['sastras] (evidently meaning the Bhavisya Purana etc). But then how did the 'sastras know? Does KV want us to believe that they actually predicted future events? Clearly the ['sastras] themselves should have been written after the ['Salivahana 'Saka] epoch, and the [Jyotirvidabharana] should be alter still. And the jumbling of people of various ages already alluded to! We are asked to take this bundle of lies as sober history!

In the same manner other romances, like the [Kathasaritsagara], Bhojaprabahdha], Vikramarkacarita etc., (there is no dearth of them) based on popular stories should be dismissed wherever they contradict what may be judged as solid evidence, for we do not know who their authors were, nor what equipment they had for giving historical facts.

Thus in all places where the word 'Saka is used for the name of an era, it is the 'Saka of 78 A.D. (what latterly came to be called ['Salivahana 'Saka) that is meant. Further, there is no evidence to show that an era was started in 550 or 551 B.C. in Persia or in India as postualted by TSN and accepted by KV, which he calls the Cyrus Era, or as postulated by VT, which he calls the Andhra ERa. It may be that Cyrus founded the Persian Empire in 550 B.C., but what evidence is there to show that he started an era then? No such era was in use in Persia itself, not to speak of India. Many great events happen in the reigns of great kings. But they are not necessarily the starting points of eras. (VT does not even mention a great event in 550-51 B.C. for the starting of his Andhra Era). Now, these people

have taken of varied historical worth. Let them by all means attempt it, for it is only too true that unconscious prejudice has had some hand in the writing of the history of our land. But what we wish to show here is that their stand on the interpretation of the term 'Saka Era, with all it sramifications, is wrong, and will not help them, as also the various other ideas of theirs which we have shown to be wrong. Also we wish to point out that attributing base motives and questioning the bona fides of people (the writings of TSN and KV are replete with these) will not only not help, but may also be "Paid back with interest", as Dr.P.V.Kane says.

DETERMINATION OF THE DATE OF THE

MAHABHARATA: THE POSSIBILITY THEREOF *

Hindus generally believe that the story of the [Mahabharata} (MBh.) is a narrative of events that actually happended, and that they all took place near the end of the [Dvaparayuga] and the beginning of the [Kaliyuga]. Some hold that the War ended with the [yuga] and many, supported by the [Puranas], say that [Krsna] passed away at the end of the [yuga] and so the War took place a few years earlier. About the question of the time when the [Dvapara] ended, there is difference of opinion. The popular view is that [Dvapara] ended and Kali began at the time fixed for it by the astronomical [siddhantas], 3179 years before the 'Saka era of 78 A.D., which corresponds to Friday, 18th February, 3102 B.C., sunrise, or the previous midnight according to some schools. We do not know the exact grounds on which the [siddhantins] fixed the date as 3179 years before the 'Saka era of 78 A.D. Most probably, the first [siddhantins], like the author of the 'Old' [Suryasiddhanta] and [Aryabhata], fixed the point of time as a convenient epoch, when the mean planets, according to them, coincided with the zero-point of the zodiac, and the later astronomers accepted it, and adjusted their own planetary cycles to agree with the epoch exactly, or nearly there, finding the difference to be small.

But there are other dates fixed for the end of [Dvapara] by people like [Varahamihira], on the authority of the astronomical [Samhitas] and tradition current in their times. [Varahamihira] fixes the date as c. 2449 B.C., which can be known from his statement in hsi [Brhatsamhita] that the [Saptarsis] stood at [Magha] when [Yudhisthira] was ruling and that the year in his era

can be got by adding 2526 to the years of the 'Saka era. The authority for this is [Vrddha-Garga's] statement:

EòʱÉuùÉ{É®úªÉÉä& ºÉxvÉÉè κlÉiÉɺiÉä Ê{ÉiÉÞnèù´ÉiɨÉÂ*

Kalhana, in his [Rajatarangini], giving the chronology of the Kashmir kings in the [Saptarsi] or [Laukika] era current in Kashmir and the Himalayan regions, accepts [Varahamihira's view in toto, saying that people who fixed others dates were misguided:

¦ÉÉ®úiÉÆ uùÉ{É®úÉxiÉä%¦ÉÚuùÉiÉǪÉäÊiÉ Ê´É¨ÉäʽþiÉÉ&*

EäòÊSÉnäù¹ÉÉÆ ¨ÉÞ¹ÉÉ iÉä¹ÉÉÆ EòɱɺÉÆJªÉÉ |ÉSÉÊEò®äú**

The Jain tradition, giving 2634 B.C. for the [Yudhisthira] era is only a variation of [Varahamihira's view. The [Saptarsicara] of [Para'sara] and the [Arysiddhanta] of [Aryabhata]II, giving [Magha] for the sages in the seventh century of astronomical Kali, and the [Matsyapurana], giving [Krttika] for the beginning of Kali, support [Varahamihira].

But many think that both c.3100 B.C. and c. 2450 or c. 2600 B.C. for the [Bharata] events are periods too early, considering the state of society and the political conditions depicted in the MBh. They try to fix the Kaliyuga epoch coupled with [Yudhisthira's] rule, by reckoning backwards from the time of the Nanda dynasty, which historians have fixed at c. 400 B.C. onwards. The [Visnupurana] and the [Bhagavata] state:

¨É½þÉ{ÉZÉÉ%ʦɹÉäEòÉkÉÖ ªÉÉ´ÉVVÉx¨É {É®úÒÊIÉiÉ&*

(or ªÉÉ®úi{É®úÒÊIÉiÉÉä VÉx¨É ªÉÉ´ÉzÉxnùÉʦɹÉäSÉxɨÉÂ*)

B´ÉÆ ´É¹ÉǺɽþºjÉÆ iÉÖ YÉäªÉÆ {É\SÉɶÉnÖùkÉ®ú¨ÉÂ**

Variants: (1) ¶ÉiÉÆ {É\SÉ nù¶ÉÉäkÉ®ú¨É (1510), (2) YÉÉäªÉÆ {É\SÉnù¶ÉÉäkÉ®ú¨É (1015), (3) YÉÉäªÉÆ {É\SɶÉiÉÉäiÉäiÉ®ú¨É (1500).

The above would mean that between [Pariksit's] (grandson of the Pandavas) birth and Nanda's coronation, the interval is 1053 years (variants : 1510, 1015, 1500). From this, they fix [Pariksit's] time as c. 1500 B.C. (or 2000 B.C.) and thence the time of the [Bharata] story.

Besides these four main periods, several other periods are fixed based on various hypotheses, some plausible, some grotesque. For e.g., some scholars take the yuga measure of 12000 years as human years instead of divinde, and fix a date accordingly. One interprets the word [sama] and [varsa] used in the [Puranas] as half-years and brings down the story to c. 1200 B.C. But few scholars make any clear distinction between the period of the 'events' and the period when they were written down in the form of the epic [Mahabharata], while the orthodox traditional belief is that [Vyasa], grandfather of the [Pandavas] and Kauravas, wrote the work, and his pupil [Vai'sampayana] narrated it to king Janamejaya, grandson of the [Pandavas].

Determining the period thus, each in his own way, these scholars try to fix the year and exact date of the war from the calendrical details and various astronomical phenomena mentioned in the context of the War, like certain planetary combinations, occurrences of eclipses etc. This is not an easy matter, because there is a lot of contradiction between various sets of planetary combinations themselves and among the other phenomena mentioned. Some of these passages may be set out here:

1. ʴɶÉÉ®ú´ÉɪÉÉ ºÉ¨ÉÒ{ɺlÉÉè ¤ÉÞ½þº{ÉÊiɶÉxÉè¶SÉ®úÉè**

(MBh., Bhisma, 3.27)

2. ´ÉEòÉ%xÉÖ´ÉGÆò EÞòi´ÉÉ SÉ ¸É´ÉhÉÆ {ÉÉ´ÉEò|ɦÉ&*

¤ÉÉÀ®ú覃 ºÉ¨ÉÉ´ÉÞiªÉ ±ÉÉäʽþiÉÉRÂóMÉÉä ´´É´ÉκlÉiÉ&** (Bhisma, 3.18)

3. ¦ÉÞMÉÖºÉÚxÉÖvÉ®úÉ{ÉÖjÉÉè ¶ÉʶÉVÉÉäxÉ ºÉ¨ÉÎx´ÉiÉÉè*

SÉ®ú¨ÉÆ {ÉÉ{bÖ÷{ÉÖjÉÉhÉÉÆ {ÉÖ®úºiÉÉiºÉ´ÉÇ (EÖò°ü) ¦ÉÚ¦ÉÖVÉɨÉ ** ('Salya, 11.17)

4. ¶ÉÖEò& |ÉÉä¹`ö{Énäù {ÉÚ´Éè ºÉÆ#É°üZÉ Ê´É®úÉäSÉiÉä *

=kÉ®äú iÉÖ {ÉÊ®úGò¨ªÉ ºÉʽþiÉ& ºÉʨÉnùÒIÉiÉä** (Bhisma, 3.15)

Another gives:

5. ¤ÉÞ½þº{ÉÊiɺºÉÆ{ÉÊ®ú´ÉɪÉÇ ®úÉäʽþhÉÔ ¤É¦ÉÚ´É SÉxuùÉEÇòºÉ¨ÉÉä ʴɶÉÉÆ{ÉiÉä**

(Karna, 100.17)

6. ¨ÉPÉɺ´ÉRÂóMÉÉ®úEòÉä ´ÉHò& ¸É´ÉhÉä SÉ ¤ÉÞ½þº{ÉÊiÉ&*

¦ÉMÉÆ xÉIÉjɨÉÉGò¨ªÉ ºÉÚªÉÇ{ÉÖjÉähÉ {ÉÒb÷ªÉiÉä** (Bhisma, 3.14)

7. |ÉÉVÉÉ{ÉiªÉÆ Ê½þ xÉIÉjÉÆ OɽþºiÉÒIÉhÉÉä ¨É½þÉtÖÊiÉ&*

¶ÉxÉè·É®ú& {ÉÒb÷ªÉÊiÉ {ÉÒb÷ªÉxÉ |ÉÊhÉxÉÉä%ÊvÉEò¨ÉÂ**

EÞòi´ÉÉ SÉÉRÂóMÉÉ®úEòÉä ´ÉGÆò VªÉä`öɪÉÉÆ ¨ÉvÉÖºÉÚnù¨É*

+yÉÉvÉÉÆ |ÉÉlÉǪÉiÉä ¨ÉèjÉÆ ºÉÆMɨɪÉÊzÉ´É ** (Udyoga, 143. 8-9)

In the first set cited above (i.e., 1-4), we are told that Jupiter and Staurn are near the asterism [Visakha]. Mars is near [Uttarasadha], Abhijit (Brahmarasi) and ['Srvana]. In the second set, (5-7), Jupiter is said to be near Rohini. Mars is retrograde in [Magha]. Jupiter is in 'Sravana. (This contradicts two other statements.) Saturn is said to be in [Purvaphalguni]. Saturn afflicts (?) Rohini. Mars is retrogtrade in Jyestha and is about to go to [Anuradha]. To add to the confusion, many people interpret the comets of different colours mentioned in [Bhismaparva], chapter 3, as planets and, that too,each one differently.

Among the contradictory phenomena we can give the eclipse mentioned:

8.SÉxuùºÉÚ´ÉÉǦÉÉè OɺiÉÉè BEò¨ÉɺÉÔ jɪÉÉänù¶ÉҨɠ* (Bhisma, 3.33)

Here a lunar eclipse, and next a solar eclipse are mentioned as having occurred before the war. Then, at the time of [Duryodhana's] death the statement occurs:

®úɽÖþ®úOɺÉnùÉÊnùiªÉÆ +{É´ÉÇÊhÉ Ê´É¶ÉÉÆ{ÉiÉä* ('Salya, 27.10)

mentioning another solar eclipse so near, when a lunar eclipse had occurred before the first solar eclipse.

Again, several impossible and some very rare phenomena, mentioned merely to indicate that these phenomena presage evil, are taken by many as having actually occurred, adding to the difficulty_

9. jɪÉÉänù¶ªÉɨÉÉ´ÉɺªÉÉÆ iÉÉÆ ]Þõ¹]Âõ´ÉÉ |ÉÉ %¥É´ÉÒÊnùnù¨ÉÂ*

SÉiÉÖnù¶ÉÔ {É\SÉnù®úÒ EÞòiÉäªÉÆ ®úɽÖþhÉÉ {ÉÖxÉ&**8*.

|ÉÉ{iÉä ´Éè ¦ÉÉ®úiÉÉä ªÉÖuäù |ÉÉ{iÉÉ SÉÉtIɪÉɪÉxÉ&**9**

SÉxuùºÉÚªÉÉǤÉÖ¦ÉÉè OɺiÉÉä BEòɼxÉÉ Ê½þ jɪÉÉänù¶ÉÒ¨ÉÂ*

+{É´ÉÇÊhÉ OɽäþhÉèiÉÉä |ÉVÉɺÉÆIɪÉʨÉSUôiÉ&**28**

<¨ÉÉÆ iÉÖ xÉÉ%ʦÉVÉÉxÉä%RÆó +¨ÉÉ´ÉɺªÉÉÆ jɪÉÉänù¶ÉÒ¨ÉÂ**32**

(Mausala, Ch, 2)

SÉxuùºÉÚªÉÉÇ´ÉÖ¦ÉÉè OɺiÉÉè ʽþ (BEò¨ÉɺÉÔ) jɪÉÉänù¶ÉÒ¨ÉÂ**

(Bhisma, 3.3)

ºÉÉä¨ÉºªÉ ±ÉI¨É ´ªÉÉ´ÉÞkÉÆ ®úɽÖþ®úEÇò¨ÉÖ{ÉèÊiÉ SÉ** (Udyoga 143. 11)

½þiÉä EòhÉæ.... ºÉÉä¨ÉºªÉ {ÉÖjÉÉä%¦ÉªÉÖÊnùªÉÉªÉ ÊiɪÉÇEÂò** (Karna 94.51)

+±ÉIªÉ& |ɦɪÉɽþÒxÉ& {ÉÉèhÉǨÉɺÉÔ SÉ EòÉÌiÉEòÒ¨ÉÂ*

SÉxuùÉä%¦ÉÚnùÊOÉ´ÉhÉÇ·É ºÉ¨É ({ÉZÉ)´É{ÉÉèxɦɺºÉlɱÉä** (Bhisma,2.2)

SÉiÉÖnÇù¶ÉÓ {É\SÉnù¶ÉÓ ¦ÉÚiÉ{ÉÚ´ÉÉÈ iÉÖ ¹ÉÉäb÷¶ÉÒ¨ÉÂ*

<¨ÉÉÆ iÉÖ xÉÉʦÉVÉÉ´Éä%½Æþ +¨ÉÉ´ÉɺªÉÉÆ jɪÉÉänù¶ÉÒ¨ÉÂ**(Bhisma 3.32)

®úɽÖþ®úOɺÉnùÉÊnùiªÉÆ +{É´ÉÇÊhÉ Ê´É¶ÉÉÆ{ÉiÉä** ('Salya, 27.10)

ÊjɹÉÖ ºÉ´ÉÉè¹ÉÖ xÉIÉjÉxÉIÉjÉÉä¹ÉÖ Ê´É¶ÉÉÆ{ÉiÉä*

MÉÞwÉ& ºÉÆ{ÉiÉiÉä ¶ÉÒ¹Éæ VÉxɪÉxÉ ¦ÉªÉ¨ÉÖkɨɨÉÂ** (Bhisma, 3.31)

Scholars trying to establish their conclusions interpret these verses differently, some neglecting one set and some another, some giving acceptable meanings and some far-fetched and extremely strained ones. A few examples will show to what extent these people go.

Passage 2, cited above, is interpreted thus: The planet Mars moved retrograde again and again, towards the constellation 'Sravana, and occupied the constellation of [Brahma], i.e., Jupiter. The interpreter is unaware that [Brahmarasi] must mean 'the group presided over by a technical term used in astronomy and not 'again and again'. Passage 3 is interpreted thus: 'The planets Mars, Venus and Mercury were in fromt or to the east of the eldest of the sons of Pandu who were the masters of the whole land. To the interpreter, [caramam Panduputranam] means Yudhisthira, being the last counted from the last of the sons of [Pandu], while it means, simply, 'behind the sons of [Pandu] and in fromt of the Kuru kings'. Line 9 of passage 9 is interpreted: The planet Mercury arose concealed, (invisibly). The meaning 'invisibly' is given to [tiras], not realising that [anc] with [tiras] means, only 'across or obliquely'. Passage 5 is interpreted: 'Jupiter, having made [Rohini] to conceal herself (i.e., set), became like the sun or moon'. The passage means only that, Jupiter by his lustre hid Rohini. Passage 6 is interpreted: Mars is retrograde in [Magha], and Jupiter in 'Sravana. Saturn is afflicting [Purvaphalguni]. In the next verse (not quoted here) there is the word [sahita] which this interpreter takes to mean 'waiting', and cites as an example the [Raghuvam'sa] verse, [dvitranyt ahany arhasi sodhum arhan].

In 9, line 8 is said to mean: 'The lunar eclipse has already happened (in Karttika Purnima) and a solar eclipse is going to happen in the next [Amavasya]. Actually, the first part means that the dark patch on the moon is inverted (vyavrttam, not nivrttam). In line 16 of passage 9, [grdhra] is interpreted as "an evil planet", instead of 'eagle' which itself indicates an evil omen.

Thus, different years are fixed by different persons as follows:

N.Jagannatha Rao 3139 B.C.

T.S. Narayana Sastri c. 3126 B.C.

K.V. Abhayankar c. 3101 B.C.

C.V. Vaidya Do.

P.C. Sengupta 2449 B.C.

Karandikar 1931 B.C.

P.V. Kane c. 1900 B.C.

S.B. Dikshit c. 1500 B.C.

K.G. Sankara Aiyar 1198 B.C.

K.L.Daftari 1191 B.C.

V.Gopala Aiyar 1194 B.C.

Within the year, the dates are fixed for the different occurrences by the day's [naksatra] or [tithi], and the interval in days between one occurrence and another, given. Here, too, there are discrepancies and misinterpretations,leading to different dates. Though many have concluded that the war began on [Karttika] New Moon day, some say that it began on [Margasirsa 'Sukla Ekada's] day. The day of [Bhisma's] death is stated at places as [Magha 'Sukla Astami], while at others as [Ekadasi]. Cf.:

EòÉè¨ÉÖnäù ¨ÉÉ漃 ®äú´ÉiªÉÉÆ ¶É®únùxiÉä ʽþ¨ÉÉMÉä*

º¡òÒiɺɺªÉºÉÖ®ú´Éä EòɱÉä......**(Udyoga,80.7)

¶É´ÉÉê¹ÉÊvÉ´ÉxÉ°ü¡òÒiÉ& ¡ò±É´ÉÉxɱ{ɨÉÊIÉEò&*

Êxɹ{ÉRÂóEòÉä..... ...........**

ºÉ{iɨÉÉSSÉÉ%Ê{É Ênù´ÉºÉÉiÉ +¨ÉÉ´ÉɺªÉÉ ¦ÉʴɹªÉÊiÉ*

ºÉÆOÉɨÉä ªÉÖVªÉiÉÉÆ iɺªÉÉÆ iÉɨÉɽÖþ& ¶ÉGònäù´ÉiÉɨÉÂ**

(Udyoga, 143.18)

SÉi´ÉÉË®ú¶Énù½þÉxªÉt uäù ¨Éä ÊxɺºÉÞiɺªÉ ´Éè*

{ÉÖ¹ªÉähÉ ºÉ¨|ɪÉÉiÉÉä%κ¨É ¸É´ÉhÉä {ÉÖxÉ®úÉMÉiÉ&** ('Salya, 5.6)

xÉ EÖò´ÉÇÎxiÉ ´ÉSÉÉä ¨ÉZÉÆ EÖò®újÉ& EòɱÉSÉÉäÊnùiÉÉ&*

ÊxÉMÉÇSUôv´ÉÆ {ÉÉ{b÷´ÉäªÉÉ& {ÉÖ¹ªÉähÉ ºÉʽþiÉÉ ¨ÉªÉÉ** ('Salya, 35.10)

¶ÉÉ乪ÉÉä%½þ¨ÉºªÉÉÆ ¶ÉªªÉɪÉÉÆ ªÉÉ´ÉnùÉ´ÉiÉÇxÉÆ ®ú´ÉäUô.....

Ênù¶ÉÆ ´Éè¸É´ÉhÉÉGòÉxiÉÉÆ ªÉnùÉ MÉxiÉÉ Ênù´ÉÉEò®ú&*

ʴɨÉÉäIªÉÉä %½Æþ iÉnùÉ |ÉÉhÉÉxÉ ºÉÖ½þnù& ºÉÖÊ|ɪÉÉÊ´É´É**(Bhisma, 20.51-53)

iÉjÉ iÉä ºÉ֨ɽþÉi¨ÉÉiÉÉä xªÉ´ÉºÉxÉ EÖò°üxnùxÉÉ&*

¶ÉÉèSÉÆ Ê´É´ÉÇiÉ ÊªÉ¹ªÉxiÉÉä ¨ÉɺɨÉäEò ¤Éʽþ& {ÉÚ®úÉiÉÂ** 'Santi,1.2)

+É´ÉÞkÉä ¦ÉMÉ´ÉiªÉEäò ºÉ ʽþ ±ÉÉäEòÉxÉ MÉʨɹªÉÊiÉ **(Ib., 46.29)

ÊxÉ´ÉÞkɨÉÉjÉä i´ÉªÉxÉä =kÉ®äú ´Éè Ênù´ÉÉEò®äú*

ºÉ¨ÉÉ´Éä¶ÉªÉnùÉi¨ÉÉxÉ&..... ... ..... **(Ib., 47.3)

{É\SÉɶÉiÉÆ ¹É]Úõ SÉ EÖò°ü|É´ÉÒ®ú ¶Éä¹ÉÆ ÊnùxÉÉxÉÉÆ iÉ´É VÉÒÊ´ÉiɺªÉ**

(Ib., 51.10)

+ʹÉi´ÉÉ ¶É´ÉÇ®úÒ& ¸ÉÒ¨ÉÉxÉ {É\SÉɶÉzÉMÉ®úÉäkɨÉä**

(Anu'sasana, 167.5)

+¹]õ{É\SÉɶÉiÉÆ ®úÉjSÉ& ¶ÉªÉÉxɺªÉÉt ¨Éä MÉiÉÉ&*

¶É®äú¹ÉÖ ÊxÉʶÉiÉÉ%OÉä¹ÉÖ ªÉlÉÉ ´É{ÉǶÉiÉÆ iÉlÉÉ**

¨ÉÉMÉÉä%ªÉÆ ºÉ¨ÉxÉÖ|ÉÉ{iÉ& ¨ÉɺɺºÉÉ訪ÉÉä ªÉÖÊvÉι]õ®ú*

ÊjɦÉÉMɶÉä¹É& {ÉIÉÉä%ªÉÆ ¶ÉÖC±ÉÉä ¦ÉÊ´ÉiÉ֨ɽÇþÊiÉ**

(Anu'sasana.107.5)

But most of the scholars do not seem to have gone to the heart of the matter, placing before themselves clearly the two things that have got to be investigated, viz.: (1) How much of the [Bharata] story is true history, and when could it have happended. (2) When was it actually written down. Scholars who have studied the problemcritically are of opinion that there is a historical core in the story, but much fictitious matter has been added to it in course of time. The [Bharata] war must be true history, and the personages taking part in it, together with the line of the [Bharatas] and [Yadus], whose names occur frequently in Vedic literature, even as early as the Rgveda, not to speak of the [Brahmanas] like the 'Satapatha. The state of society and the political conditions point to a time earlier than the [Chandogya], one of the earliest of the upanisads, as can be seen from two statements in the work:

"EÞò¹hÉÉªÉ näù´ÉEòÒ{ÉÖjÉɪÉÉäCi´ÉÉä´ÉÉSÉ- ªÉt{ªÉäxÉSUÖô¹EòÉªÉ ºlÉÉhÉ´Éä §ÉªÉÉiÉ VÉɪÉÉä®úzɺªÉ ¶ÉÉ®ú´ÉÉ&, |É®úÉä½äþªÉÖ& {ɱÉɶÉxÉÒ <ÊiÉ**" "·ÉäiÉEäòiÉÖ½Çþ +É°ühÉäªÉ& EÖò°ü{ÉÉ\SÉɱÉÉxÉÉÆ ºÉʨÉÊiÉʨɪÉɪÉ*......"

The latter of the above statements shows that, at the time of the Upanisad, the Kuru and the [Pancala] country had coalesced, while at the time of the [Bharata] war they were different, the [Pancalas] being the allies of the [Pandavas]. It is quite natural for stories to gather accretions when they are repreated generation after generation. Most of the Superhuman and obviously exaggerated portions must have been added later. The core is generally placed between the eleventh and the nineth centuries B.C. Other story matter could have been added during a few subsequent centuries, when Krsna came to be deified. The lot of [Dharma'sastra] matter with the illustrative stories must have been added last, in the course of several generations. Anyhow, by the first or second century B.C. or A.D., the [Mahabharata] must have arrived at its present form, with a few bits of interpolations here and there, made later.

As for its writing, the language is that of the early classical period, for it is clearly later than that of the genuine upanisads . The addition of the later matter and the development of the classical language must have, naturally enough, gone on together. By the first or second century A.D. most of the whole [Mahabharata] must have attained the present form.

It is natural for story writers to incorporate into their storiesideas current in their own time. A lot of the astronomical facts found in the work, especially in the context of the war, must have been cooked up by these later writers in the light of their own knowledge, and added by different people at different times. That explains the contradictions. It must be clearly noted that the astrological ideas mentioned in the [Samhitas] which developed from the 2nd century B.C. could not have been current as early as the 11th to 9th century B.C. and , even if current, are not likely to be remembered after so many generations. By the first century A.D. or B.C., the astronomical [Samhita] had mostly been written, and naturally the ideas in them find a place in the work. The calendric system of the [Vedanga Jyotisa] continued to be current in this [Samhita] period, as can be seen from the [Garga Samhita], and ideas showing ['Sravistha] as being the first star (beginning the winter solstice) are in evidence, together with its shifting to 'Sravana, (c.third century B.C.) as can be gathered from the [Visvamitra] episode. The MBh. in its [Virataparva] ch. 52 contains the [Vedanga]calendric system.

Again, in the context of the war, it is natural for writers, especially of epics, to describe portents as happening to presage evil. The [Samhitas] devote chapters to describe these portents. The [Ketucara], on the appearance of comets, is full of portents, as also separate chapters devoted to portents like rare or unnatural, impossible or terrible phenomena. These have been included in the work. But most investigators have not interpreted these portions properly, for which a detailed study of the chapters on [Ketucara] and [Utpatas] in the [Brhat samhita] of [Varahamihira] would be advantageous. For example, the mention of the new moon together with solar eclipse occurring on [Trayodasi], the sun and the moon being eclipsed on the same day (the same month), and that on [Trayoda'si], Mercury moving across the sky, (i.e., north-south), the dark patch on the moon being inverted, the lunar eclipse at [Karttika[ full moon, the solar eclipse at [Karttika] new moon, and again the solar eclipse at the time of the mace-fight, are all intended by the writer to be impossible things occurring. The mention of the red moon indistinguishable from the red sky (digdaha), eagles falling on the flag, appearances of comets of different colours and in groups are all portents. Ignorance of teh fact that the 'grahas' of different colours mentioned in [Bhismaparva], chapter 3, are not planets but comets, has added to the confusion, because these scholars do not realise that, in the [Samhitas, the word 'graha' means primarily comets, (vide the chapter on Ketucarain the Brhatsamhita).

It would be clear from the abvoe, that all the skill shown in distorting the meanings of words and trying to show when these impossible or rare phenomena and contradictory planetary combinations would actually occur, has been wasted. Excepting the time of the year when the war might have happened, there is nothing in the [Mahabharata] to fix the year definitely. We do not have adequate data to fix either the happenings or when the work, even part, was written.

A BRIEF HISTORY OF TAMIL ASTRONOMY*

1. Prefatory

By Tamil astronomy, I mean astronomy written by a Tamilian, whether the language is Tamil or Sanskrit. But, as far as I know, few outstanding works on astronomy have been written in Tamil Nadu, unlike in Kerala, where there has been a succession of sholars writing original astronomical works or commentaries, amking their oen extensive contributions. Two Tamil works on Mundane and Electional Astrology, [Cudamani Uttamudaiyan](13th century), and [Vime'svara Uttamudaiyan](17th century), contain small portions of astronomical computations of the planets, taken chiefly from the [Parahita] system of [Haridatta] of Kerala, as it is , or with the ['Sakabda] or [Parahita] [Bija] corrections carried out. Excluding two Sanskrit commentaries, one of which is by [Suryadeva-yajvan], only one work, called the [Vakyakarana] (c.1300), is wholely devoted to astronomy, and somewhat co-extensive with its content. It s source is mentioned to be the [Bhaskariya] as studied in Kerala, with the [Parahita] correction, and with [Haridatta's] [Parahita] method used. There is no original material in it. But being a [Karana] or manual,it has invented devices for ease of computation, especially by almanac-makers. It is in these devices that it is original, in so much as perpetually repreating planetary tables are given, with mnemonic phrases (vakyas), being used for numbers. It is in Sansrit. All later almanac-makers and computers of the Tamil country use this or Tamil adaptations of this.

About 1800 AD, first Le Gentil, and then Warren, in order to learn how the Tamils computed, asked some natives of Pondicherry to demonstrate to them. The methods and constants were almost exactly those of the [Vakyakarana]. Le Gentil and Warren reported the demonstration in their works, not knowing the source. Two periods, 248 days and 3,031 days, occur in the moon's tables. Any table-maker must get these as full days closely approximating to 9 and 110 anomalistic revolutions of the moon, on which the values must depend. But not considering this point, and since these periods occur in the [Vasistha Siddhanta] of the [Panca siddhantika], exhibiting Babylonian connections, Assyriologists and Indologists like Neugebauer began to declare a Babylonian derivation for the method, a mere surmise. Neugebauer publised a paper examining the methods of the pondicherry informant of Warren. The paper exhibits ignorance of the fact that the source for the informant was [Vakyakarana], since it was unknown to scholars at that time, not having been printed yet, and that this work itself is based on the works of [Bhaskara] I and Haridatta. Seeing Neugebauer's paper recently, I wrote to him explaining the position and incidentally clarifying certain difficulties expressed by him. But I thought I must write a history of Tamil astronomy, however meagre or insignificant it may be to lay the Babylonian ghost to rest. This short paper is the result. I consider this only as a beginning, so that others may follow up make their own contributions.

II. Introduction

By the word [Jyotisa], Hindus mean both astronomy and astrology combined. It is in three divisions, (1) [Siddhanta-skandha]: This consists of two parts, [Ganita], giving the positions of heavenly bodies, and [Gola], dealing with general astronomy like cosmogony. (2) [Samhita-skandha]: This too consists of two parts. In one, called Mundane Astrology, predictions for the whole world or regions of the world are made, based on planetary positions. Several other things considered useful to man, like knowledge of omens and indications of weather, are also given. In the other part called Electional Astrology auspicious moments for religious rites and ceremonies and journeys are given, together with rules for compatibility in marriage etc. (3) [Horaskandha] gives life predictions for individuals, based on the planetary positions at conception or birth or the moment when the astrologer is approached and requested to make the Predictions. The origin of the first two divisions can be traced to the Vedas and Vedic times. The last is declared by scholars to be foreign in origin, as can be inferred from the large number of Greek and Babylonain words used, and the period it appeared in India, namely the first or second century A.D.

From words like [Naksatradar'sa] and [Ganaka] occuring in the [Yajurveda], and a [Naksatra-vidya] being mentioned in the [Chandogya-upanisad], and the fact that a knowledge of the positions of the sun and the moon is required for fixing various Vedic rites and rituals, we can infer that the [Siddhanta] division must have originated very early. The [Vedanga-Jyotisa], by one [Lagadha], whose content points to the 12th century B.C. though re-written later, is the earliest astronomical work extent. The [Samhita] and the [Hora] [Skandhas] require planetary positions as a pre-requisite for [prediction]. So, by the period of the [Samhitas] which are earlier and flourished in the first few centuries B.C., astronomy proper must have been tolerably well developed. [Varahamihira] (first part of the 6th century A.D.) has condensed fice [Siddhanta] current during his time or earlier, in his [Pancasiddhantika. Of these the [Paitamaha] is the system of the [Vedanga-Jyotisa itself. The [Vasistha] and the [Pauli'sa] show clear connections with the [Babylonian] astronomy of the [Seleucid] period, and the [Romaka] with that of Alexandria. The [Saura] is indigenous, and a model of the Hindu [Siddhantic] astronomy. Being necessary to fix the times of the hundreds of Hindu religious rituals and the numerous fasts, feasts and festivals, as also for the use of the horary astrologers, scores of [Siddhantas] and [Karanas] or manuals were written by astronomers like [Aryabhata], [Varaha] and [Brahmagupta], together, with commentaries or expansions of these, which are themselves astronomical works. Together with the Sanskrit cultural migration to South India beginning with the last centuries B.C., astronomy too spread to the South.

As far as astronomical works are concerned, it seems that the Kerala country was the seat of its developments in the South. It is all based on teh [Aryabhatiya], with or without corrections called [Bijas], though several later astronomers like [Parame'svara], Nilakantha etc., made their own original contributions. How [Aryabhata] cam eto be connected with the Kerala country is yet to be explained. He is called [A'smaka] (i.e., one born in the A'smaka region) and some say that an early name of the erstwhile princely state of [Travancore] was [A'smaka] (Apte's Dictionary). But many say that the region near the Vindhyas was called the [A'smaka] country (i.e., the region of the [A'smaka] people), and [Aryabhata] was a native of this country. [Bhaskara] I, (c. A.D.600), who was an exponent of his school, seems to have belonged to the [Valabhi] regions in Gujarat. The [Aryabhata] school must have migrated to Kerala from this region, throguh some Keralite who had learnt astronomy at Valabhi or some [Valabhi] astronomer settling in Kerala. Anyhow, before the end of the 7th century the first well-known astronomer of Kerala, Haridatta, had appeared. He has written two works. [Mahamarganibandhana], and [Grahacaranibandhana]. The former must be a full treatise, but manuscripts of this are yet to be found. But the latter is well known as the [Parahitaganitam]. It gives an easy method for computers to use. Tables are given for each planet to find the equation of the centre and the equation of conjunction, at intervals of [3 3/4o] of the respective anomalies. Haridatta uses the [Aryabhatiya] constants in this without any corrections on the [Aryabhatiya], called the [Vagbhava] or ['Sakabada], or the [Parahita] corrections. It is said that he promulgated hi [Parahita] system of comuputation together with his corrections in 683 A.D. at [Tirunavay], to the astronomers who had assembled there for the ["Mamankam]" festival occurring once in twelve years.

From Kerala [Parahita] system and the [Aryabhatan] school of astronomy spread to the Tamil part, and the Tamil astrologers have been using the [Parahita] when they desired to compute the sun, moon and planets for predictions.

II. Tamil Astronomy

The first astronomers of the Tamil country, known to us at [present, was [Suryadevayajvan]. He has written many fine commentaries in astronomy and astrology in Sanskrit, but has not produced any independent work. In these he says he was born in [Gangapura] in the Cola country, which can be indentfied with [Gangaikondacholapuram] (N.Lat 11o 13', E. Long 79o 30'), about 40 miles north of Tanjore. He gives its equinoltial shadow to be 2.4 angulas, which corresponds to N.Lat 11o 17', and its distance, east of Kharanagara to be 11 [Yojanas], which is 1.2o according to the [Aryabhata] measure. Kharanagara is said to be on the Ujjain meridian, as given by [Bhaskara] I in [Mahabhaskariya], ch.II. So, the east longitude comes to about 77o, but actually it is 70o 39'. Either the eleven [yojanas] given is a scribal error, or [Suryadevayajvan's] calculation of the number of [yojanas] is different. He says he was born in 1113 'Saka (1191 A.D.), and learnt astronomy from his maternal uncle [Suryadeva], whose protege he was. He was a devotee of Krishna, whose grace he invokes throughout. The following are his works: (1)The [Aryabhatiya-Bhasya], (2)Laghumanasa,(3) [Jatakapaddhati-vyakhya], a commentary on the astrological work, [Jatakapaddhati], of ['Sripati], (4) [Govindasvamibhasya vyakhya], a supercommentary on the Govindasvami-bhasya on the Mahabhaskariya], (5)A commentary on the [Yogayatra] of [Varahamihira], (6) Khandakhadyaka-vyakhya on the Khandakhadyaka of Brahmagupta. The last three works are now known only from reference.

Astronomy porper (next) appears in the Tamil region as a small part of a mainly astrological work dealing with mundane and electional astrology, by name [Cudamani Ullamudaiyan], the name appearing in verse 7. The name and time of the author is found in the last verse:[ ]

The name of the author is [Tirukkottinambi] son of '[Mamuniari]', he lived in [Pandamangalam], near [Uraiyur] in [Triuccirappalli]. The period mentiod is after 1100 'Saka. But the [Khanda[ for the moon giving c.1107 'Saka, and the [Parahita Khanda] of all planets c. 1167 'Saka shows that the work must be alter than 1245 A.D. (From this it can be inferred that the author was a junior contemporary of Suryadevayajvan). It says that the source of all its content is from a Sanskrit text, and its computation of eclipses is according to a Sanskrit work called [Jayantamala], of which manuscripts are yet to be discorered.

The meagre astronomical matter of [Cudamani Uttamudaiyan] consists of : (1) Days from Kali epoch using the 'Saka year reckoned in Solar years. The multiplier and divisior to get this gives 365-15 29 29 days which is less by about 2 [vinadis] compared with the value given by all[ Hindu Siddhantas]. Perhaps the muliplier given by -["------------------------"](verse 381) is ["-----------"]. The sun's motion in [rasis] are to be found by using monthly periods, and the motion within the month, by the true motion per day given roughly for each month. (2) The sixty year cycle and the division of each in to three parts are given. (3) A method to find the true moon using the two anomolistic cycles of 3031 and 248 days, together with the so-called [Vararuci-cakyas] giving the true anomaly for each day of the 248 days period, is given. No correction is given for the error in the two periods, with the result that the error will accumulate in course of time. (4) Haridatta's [Parahita] method is given to compute all bodies, i.e., the sun, moon, and star planets. There is a very small difference in the constants, from those used by Haridatta, but the 'Parahita correction used by later computers, is not used. (5) As mentioned already, there is a section devoted to eclipses, taken from an earlier Sanskrit work, [Jayantamala]. It is seen from the method for Days from Kali, given in this work, that the solar sidereal year, with the dates of the solar month, had already come to be used in the Tamil country for calenderical purposes. It is more advantageous than the luni-solar calendar, used everywhere in earlier times, and still used in all parts of India except Kerala. Tamil nadu, Bengal and Orissa. Its ease to get the Kali-days is obvious as against the complicated luni-solar method of using the tithi and lunar month. When did it come to be used? Haridatta in his [Parahita-ganita] uses only the luni-solar method,as also earlier perople like Bhaskara I. With the solar year is associated reckoning the 'Saka years and the 60 year cycle years, which are Jovian, in solar years. The cycle years have fallen back by now in the south by 12 years. From this we can reckon that the practice has originated with the Kollam era of 825 AD, which is a landmark in Kerala astronomical history. Perhaps the astronomers used it first, for its obvious advantages, and the laymen and civil administration fell in line. The origin of its use in inscriptions will be revealing.

The next work coming to view is the [Vakyakaranam] or [Vakyapancadhyayi], by [Sundararaja].

This is the one work that can really be called astronomical, though it is only a manual, using rought methods for ease of computation. It is in Sanskrit. It states that it follows the Bhaskariya of BhaskaraI mentioned above, and the [Parahita-ganita] of [Haridatta] for its methods and constants, but we can see that the [Parahita] corrections have also been used. From the subtractive days given for commencing computation it can be inferred that is date is near the end of the 13th century A.D. The author [Sundararaja] must have hailed from the region of [Kanchipuram] in Tamil Nadu, as can be inferred from his invoking the grace of [Varadaraja] of [Kanchi] in the first verse, and using words and phrases reminiscent of [Kanchi] and the [Cholas], as mnemonics of tabular values.

This is the basis for the so-called [Vakya] [almanacs] extensively used in Tamilnadu. It can be seen that the informants of Warren and Le Gentil, c. 1800, at Pondicherry, were using the methods and constants of this work for computational demonstration.

Since the true longitudes of bodies repeat in periods, short or long, as the case may be, the actual values of convenient segments of the periods are computed and given by groups of words or [vakyas] being used for numbers. Sicne the sun's apogee has no motion according to [Aryabhata's] system, the sun's true positions repeat in the year, and are easily computed. Since there are 9 anomolistic revolutions of the moon in 248 days, with an error of only 8', true anomalies for each day has been given as sufficiently accurate, and a correction given for each period of 248 days. In a period of 3,031 days of 110 revolutions, the error would be only 2' and in 12,372 days of 449 revolutions, it is taken as zero. Thus the error can be prevented from accumulating. Adding the mean moon at the beginning of the period to the true anomaly and corrections, the true moon is got. These 248 [vakyas] called [Candravakyas] or [Vararuci-vakyas], are said to have been first formed by an ancient astronomer of Kerala called Vararuci.

In the case of the fice planets, the true longitudes repeat when a number of synodic periods are also whole solar years, here too there being no motion of apogees as taaken. These are very long periods. But sufficient accuracy can be secured by using a smaller number of synodic revolutions, using corrections for the small error, which can be prevented from accumulating as in the case of the moon. These are the [mandala-vakyas] and [samudra-vakyas] of the planets. The explanations of their formation and use is found in Chapter II, while Chapter I is devoted to the sun, the moon, and Rahu (the nodes).

The third Chapter is devoted to matters depending on the solution of spherical triangles, like daylight, rising ecliptic point, time and shadow. The fourth chapter deals with eclipses, and the fifth with heliacal rising and setting as also the [Mahapatas]. Thus the astronomical matter given is practically full. The results are also sufficiently accurate. The much-spoken-of error in the [Vakya-almanacs] now-a-days is due to certain conventions followed to conform to the [Dhrama'sastras] (which is therefore not error at all) and the error in the astronomical constants used accumulating over the course of so many centuries now. [See the printed edition (loc.cit.) for details. Being well suited for almanac making, this or Tamil adaptations of this are used by computers.

The Sanskrit commentary mentioned gives some interesting details about the subsequent history of the [Vakyakarana] in the colophon ending the commentary. A king by name [Tipparaja] is mentioned here who can be identified with [Gopendra Tipparaya] of the [Saluva] dynasty known from his inscription of 1475 A.D. to have ruled the southern districts of the [Vijayanagar] empire, as a feudatory.

Tipparaja. An astronomical work of [Tipparaja] is mentioned in the abvoe mentioned colophon, bearing the name [Tippa-rajiyam]. He has written three works in astronomy, the [Tantraratna], a set of [Candra-vakyas], and [Uparagadarpana]. It is the [Tantraratna] that must have been referred to as the [Tipparajiyam]. Besides these, he has written a commentary called [Kamadhenu] on [Vamana's] [Vakyalankara-sutra-vrtti], and the [Taladipika], on the [Tala] branch of muscic. All are in Sanskrit.

The [Tantraratna] is a brief work, covering the whole field of astronomy. It follows the later [Suryasiddhanta] in its yuga elements and other constants. It is in eight chapters. The first chapter deals with the computation of the chief items of the Hindu almanac, [Vara], [Naksatra] of the sun and moon, [Tithi], [Yoga], [Karana] and [Tyajyam], following the method of the [Vakyakarana]. In the place of the [Bhupadi-vakyas] for the sun, it gives the [Yogyadi-vakyas] which are easier to use, and appear here, like [Goparaja-tanaya], [kathari-Saluva], [Sangita-rasa-bhavajna], [Ubearadityasamvardhita], [Camburaya-sthapaka], [Canura-malla], [Talajna], and [Kalyarapurendra]. The colophon contains the name [Saluva-Tipparaja]. Chapters I to VIII are a regular Tantra work, of course, following the [Surya-siddhanta] in its elements, methods and contents. But the work is only mediocre in merit.

The [Candravakyas]are expected to be used as an appendic to the first chapter taking the place of the so called [Vararucivakyas] used in the computationof the moon. But the [vakyas] go up to seconds, instead of stopping with minutes. At the end of the work there is some prose matter, in which an example to get Kali days is given. This is 16,74,709 days expired, Kali 4585=1484 A.D., the time of [Tipparaja] himself. His [Uparagadarpana], on eclipses, is available manuscript form.

Next, in an astrological work by name Vi (Bi) [me'svara Uttamudaiyan], a small section on astronomy appears. An alternative name given is [Jodidagrahacintamani]. The main matter is what is found in the [Cudamani Uttamudaiyan], i.e., mundane and electional astrology with a little horary astrology added. The author does not give his name, but in every verse he mentions Vi(Bhi) me'svara, the Lord of [Todukkadu] in [Tanjore] District, often invoking the grace of [Vime'svara's consort, Nahaimuhavalli. The date of the work given in verse 10 is Kali 4728, i.e., 1627 A.D. The subtractive given for beginning the moon's computation is 16,35,565 days (got by adding 248 and 3031)and 248 days. To bring the moon to true sunrise caused by the sun's equator of the centre, the change in daylight and the difference in longitude called [de'santara] (the reduction to the equator being omitted, as in the original [Vakyakarana itself), subtractive or additive minutes being given for every 8 days of each month. The whole correction is wrongly called [de'santara] correction by the commentator, though the actual work does not make this mistake. Since the average of all the corrections given is zero, we have to assume that either the [de'santara] itself (about -7' for Tanjore) is not given, or that a correction is given to teh mean moon, equal to 8'. This is the dawn of the [manyadi] correction to the moon which includes a correction to the mean moon of about 26', used by the informants of Gential and Warren, c. 1800 A.D., and later. upto this day. For finding the true sun with in the month, the [Yogyadivakyas] are given, appearing in this work for the first time instead of the [Bhupadivakyas] of the [Vakyakarana], which are more difficult to use. Here too the commentary makes mistakes in teh instructions to add or subtract, not to speak of the printing mistakes of the constants.

To compute the five planets, the subtractive to begin work is given as 1368 'Saka. Very rough values for the mean motions per annum are given, showing that the author intended them to be used for the author's own lifetime. But these values varry the [Parahita] corrections used by the [Vakyakarana]. To correct the mean planets for the equation of the centre and equation of conjunction, values are given for each [rasi], a rough approximation instead of the intervals of [3 3/4o] as in the [Parahita-vakyas] of Haridatta, repeated in the [Cudamani Uttamudaiyan].

During the last decades of the 19th century, several such astrological-cum-astronomical works appeared for the use of almanac makers. One is the [Muruga-'Sekharam], which gives the [Vakyakarana] method for the sun and the moon and the corrected Parahita method for all. It mentions how to use different calendars for use by Muslims and others. It includes also [Dharma'sastra] matter to determine dates for offerings to the manes, [Vratas], fasts, feasts, etc. It is by one [Muruhaiya Josyar], and reprinted by Ratnanaicker and Sons in 1932.

During the same period, a work called [Jodidagrahacintamnni], popularly known as [Varsadinul], appeared, for the same purpose. It is practically the [Vime'svara Uttamudaiyan] with some ramifications of the astrological portion.

One [Swami Iyengar] of [Karaiyur] issued a [Parahita Ganitam] in Tamil, with his own corrections, mostly following the [Vakyakaranam] constants and a subtractvie from Kali days of 16,83,112 days, which is equal to eight times the sub-yuga used by Haridatta, etc. 2,10,389 days, the time being 1507 A.D. Why it uses this subtractive instead of a larger one, by which it could have obviated labour, is a mystery. It also gives methods to find day-time and ascensional difference for different latitudes.

A very important work c. 1880 is the [Jotisaganitasastam] of [Munampannai Krsna Josyar of Nanguneri] (Tirunelveli District). It is in Tamil and is extensive used by [Vakya] almanac makers, many of whom own copies it. It follows the [Vakyakarana], but uses the [manyadi] to bring the moon to true sunrise, with a [bija]. The author's knowledge of astronomy is good, and he brings it to bearon his work. It is in five parts: (1) The sun, moon and RAhu; (2) The fice Star-planets; (3) Eclipses; (4) Daylight, Ascension etc., (5) Miscellaneous matters. For eclipses he gives different ancient methods and [bija] corrections to secure tolerable accuracy. He also gives rules to determine dates for offerings to the manes and Vratas.

IV. Modern Times

We now come to out own century. A desire to know the correct positions of planets,for the sake of predictions, is evinced by the educated. A new type of almanac, called [Drgganita], whose calculation is based on modern Nautical almanacs and ephemerides is becoming popular. One such almanac-maker, C.G. Rajan, a Tahsildar in the Chiglepet district, compliled tables based, fairly accurate positions of planets can be found for any day from 3000 B.C. to 3000 A.D., which may be used by astrologers and research scholars.

Rajan has named the work [raja-Jyotisa-ganitam]. It is in English, and there is also a Tamil translation by himself. It was published in 1935. In his explanation and discussion, he exhibits good knowledge and gransp of the subject. In 1959, he published three booklets for the benefit of [Vakya-almanac-makers, based on teh [Vakya-karana]one, on the sun, moon and Rahu, one ont he computation of the five star planets, and one on calculating eclipses, using earlier cycles. He also gave the manner of computing the elements, in the usual method. In 1961, he published a booklet on the claculation of the solar eclipse, computing and using modern elements. Till he died a few years ago, he was preparing the [Rastriya Panchang in Tamil], for the Government of India.

There was also one L. Narayana Rao, a native of Tanjore and a retired Central Government Officer, who was publishing a [Siddhanta Panchangam], till he died recently. He has collected an dpublished the planetary ephemerides for various groups of years from 1800 A.D. to 1950 A.D. Being a master of modern astronomical calculations as well, he has shown how the set of ephemerides given can be used for any time earlier or later outside the given periods, provided the time is not too far away. That is , he has shown how the sets of ephemerides can be used as perpetually repeating tables like the [Candra and Samudra-vakyas] fo the [Vakya-karanam]. If he had given longer periods to prevent the accumualtion of the inevitable small errors at the end of each given period, it would have been useful to research workers desiring to compute the correct positions of planets at very early times. Also, he could have minimised the labour of reducing the given positions to the times required earlier or later, provided they are not too far away, by giving the instruction for changing the equation of the centre by differentiation. He has also given some tables useful for astrologers.

Lastly, the author of the present paper too has made some modest contributions. He has translated the [Vedanga-Jyotisa] with an Introduction. He has also translated the [Vakyakarana] in Tamil, with elaborate notes and worked examples. In the notes he has pointed out the merits and defects of the work, with modern astronomy. It is being used by almanac-computers and some of his friends.

He thought it would be good if, instead of the various instructions given in his notes to modernize the [Vakya-karana] results, he added two appendices to his tanslation, one to compute the correct positions of planets, and the other to give the correct circumstances of eclipses. The former he put in the [Vakyakarana garb], just for the pleasure of it, though the planning and computation of the tabular values demanded a lot of ingenuity and enormous labour. In the case of the moon he has taken into account all the equations necessary for tolerable accuracy. The latter is a compendium in Tamil of a larger work he has writen in English. He finished these two works by 1953 and 1956. They are still in manuscript form.

He has revised his old work (in English) on elcipses, making a better arrangement of the matter and the tables more easy to use. Its merit is that it is complete in itself, where the elements used in computation, normally taken from the almanac themselves, are computed by himself and given. To do it easily, he has devised methods to combine the large number of the respective equations and hss given them in a smaller number of tables. He has given three methods of computing the circumstances, viz., the nonagesimal method used by teh earlier astronomers, the right ascension declination co-ordinates method, and [Bessel's] methods, explaining all of them fully.

In 1957, he critically edited the [Mahabhaskariyam] of BhaskaraI with the commentary of Govindasvamin and a super-commentary by [Parame'svara], for the Government Oriental Manuscripts Library, Chepauk, Madras. In 1962 he brought out critical edition of the [Vakyakarana], with commentary jointly with K.V.Sarma.

In 1962, he began the preparation of a critical edition of the [Pancasiddhantika] with an elaborate Sanskrit commentary and worked examples and an English translation of both the text and the commentary. He has also prepared a bookon the theory and practive of modern astronomy, with tables and some basic mathematics and otehr matters useful to astronomers. This too remains in manuscript form.

THE AGE OF 'SANKARA

1. A REVIEW OF THE AGE OF SANKARA

by T.S. Narayana Sastry (TSNS)*

The Age of Sankara was first published in 1916, and a second edition has come out omitting certain parts of the Appendix. One of the important parts omitted is the ''Sakakala or 'Saka era', treating about the 'Saka era mentioned as No. VI of the eras listed on page 22.

Thsi work gives an interesing account of the life of the famous [Adi-'Sankaracarya], detailing the various incidents of his life, in the manner of a historical fiction. [Vamar'sa], a Sanskrit work on the same subject, written in 1898 by [Rajaraje'svara] ['Sankaracarya] ['Svami] of the [Dwaraka Mutt], has shown the author this method of treatment, together with many other things. The author says that the date of 'Sankara is from 509 B.C. to 477 B.C. To see such an early date determined for 'Sankara must be very pleasing to Hindus in general, and teh [Advaitins] in particular. But when historians and research scholars determine from internal and external evidences that ['Sankara] could have lived only several centuries after this time, and it might be even more than a thousand years later, we have to accept it as true, because the evidences put forth by the author for his date are, I am afraid, untenable.

I shall set out the true position first. Al the Mutts established by 'Sankara have constructed their [Parampara]('line of succession') from tradition and inadequate records, long after they were first founded, after they gained worldly importance, and felt the need for an uninterrupter pedigree from 'Sankara downwards. There were several [Vikramaditya-s] in whose names records dated in [Vikramabda] are found, confusing them with the well-known Vikrama Era of 58 B.C., and this has led to several contradictions. The line of succession may be expected to be correct for a few pontiffs before the attempt at reconstruction was made, when memory would have been fresh. Generally speaking, uncertainty would increase as we go further up. The [Sringeri Mutt], having had connections with the [Vijayanagar] and later empires and had become famous from the 13th and 14th century onwards, it sline can be expected to be the most authentic. But mistaking 'Sankara to have lived in the first century B.C., owing to the confusion in the [Vikramaditya] names, has resulted in giving 800 years to 'Sankara's successor [Sure'svaracarya], to connect him with the next in succession in that line, beginning from whom the succession has been quite authentic. If ['Sankara] is taken to have lived in the 8th century A.D., or a little earlier or later, on which there is an almost consensus of opinion among historians, not only will all discrepancies be resolved, but it will also fit in with the internal evidence of 'Sankara having known [Dinnaga], [Bhartrhari], [Dharmakirti] and [Kumarila], and having condemned the [Vijnana] and ['Sunyavada-s] of the Buddhists and several Jain tenets of later growth.

Like the line of succession of 'Sankara, the writing of his history too must have been attempted long after he had lived, after the desire to write it had cropped up in peoples' minds. Therefore, some of the incidents mentioned therein may be authentic, but others that are impossible or seem improbable to the fair-minded historian, might not be true, and are the product of the poet's [Kavya-style] of writing, a mixture of fancy and fact. That 'Sankara was born on a [Vai'sakha-'suddha-pancami] must be true, and that the birth-star was [Ardra] may also be true. But the planetary combinations and auspicious birth-times given variously by various authors must have been drawn from their imagination, fancying that such a great man must have been born under such a combination.

But one thing we can say. The ['Sankaradigvijaya], popularly known as [Madhaviya-'Sankaravijaya], written by [Vira Vasanta-Madhava], better known as [Madhava-mantrin], of [Angirasa gotra], minister of Immadi Bukka II, in the early 15th century, and publised long ago with the two commentaries [Dindima] and [Advaitarajyalaksmi] and referred to even by teh [Susama] as [Sanksepa-'Sankara-vijaya], being the most ancient extant text on 'Sankara], may taken to be the most authentic, saving the inevitable [mahakavya] style.

This work mentions a prior work, [Pracina-'Sankara-vijaya], of which this is a compendium, but that work does not seem to be available now, abotu which we shall see later. some Sankara Mutts have caused to be written or patronised several 'Sankara-vijaya-s, literary works and [Guru-parampara-s] to establish each its own importance and, in some cases, when disputes arose about property or jurisdiction, and have even concocted new evidence like copper-plate inscriptions etc.

In thus concocting evidence, it is patent that contenders entering in recent times have certain advantages. They can use the better knowledge of history and science to make more plausible concoctions, and even hold up the earlier mistakes of their adversaries to ridicule. But if truth will triumph, the very excessive greed of these concoctors will expose them; the very knowledge that theyuse, being insufficient to cover them up, willexpose them. The work which we are reviewing serves as a good example of what I am saying.

The author of the work reviewed here, T.S. Narayana Sastri, asserts in this work that 'Sankara was born in 509 B.C. corresponding to Kali 2593, in the year Nandana, on [Vai'sakha-'suddha-pancami], Punarvasu, Sunday, in [Karkataka-lagna]. The evidence he adducess for this can be classified under four heads:

1. Bits of a [Brhat-'Sankaravijaya (BSV) attributed to the famous [Citsukhacarya], quoted in the commentary called [Susama] (alleged to be by one Atmabodhendraa pontiff of the Kumbakonam Mutt), on the [Guru-ratna-malika], giving the line of succession of the [Mutt], alleged to be written by the famous [Sada'siva] Brahmendra, together with a [Punya'slokamanjari] also alleged to be written by a pontiff of the same Nutt.

2. Bits of a [Pracina-'Sankarajaya] (PSJ) attributed to the famous [Anandagiri], alleged to be the seventh pontiff of the Mutt, belonging to the second century A.D.

Even according to the author, the [Brhat-'Sankara-vihaya] and the [Pracina-'Sankarajaya] are not extant now but for the bits quoted in the [Susama]. Of these two, regarding the former, only the name [Citsukha] is true as the author of the famous [Citsukhiyam]. There is no evidence for the story that he was born 'Sankara's oen village, had been his mate as a boy, lived always with him and long after his death, knew every detail of his life, became a pontiff of the Mutt and wrote the [Brihat-'Sankaravijaya]. All this is known only to the [Gurugranthamalika-Susma-Punya'slokamanjari] group, and to nodody else. These have concocted, for the Mutt, a line of succession with precise dates of accession and death, from 'Sankara downwards, using the names of authors of famous advaitic and other works, who have been established by research scholars to have lived in quite other times and at other places.

As for [Anandagiri], the name is well known as the writer of the gloss on 'Sankara's] works. [Pracina-'Sankara-jaya] may also be taken as having existed, if [Madhava] had meant by the word the name of a prior work instead of an earlier ['Sankaravijaya] in general. But there is no evidence for believing that [Anandagiri] wrote the work, or that he was the seventh pontiff of the line, and lived in teh second century, excepting the [Gururatnamalika-Susama-Punya'slokamanjari] set, written solely for the glorification of the Mutt. If what TSNS says about the [Brhat-'Sankaravijaya] and [Pracina-'Sankarajaya] were true, then they would be the strongest eveidence for 'Sankara's life history.

In the BSV all incidents are given in a [Yudhisthira] 'Saka (YS), 'Sankara's birth-date being given as 2631 of the era, and in [Anandagiri's] PSJ in the Kali era, the birth year being given as 2593. This Ys itself, according to TSNS, began in 3140 B.C., (sometimes he says it is 3141 and sometimes 3139, according to his convenience), which date itself he has invented for his own purpose, no one else having mentioned it before as "beginning from the first coronation of Yudhisthira at Indraprastha." But the name itself is well-known and used by teh Jains as beginning from 648 Kali. But the author asserts that the Hindus all use his YS of 3140 B.C. He is not speaking the truth here. While his YS is not used by anybody else, there are two other YS which have been in use among Hindus. The well-known [Jyotirvidabharana], in its list of eras current in Kaliyuga, says that the YS began with the Kaliyuga, in 3102 B.C., and ended with the Vikrama Era beginning in 58 B.C., and ended in 78 A.D., when the 'Saka Era proper, later known as the 'Salivahana 'Saka, began. Again, [Varahamihira] in his [Brhat-samhita], in the context of the [Saptarsi-cara], says that the era of [Yudhisthira's] reign began 2526 years before the 'Saka ERa of 78 A.D.:

[sad-dvika-panca-dvi-yutah (2526)

'Sakakalas tasya rajyasya

The years of the [Saptarsi-cara], omitting the centuries, and called the [Laukika Era], was current in Kashmir and the near by Himalayan regions. It is thsi that is used by Kalhana in his [Rajataranginin]. If is this also taken as YS, sicne it began with Yudhishthira, we have it that all Hindus have been using these two, and none else was known. When this is the truth, TSNS asserts on page 22, SEc.V, that his concocted YS of 3140 B.C. was the one used by all the Hindus. I shall explaing why he was constrained to invent this, when dealing later with the [Sudhanva] copper plate. For the present we shall accept what he says and proceed.

The [Gururatnamalika] and the [Susama] with the quoted portions of BSV and [Anandagiri], and the [Punya 'slokamanjari], all state that 'Sankara was born in Kali 2593 (the author's Y.S. of 2631) on [Vai'sakha-'suddha pancami, Karkataka-lagna], with the five heavenly bodies, the Sun, Mars, Jupiter, Venus and Saturn in exaltation (ucca), as shown in the following birth chart I, as given on page 243 and also 288 by the author.

Chart I (given) Chart II (actual)

Venus Sun Mercury Moon Saturn Sun Mercury Moon

Venus

Jupiter

Lagna

Rasi Rasi

Mars

Saturn Mars Jupiter

But actually on that date the positions were as in Chart II. Note how far away are the four planets Jupiters, Saturn, Mars and Venus from the exaltation positions given by the author. The differences are so great that they cannot be accounted for by the difference in the [Siddhantas] used, or mistakes in them. Could the great [Citsukha], alleged to be a contemporary, close friend and later a successor, have given that chart even by mistake, or [Anandagiri], who is alleged to have followed in the line close after? As the line of succession and dates of accession have been constructed in the :[Punya'sloka-manjari] and [Sisama] exactly following this date as origin, and as the line can neither be contracted nor expanded in time, giving as they do specific dates for each succession, they must all be equally unreliable, as also the existence of these two ['Sankaravijaya-s], since they exist only in the quotations in these works. It is also obvious that works like the spurious [Vyasacaliyam] quoted by the author on p. 247, giving for ['Sankara's] borth the same year 2593 [Kali], Nandana], with the same five-planet exaltation, and for his death 2625 [Kali], [Raktaksa] in p. 230 must be considered as unreliable as the above works.

The probable genesis of these works is the publication of the [Vimar'sa] in 1898, by the said [Swamiji] of the [Dwaraka Mutt] with a copper-plate inscription in support of ['Sankara's] date, alleged to have been issued by a contemporary king, [Sudhanva], to ['Sankara] himself, and said to have been preserved in the Mutt. (we shall examine the genuineness of this [plate later.) In it the date of issue of the plate is given as 2663 YS, the wording showing that ['Sankara] was alive at that time. Our author resolved to use this for his purpose. Even if YS 2663 is the last year of ['Sankara's] life-span of thirty-two years accepted by all, he should have been born not earlier than 2631 Y.S. If this YS is taken as the accepted YS of [Jyotirvidabharana], ['Sankara's] birth must fall just after 2631 Kali, since both eras are synochronous. The author also wanted to give a birth-date for ['Sankara], with five planets exalted, for the sake of plausibility, and his computers fixed a date in 2593 Kali (Jyotirvid. Y.S. 2593) thinking that date answered to the specification, and no other date nearby. (This is the real meaning underlying the author's statement on page 245 to the effect, " We have ascertained from two of the greatest astrologers of South India that this particular combination of the planetary bodies did actually occur on ['Vai'sakha] (Mesa) ['Sukla Pancami] of the year Nandana in 2593 of the [Kaliyuga], corresponding to 509 B.C."). But since this is too low down from YS 2631, which is lowest limit agreeing with the date of the copper-plate mentioned above, the author had to invent a YS beginning 38 years earlier, with the first coronation of [Yudhisthira] in 3140 B.C. according to him, and asserting without an iota of truth that was the only YS known to the Hindus. But fortunately for truth, the concocters have gone wrong, and the author's edifice built upon their date has collapsed, as we have already mentioned.

Another mistake of the concoctors that exposes them is giving Nandana as the year of birth in both the ['Sankara-vijayas], while actually it must be [Dhata] for 509 B.C. according to the astronomical [samhitas] and [Siddhantas]. They have got it by counting backwards one year of the Jovian cycle for each solar year and arriving at Nandana for 2593 Kali, on the mistaken practice now current in the extreme south of India, since the solar calendar was adopted about a thousand and one hundred year back, (the rest of India following the correct procedure), as evidenced by the twelve-year advance there in the Jovian year from what is given in the South. (For details see the Bulletin of the Institute of Traditional Cultures, October 1967, Part I. Sect. ii, p.45). The reason given for the lagging behind in the South, and advance elsewhere in the Introductions to some Tamil [Pancangas] is mere bluff, not supported either by the astronomical works or the Dharma ['sastras].

Further, historians and scholars agree that at such an eargly age as 509 B.C. weekdays like Sunday, names of [rasis] like [Mesa], ideas like exaltation, and giving numbers in [bhutasankhya] were not in use among Indians. But these are freely used inthe said quoted portions. Mistakes in the use of the [bhutasankya] indicate that novices not conversant with it (like the people of the South accustomed to using mostly the Ka-ta-pa-ya-di-sankhya) are handling it . They are unaware that there is practice governing its use. [Adhva[ for 6, [vara] for 7, ['sastra] for 6, and [anga] for 4 are not used in the [bhutasankhya]. While the first three are simply not in use, the last one, [anga], used for 6 by custom, cannot be used for 4, since the resulting ambiguity would play havoc with the astronomical constants. Thus too the concoctors stand exposed.

Against the argument of anachronism given in the last paragraph, the author may say that the historians have all gone wrong in dating the past events, that actually the early periods like the rise and fall of the [Magadha kingdom] etc., should be put backby about 700 years, and if done so what they fix for the second century A.D. will be possible for 509 B.C. (sometimes he makes it 575 B.C. and sometimes even 550 B.C. according to his convenience), and that the early records like the [Aihole] Inscription and astronomers from [Varahamihira] to [Bhaskaracarya], mean only this 'Saka when they mention it. I have exposed the baselessness of his arguments in detail in a paper on the subject (see above pp.255-87).

Another thing has got to be mentioned. In describing the planetary postions in verse 12, on page 273, we find the words, "¹ÉbÚ÷ʴɶÉä ¶ÉiÉEäò, ¸ÉÒ¨Étι]õ®ú¶ÉEòºªÉ ´Éè". This means that the birth was in the 26th century of the YS and the date would have to be taken as 2531 YS (equal to 609 B..) and not 2631 YS. But this year cannot be taken, for the following reasons: Mars cannot be in exaltation, being nearly four [ra'sis] away. This wil also go against the year of death given in the same work as 2663 YS and Kali 2593 for birth, and the cycle year Nandana given for it, as also the [Gurugranthamalika-Susama-Punya'slokamanjari] group, giving definite dates beginning from 2593 Kali alone. This would also contradict the concocted [Jinavijaya] agreement -(we sahll be dealing with this presently)- and also the evidence of the [Sudhanva] plate. Another thing may also be added. In the quoted verses, there is a confusion that [Saugatas] and [Jainas] are synonymous. Do the concoctors mean that [Citsukha] and [Anandagiri] did not know that Jainas are different from [Saugatas], viz., Buddhists?

Another important point: On page 227 there is an alleged quotation from the PSJ. descibing ['Sankara's] death, saying that in Kali 2625, ['Sankara] placed [sure'svara] on the [Pitha] renowned as [Kamakoti] to take care of [Saravajna], and leaving his body before [Kamaksi] attained [moksa]:

Eò±ªÉ¤nù¸´É ¶ÉÉäIÉhÉÉv´ÉxɪÉxÉè& ºÉiEòɨÉEòÉäÊ]õ|ÉlÉä

{ÉÒ`äö xªÉºªÉ ºÉÖ®äú¸´É®Æú ºÉ¨ÉÊ´ÉiÉÖÆ ºÉ´ÉÇYɺÉÆYÉÆ ¨ÉÖÊxɨÉÂ*

EòɨÉÉIªÉɺºÉÊ´ÉvÉä ºÉ VÉÉiÉÖ ÊxÉʴɶÉzÉÖx¨ÉHò±ÉÉäEòº{ÉÞ½þ&

näù½Æþ º´ÉÆ ´ªÉ{ɽþÉªÉ näùZɺÉÖMɨÉÆ vÉÉ¨É |É{Éänäù {É®ú¨ÉÂ**

This is an anachronism because the [Kamaksi] cult itself did not develop till after several centuries A.D. and archaeologists and historians agree that the [Kamaksi] temple itself arose on the ruins of the Jain and Buddhist temples situated in the same place, with many of the old [murtis] transformed into the present ones, (as seen from the old vestiges still visible), and the worship of [Kamaksi] commenced about the ninth or tenth century A.D., the temple itself being later still. When such is the case, how could there be a Math called the [Kamakoti Pitha] there or the mention of [Kamaksi] in the 5th century B.C.? (For details see tha booklet, Devi Kamakshi) in [Kanchi-A histrorical study by K.R. Venkataraman, Second edition, 1973). Thus we see the author's edifice collpsing at every line of examination.

III. A corroborative evidence advanced by the author is an alleged extract from a work called [Jivavijaya], apparently written for the glorigication of Jina, i.e., [Vardhamana Mahavira], the 24th [Tirthankara] and founder of the Jain sect. It may be mentioned even at the outset that such a work is not found in the [Jinaratnako'sa] a bibilography of known Jain works, nor in [Aufrecht's] [Catalogus Catalogorum]. There is a work of this name in the [Madras University's New Catalogus Catalogorum], of Sanskrit works, but it deals only with the tenets of the Jain sect and nothing more. The author must have invented the name and concocted the quotations as there-from, thinking that evidence from a rival faith's work would be more convincing. On p. 150, he himself says that he has no firsthand knowledge of this evidence, but saw it mentioned in an issue of the journal ['Sankrita Chandrika'] of one Appa Sastri of Kolhapur that 'Sankara was born when [Kumarilabhatta] was forty-eight years of age, and so he wrote to Appa Sastri for authority and that Appa Sastri sent him by post the verses quoted by the author, saying they were from the [Jinavijaya]. We have only to believe the author's words for this. Be it so. But why should he give references under the verses (like "vide p.6,Sanskrita Chandrika etc.") as if these verses are taken from the journal itself? Further, an examination of the verses would show that the verses have been written by a person having very little knowledge, especially of Sanskrit. For example, but for the only statement that [Kumarila] was defeated in argument (by Jina himself!!) the gist of the verses glorifies [Kumarila], against the ostensible purpose of the work. What is the relevance of ['Sankara's] year of death as [Raktaksa] being given in this work, where even giving [Kumarila's] full life will be irrelevant? In the context of 'Sankara meeting [Kumarila], a verse (as translated by the author himself) says "when Sankara was fifteen years of age, Siva met [Bhattacharya Kumarila"]. Did the Jains too consider [Sankara] as an incarnation of ['Siva]? Further,the wording is such that ['Sankara] and ['Siva] in the sentence are different persons. In mentioning Sudhanvan killing Jains, the verse says ÊVÉ (?VÉè)xÉÉxÉÉÆ ªÉÉäxÉ ºÉvÉÚxÉÉÆ SÉGäò EònùxɨÉnù¦ÉÖiɨÉÂ*, the word +nÂù¦ÉÖiɨÉ in the verse indicating that the narrator mentions the fact with exultation. Would a Jain speak like this ? Here, too, in giving [Bhutasankhya], words not current then are used. For 'two' the author seems to be afraid of using aksi and always uses the expression [maryaksau] as if there would be some doubt in the numer if [martya] is not used together. The same fear, bespeaking a modern novice of South India is seen in giving the instruction [vamamelanat], again and again, as if otherwise the numbers would be used in the order he is accustomed to, beginning from the highest value, proceeding to lowere and lower. Here, too, is the same mistake, in getting the Jovian cycle-year [Raktaksa] as ['Sankara's] year of death. The mistaking of [Saugata] for Jain is also found, all pointing to the same person or group as concoctors. Again, while the ['Sankaravijayas] in general give as the opponents of [Kumarila], the [Bauddhas] with whom he lived incognito, learnt their ['Sastras] and ultimately vanquished them, why should the Jains take their place here? Here is the hand of the Dwaraka Math, latterly obsessed with teh Jains predominant in the region around it. The Sudhanva plate also exhibits this obsession and is perhaps the genesis of the substitution of the Jains for the Buddhists. In conclusion, since the alleged birth-date by the Jain reckoning also exactly coincides with the discredited 509 B.C., it is evident that the former has been concocted simply for corroboration with the latter.

IV. We shall noe take up for scrutiny the evidence of the Sudhanva copper plate inscription, given on pp. 220-21. For a detailed study, I refer the reader to the speech, exhibiting deep critical acumen, delivered by Prof. V. Venkatachalam, Head of the Department of Sanskrit, Vikram University, Ujjain, at the Seminar held at Sringeri, on the occasion of the Kumbhabhishekam renovating the [Adhisthana] of [Sure'svara] on May 10, 1970, and printed in teh Commemoration Volume, Seminar Section, pp. 86-105, under the title 'The Sudhanva copper-plate : A dispassionate re-appraisal'. I shall give the substance alone here, adding my own comments thereon.

(1) In 1898 A.D. the then ['Sankaracarya] of the Dwaraka Math published a Sanskrit work by name [Vimar'sa] in which there was a copy in modern Devanagari of an alleged copper-plate inscription issued by King Sudhavan to 'Sankara in YS 2663. If genuine, the original should have been in teh pre-Asokan [Brahmi] script, examining which we can ascertaing its genuineness. But it is not available at the Math noe for examination. They say that it had been submitted as an exhibit in a Court of Law for evidence in a dispute, and was not taken back. It is not likely that they would have failed to get back such an important document undless it was a fake, and, so, was nto claimed back, for fear of exposure by historians.

(2) Its language is different from that of c. 500 B.C. when it was alleged to have been issued, and contains modern provicialsms current in the Gujarat region. For example, the word [satta] is used int eh sense of power or suzerainty.

(3) The word ['Saka] is used in the sensa of era. Thsi sense originated from the word ['Sakabda] meaning the to indicate the era named after the ['Sakas', and later extended by a sematic chage to indicate an era in general, like the word[-----------------] in Tamil, first meaning an oul expressed from sesamum, (------), and later used for any oil, so that we have words like (--------)etc. (Even if the author's plea for an older 'Saka referred to already be accepted, a hundred years is too short a period for such a generalisation to happen.)

(4) The date if issue of the plate is given as YS 2663. Whether there was such an era current at that period or not, whether this plate is a fake or not, the [Swamiji] of the Math that published it in his book and used it for his Life of Sankara has taken the YS to mean the well-known one given in the [Jyotirvidabharana], beginning with Kali 3102 B.C. and dates the events of 'sankara's life and the accession of the subsequent pontiffs. This can be seen from the synchronism in the date of composition given by him at the end of the [Vimar'sa]"¸ÉÒ¨ÉSUÆôEò®ú-¦ÉMÉ´Éi{ÉÚV{ÉÚVªÉ{ÉÉnùÉSÉɪÉÉÇhÉÉÆ +´ÉiÉÉ®ú¶ÉEòɤnùÉ& {ÉÉè¹É¶ÉÖC±É{ÉÚÌhɨÉɪÉÉÆ........." This gives 'sankara's year of birth as 414 years before the Vikrama Era of 58 B.C. i.e. as 472 B.C. Even if the plate had been issued in the last year of Sankara's life, the birth year in YS should be 2631, i.e. the YS began in 3102 B.C., i.e., with Kali. Now, out author says that 'Sankara was born in 509 B.C., 37 years earlier than 472 B.C. which we get by the YS of the [Jyotirvidabharana] used in the [Vimar'sa]. It is to make up for this that the author has invented his YS beginning 37 years earlier, in 3140-39 B.C. (about which we have spoken already).

(5) We shall now take up the contents of the plate for scrutiny.

(a) The only original authority for the exi9stence and content of the plate is its publication in the [Vimar'sa]. But in taking it and printing it in his work (p.222) the author has silently changed the words "Ê´É·É°ü{ÉÉ{É®úxÉɨÉ-ºÉÖ®úÉä·É®úÉSÉɪÉÉÈ·É" into "¨Éhb÷xÉʨɸÉÉ{É®úxÉɨÉvÉäªÉ-ºÉÖ®äú·É®úÉSÉɪÉÉÇ·É" for his own purpose. This is trick to appropriate for the Kumbakonam Math the [Sure'sveracarya], whom the Dwaraka Math claims for itself, and who was really in the Srigeri Math as evidenced by his ancient [Adhisthana] (sacred tomb) there, for whcih the renovation Kumbhabhishekam ceremony was performed in 1970.

(b) Sudhanva states in teh plate that Totaka was the least in knowledge among the four disciples, and therefore 'Sankara appointed him to [Jyotirmatha] in the north where would not be many controversialists to meet, and that [Sure'svara] was the greatest in knowledge and was appointed to the Dwaraka Math in the west, where there would be the greatest need to dispute with opponents. If Sudhanvan had this estimate of Totaka, would he mention it to 'Sankara himself, and, even if 'Sankara had estimated the ability of his pupils like this, would he have confided this to sudhavan, and even then would Sudhanvan mention this in a copper plate issued to him ?

Such is the credibility of the plate, and out author Narayana Sastry gives gives this as an important piece of evidence for his date of 'Sankara. I doubt if he himself believed it to be genuine, because his mind and that of the Dwaraka 'Sankaracarya have worked on the same lines, and he has perpetrated all the tricks that the other has done. Both have fictitiously related most of ['Sankara's] peoms with the events of his life. Both have stated that biographers have confused the later pontiffs of the same name with the original 'Sankara and mixed up the events of their lives, and both give authorities from the non-existent and concocted works. While the [Swamiji] creates a [Brhadrajataranginin] to give as authority, our author creates a [Jinavijaya] to support the created [Brihat 'Sankaravijaya] and [Pracina-'Sankaravijaya].

I must also mention here a concoction named [Kaliyugaraja-vrttanta], extensively used by the author Narayana Sastry, in the first editionm, the name being chosen with the intention of creating in the minds of the readers the impression that it has been taken from the [Bhavisya Purana], [Uttarakhanda]. (The authoritativeness of even this Bhavisya, Uttara can be seen from the fact that it contains the story of the Bible from Adam and Eve to Jesus Christ, the story of Mohamed and Islam and the history of Muslim India till Shah Alam, and the fact that the personage occurring in these stories are declared to be the good and bad kings and demons of the [Dvapara] and earlier yugas that have re-incarnated themselves as the godd and bad kings and tyrants and villains in the Kaliyuga). We can guess what this means when all attempts by research scholars to trace it to any source, puranic or otherwise, have failed.

To continue, both Narayana Sastry and the Swamiji of the Dwaraka Math have tried to appropriate for their own Math famous writers of advaitic and other works, though they belong to different places and times. But even as teh 'perfect crime' was out, the author's false claims stand exposed by his own imperfect knowledge and trying to be too clever.

Besides what we have discussed in detail, there are sundry other points mentioned by the author that cannot be replied to in detail in this article, for fear of being made too long for a review. The reader is referred chiefly to the following work to get his doubts cleared: [The Kumbakonam Mutt and the Truth about it], Parts I and II, (published 1965), by R.Krishnaswami Aiyar, M.A., B.L., Advocate, Trirunelveli (latterly known as Gnanananda Bharathi Swamigal) and Sri (now late) K.R.Venkataraman, Redt. Director of Public Instruction and Historical Records Officer, Pudukkottai, respectively.

But what is the correct age of 'Sankara ? I shall discuss this in the next paper.

THE AGE OF 'SANKARA: II

The genuine ['Sankaravijayas] extant are all late works, as we have stated in the previous paper, and some of them have given dates which are only guesses from traditional stories, and are found mostly untrustworthy on examination. The [Madhaviya-'Sankaravijaya] is silent on the point of 'Sankara's date of birth, but gives a planetary combination that can give a series of dates at intervals of about 300 or 400 years. Interal evidence from the ['Sankaravijaya] of [Anantanandagiri] discredited by T.S.Narayana Sastrya, as we have already referred to, points to a date later than even [Ramanuja] and Madhva, and obviously absurd.

But there is plenty of internal evidence and some external evidence also, to show that ['Sankara] must have been born not earlier than the last part of the 7th century A.D. and not later than the first few years of the 9th century, on which point there is an almost consensus of opinion among historians and scholars of Indian philosophy.

['Sankara's] writings show that he is well acquainted with teh [Puranas] in their modern from, which were redacted during the Gupta period. He is said to have studied the [Sutasamhita] several times before he wrote his [Bhasyas and the Sutasamhita] forms part of [Skanda], one of the latest of the Puranas. He is also known to have purified and propagated the six Indian cults (Sanmatasthapanacarya) that were fully shaped only after several centuries A.D. In the [Mahayana] form of Buddhism, the two schools of [Vijnanavadins] and ['Sunyavadins] were perfected during the period from the 4th to the end of the 7th century A.D. and 'Sankara discusses and refutes them in his [Brahmasutra-bhasya], II. 2. 18-36. His direct pupil [Padmapada] refers to this in his [Pancapadika] thus:

"+iÉ& ºÉ B´É ¨ÉɽþɪÉÉÊxÉEò& {ÉIÉ& ºÉ¨ÉÉʽþiÉ&"

In II. 2.28, 'Sankara quotes the first half of a [Karika] of [Dinnaga], pupil of Vasubandu, from his [Alambanapariksa]:

ªÉnùxiÉYÉèªÉ°ü{ÉÆ iÉiÉ ¤Éʽþ´Éænù´É¦ÉɺÉiÉä*

ºÉÉä%lÉæ%Ê´ÉYÉÉ°ü{Éi´ÉÉiÉ xÉ iÉi|ÉiªÉªÉiÉÉ%Ê{É SÉ **

It is known that [Dinnaga] lived in the 5th century A.D.

[Sure'svara], another direct pupil of ['Sankara], mentions the Bauddha Naiyayika Dharmakirti, by name, in his [Brhadaranyaka-varttika]:

Êjɹ´Éä´É i´ÉÊ´ÉxÉɦÉÉ´ÉÉÊnùÊiÉ ªÉnÂù vɨÉÇEòÒÌiÉxÉÉ*

|ÉiªÉYÉÉʪÉ, |ÉÊiÉYÉäªÉÆ ½þÒªÉäiÉɺÉÉè xÉ ºÉƶɪÉ&**

He also quotes him:

+ʦÉzÉÉä%Ê{É Ê½þ ¤ÉÖuùªÉÉi¨ÉÉ Ê´É{ɪÉÉÇʺÉiÉnù¶ÉÇ´Éè&*

OÉÉZÉOÉɽþ±ÉºÉÆÊ´ÉÊkɦÉänù´ÉÉÊxÉ´É ±ÉIªÉiÉä**

And [Anandagiri], the author of the gloss, writes on this;

EòÒÌiÉ´ÉÉCªÉ¨ÉÖnùɽþ®úÊiÉ -+ʦÉzÉä%Ê{É Ê½þ etc.

This same verse occurs in ['Sankara's Upade;sa-sahasri] (XVIII. 142). Again, in his [Brahmasutra-bhasya (II.2. 28), in refuting the [Vijnanavada], ['Sankara] says,

<½þ iÉÖ ªÉlÉɺ´ÉÆ ¤ÉÉZÉÉä%lÉÇ ={ɱɦªÉ¨ÉÉxÉ& ....... +iÉ B¤É 'ºÉ½þÉä{Éɱɨ¦ÉÊxɪÉ-¨ÉÉä%Ê{É"

|ÉiªÉªÉʴɹɪɪÉÉä °ü{ÉɪÉÉä{ÉäªÉ¦É´É½äþiÉÖEò&. xÉɦÉänù½äþiÉÖEò& <iªÉÖ{ÉMÉxiÉ´ªÉ¨ÉÂ*

This ºÉ½þÉä{ɱɨ¦ÉÊxÉªÉ¨É is a reference to Dharmakirti's

ºÉ½þÉä{ɱɨ¦ÉÊxɪɨÉÉnù¦ÉänùÉä °ü{ÉiÉÊuùªÉÉä&*

and

¦ÉÉänù·É §ÉÉxiÉÊ´ÉCYÉÉxÉè& ]Þõ¶ªÉiÉäxnùÉÊ´É´ÉÉ%nÚùªÉä*

respectively, from his [Pramanani'scaya] and [Pramanavarttika]. This is again quoted by [Vacaspati Mi'sra] in the [Bhamati] under the {ÉÖ´ÉÇ{ÉIɦÉɹªÉ of ['Sankara]:

+Ê{ÉSÉ, ºÉ½þÉä{ɱɨ¦ÉÊxɪɨÉÉnù¦ÉänùÉä, ʴɹɪÉÊ´ÉYÉÉxɪÉÉä®úÉ{ÉiÉÊiÉ*""

The Chinese traveller I-T'sing, who tored India during 673-695 A.D., says in his report that [Dharmakirti] had been his own contemporary and pupil of Dharmapal, head of the Nalanda University, and class-mate of [Acarya 'Silabhadra]. Thus Dharmakirti must have written his works during the second half of the 7th century. 'Sankara] also quotes in his [Bhasya] in II. 2. 22-24, several bots from [Bauddha] works, and one of them is from the [Abhidharmako'sa-vyakhya] by Gunamati, who is placed in teh middle of the 7th century.

The celebrated [mimamsaka], [Kumarila Bhatta], who is said to be a contemporary of king [Sranga-san Gampo] (629 698 A.D.) by Lama [Taranatha], refutes in his ['Slokavarttika Dharmakirti's] definition of [Pratyaksa], in the passage "Eò±{ÉxÉÉ%{ÉÉäfø¨É§ÉÉxiÉ " etc. He quotes in his [Tantravarttika from Bhartrhari's Vakyapadiya] the verse,

+ºiªÉlÉÇ& ºÉ´ÉǶɤnùÉxÉÉÆ <ÊiÉ |ÉiªÉɪªÉ ±ÉIÉhɨÉÂ*

+{ÉÚ´ÉÇnäù¤ÉiÉɺ´ÉMÉê& ºÉ¨É¨ÉɽÖþMÉÇ´ÉÉÊnù¹ÉÖ**(II. 121)

and I-T'sing in his report says that Bhartrhari died in 651-52 A.D. Umveka Bhatta, who became famous as [Bhavabhuti] through his drama [Uttararamacarita] is said to be a pupil of [Kumarila]. Kalhana's [Rajatarangini] says that in the year 733 A.D. king [Ya'sovarman] of Kanauj, who had [Bhavabhuti] and [Vakpatiraja] as his court poets, was vanquished by the Kashmir king [Lalitaditya]. Cf. the verse:

EòÊ´É´ÉÉC{ÉÊiÉ®úÉVÉ C¸ÉÒ¦ÉÚiªÉÉÊnùºÉäÊ´ÉiÉ&.

ÊVÉiÉÉä ªÉªÉÉè ªÉ¶ÉÉä´É¨ÉÉÇ iÉnÂùMÉÖhɺiÉÖÊiɤÉÎxnùiÉɨÉÚ**

These point to the second half of the 7th century and early 8th century for [Kumarila]. ['Sankara], after Writing his [Bhasyas] by his sixteenth year, met [Kumarila], well-nigh old man, and at his suggestion vanquished his pupil, Mandanami'sra. While adopting the [Pramanas] established by Kumarila (´ªÉ´É½þÉ®ú¦ÉÉ]ÞõxɪÉ&) and his conclusions (¨ÉÒ¨ÉÉƺÉÉiªÉÉ&) as norms, as far as possible, ['Sankara] refutes other views which are specifically [Kumarila's], inconsistent with the ideas of the [Brahmasutras}.

From all these, we can see that ['Sankara] must have been born not earlier than the last quarter of the seventh century A.D. Therefore, any date earlier than this is untenable, and the date c. 655-689 determined by T.R. Chintamani (Journal of Oriental Research, Madras, III) must be considered slightly too early.

We shall now proceed to discuss what authority we have to fix the date more precisely.

Shri Jayapura Vishvanatha Rajagopala Sharma in his [Sri Jagadguru 'Sankara -mata-vimar'sa, (pp 16-27), fixes 648 A.D. for 'Sankara's birth, (which seems most plausible), giving the following reasons:

(1) The Sringeri Sarada Matha finding the 14th year of Vikrama mentioned in its records, mistool it for the Vikrama Era of 58 B.C. and determined a date corresponding to 44 B.C. But at that period this era itself was not called by the name of Vikrama, but went by the name of [Malavagana Era] or Krta Era. It was also a common practice in those days to recon dates in the ruling king's regnal years. There were several kings bearing the title [Vikramaditya] and we ahve to determine from other factors which [Vikrama] is meant. From our discussion, the Vikrama referred to msut be the [Chalukya Vikramadiya]I of [Vaapi] (son of Pulike's in Ii), who ascended the throne in 670 A.D. or therabouts. This must have been meant by the records, and ['Sankara] must have been born in his 14th regnal year, c. 684 A.D. B.G. Tilak also has accepted this view and determined this date for 'Sankara in his work on the [Gita].

(2) ['Sankara] must have finished writing his [Bhasyas] by about 700 A.D. (from his life stories) and become famous only after that time. (That explains why I-T'sing does not mention him). This agrees with certain other facts. Ramachandra Kak has concluded that "[Adi 'Sankara] visited Kashmir c. 700 A.D. to mark which the [Sankaracharya Hill] was named after him".

(3) The Hindu Religious Endowmenet Commission (Chairman, C.P. Ramaswami Aiyar), after personal examination of evidence tendered, says that "[Adi 'Sankara] reclaimed the temple of [Badrinath] in the beginning of the 8th century A.D. with the help of King [Nanak-pal] of [Gahrwal], and it is significant that the [Gahrwal State] still has control over Badrinath temple."

(4) Youthful ['Sankara's] meeting with [Kumarila] at the latter's old age agrres well with this period.

(5)Cunningham considers that the ['Sastravada] held at teh ['Sarada] temple in Kashmir during the time of King [Lalitaditya], already referred to , menitoned in [kalhana's] [Rajatarangini], might well be by 'Sankara, as it coincides with the first 8th century.

(6) Rajendranath Ghosh in his work [Sankara] and [Ramanuja], fixes 686 A.D. for ['Sankara's] birth, and D.R. Bandarkar fixes 680 A.D., both very near the date fixed.

(7) K.T. Telang fixes 688 A.D., using the verse,

ªÉÖM¨É{ɪÉÉäÊvɺÉÉä (642) Îx¨ÉiɶÉÉEäò

®úÉènÚùEò´ÉiºÉ®ú >ðVÉÇEòMÉɺÉä*

¶ÉEò®ú±ÉÉäEò¨ÉMÉÉÊzÉVÉnäù½Æþ

Rèó¨ÉÊMÉ®úÉè |ÉʴɽþÉªÉ ½þ`öÉäxÉ**

This verse gives 642 'Saka (720 A.D) as the date fo ['Sankara's]abandoing his body among the Himalayas. (The verse is said to be taken from a work called ['Sankara-Paddhati], as quoted by another work called [Dar'sana-Praka'sa] dated 1638 A.D. The year should actually be [Pingala, and Raudra] is three years off, a mistake caused through back-reckonin usually committed by south Indians, as already referred to).

(8) To illustrate a point, ['Sankara] says in his [Bhasya], "ªÉÉä ʽþ ºjÉÖ´PÉÉx¨ÉlÉÖ®úÉÆ MÉSUôÊiÉ ¨ÉlÉÖ®úɪÉÉ& {ÉÉ]õʱÉ{ÉÖjÉÆ SÉ, ºÉ ºjÉÖ´PÉÉiÉ {ÉÉ]õʱÉ{ÉÖjÉÆ MÉSUôiªÉÉä´É *" From the present tense used, we can understand that the three places mentioened were important and existed at ['Sankara's] time. Mathura is as well known now, as then. and Srughna is now the village Sughna near Mathura. But Pataliputra was destroyed by the floods of river sone by the middle of the 8th centruy and ceased to exist in later times, the city of Patna having been built near the old site of [Pataliputra] by [Sher-shah] in 1541 A.D. This shows that ['Sankara] must have passed away by teh end of the first quater of the 8th century, agreeing with his date fo birth 684 A.D.

But various other scholars have given various other dates, at the end of the 8th century orthe beginning of the 9th, taking into consideration various other factors, which they consider more trustworthy,

(1) L.Rice in the Mysore Gazetteer, Vol I, p. 300, suggests A.D. 745-769 for ['Sankara], I do not know on what grounds.

(2) J.F. Fleet, D.R. Bandarkar, Max-Muller, A.A.Macdonell, Buhler, M.Barth and Pathak give 788 A.D. as the date of birth, as also[Nagamiah], (Tracancore State Manual, Vol.II, Ch. VIII, P.99), K.P. P. Menon (History of Kerala, Vol III. P 620), W.Logan, (Malabar Manual, Vol I, Pp.155 ff. and 187 ff.). Their authority for this seems to be two chronograms. Their authority for this seems to be two chronograms. The first is a string of floating verses, source not known, discussed by Pathak, and reading as follows :-

nÖù¹ÉÞSÉÉ®úÊ´ÉxɶÉÉªÉ |ÉnÖù¦ÉÖÇiÉÉæ ¨É½þÒiɱÉÉä*

ºÉ B´É ¶ÉÆEò®úÉSÉÉ®äúªÉ& ºÉÉIÉÉiÉ Eèò´É±ªÉxÉɪÉEò&**

ÊxÉÊvÉxÉÉMÉä¦É½þSɤnäù (3889) ʴɦɴÉä ¶ÉÆEò®úÉänùªÉ&*

+¹É޴ɹÉæ SÉiÉÖ´ÉænùÉxÉ (?´ÉænùÒ) uùÉnù¶ÉÉä ºÉ´ÉǶÉɺjÉÊ´ÉiÉÂ**

¹ÉÉäb÷¶ÉÉä EÞòiÉ´ÉÉxÉ ¦ÉɹªÉ uùÉËjɶÉä ¨ÉÖÊxÉ®ú¦ªÉMÉÉiÉÂ**

(Eò±ªÉ¤näù SÉxuùxÉäjÉÉRÂóEò¤É½þlɤnäù (3921) MÉÖ½þÉ|É´Éä¶É&*)

´Éè¶É JÉÉä Î{hÉŨÉɪÉÉÆ iÉÖ ¶ÉÆEòÉ& ʶɴÉiÉɨÉMÉÉiÉÂ**

Here ['Sankara's] date of birth is given as Kali 3889 (788 A.D.) and the date of his disappearance, Kali 3921 (820 A.D.) The second is a verse from ['Sankara-mandara-saurabha]:

|ÉɺÉÚiÉ ÊiɹªÉ¶É®únùɨÉÉÇiɪÉÉiÉ´ÉiªÉÉÆ

BEòÉnù¶ÉÉ%ÊvÉEò¶ÉiÉÉäxÉSÉiÉÖ ººÉ½þºjÉɨªÉɨÉÂ(3889)*

ºÉÆ´ÉiºÉ®äú ʴɦɴÉxÉÉΨjÉ ¶ÉÖ¦Éä ¨ÉÖ½ÚþiÉæ

®úÉvÉä ʺÉiÉä ʶɴÉMÉÖ®úÉäMÉÞÇʽþhÉÒ nù¶ÉɨªÉɨÉÂ**

(Both give the Jovain cycle year Vibhava, by back reckoning, while it should properly be Prabhava).

(3) Shri (now late) K.R. Venkataraman, (Retired Director of public Instruction, Pudukkottai), and Shri K.V. Venkataraman of Erode, have taken the planetary combination at 'sankara's birth given in the [Madhaviya-'Sankara-vijaya] and the [Guruvam'sa-kavya] (Sun in Aries, Jupiter in Cancer, Sturn in Libra and Mars in Capricorn) as facutual instead or mere poetic fancy, and decide 805 A.D. as the only year answering to the combincation, during the 8th century or nearby, (also long ago fixed by Pichu Aiyer, Cochin State Astrologer). The verses are:-

±ÉMxÉä ¶É֦ɪÉÖiÉä ºÉÖ¹ÉÖ´Éä EÖò¨ÉÉ®æú

¸ÉÒ{ÉÉ´ÉiÉÒ´É ºÉÖJÉxÉÒ ¶É֦ɴÉÒÊIÉiÉä SÉ *

VÉɪÉÉ ºÉiÉÒ Ê¶É´ÉMÉÖ®úÉäÌxÉSÉiÉÖRÂóMɺÉƺlÉä

ºÉÚªÉè EÖòVÉä ®úʴɺÉÖiÉä SÉ MÉÖ®úÉè Eäòxuäù**

(Madhaviya., II. 71)

¶ÉÉv´ÉÒ ºÉÉ ÊEò±É xɴɨÉä%lÉ ¨ÉÉ漃 {ÉÚhÉæ

ºÉ±±ÉMxÉÉä%ÊvÉ{ɺÉÆκlÉiÉä EÖò±ÉÒ®äú*

EäòxuùÉä\SÉèMÉÖÇ°ü¶ÉÊxɦÉÚʨɺÉÚxÉÖºÉÚªÉÉê&

|ÉɺÉÉä¹]õÉʨɽþºÉÆ ºÉÖJÉäxÉ ºÉÚxÉÖ¨ÉÂ** (Guruvam'sa,,II. 59)

(The second verse here adds the Moon also in Cancer, which excludes the asterism [Ardra for 'sankara). Shri K'V.Venkataraman adds for confirmation, a tradition in Kerala that a contemporary Kerala king, who was in danger of losing his kingdom, was established firmly on the throne, and peace and plenty came to the land by the blessings of 'Sankara, and , in gratitude, the king strated the Kollam Era in 825 A.D. from the day he met ['Sankara]. (But there is another tradition current among Kerala astronomers that there is an astronomical event connected with the starting of the Kollam Era).

K.R. Venkataraman also mentions this , but places more credence on a tradition recorded in the [Guruvam'sakavya], that a Kerala king named [Raja'sekhara] (which is perhaps a surname of [Bhaskara-Ravivarman] of A.D. 798-834, of the [Kula'sekhara] line, who is reputed to have composed three dramas), read out his dramas to ['Sankara] for approval. Another evidence, most clinching in his view, is an inscription in [Cambodia], belonging to the regin of Jayavarman II (A.D. 878-887) which mentions the royal preceptor ['Sivasoma], "who had learnt the ['Sastras] from [Bhagavan 'Sankara]",

ªÉäxÉÉ%vÉÒiÉÉÊxÉ ¶ÉɺjÉÉÊhÉ ¦ÉMÉ´ÉUÂôUÆôEò®úɽþªÉÉiÉä*

Êxɶ¶ÉÉä¹ÉºÉÚÊ®ú¨ÉÚvÉÉÇʱɨÉɱÉÉføÉ%RÚóÊQÉ{ÉRÂóEòVÉÉiÉÂ**

(G.Coedes: Inscriptions de Cambodge, o.40)

But we think that this does not go aganist an earlier date for ['Sankara], since the term may refer to any later pontiff of the line, who are called ['Sankaracaryas], as is the practive now. Anyhow, this sets the upper limit ti 'Sankara's age. The date 898 Samvat (841 A.D.) is given by [Vacaspati Misra] in his [Nyayasucinibandha] for its composition,

xªÉɪɺÉÚSÉÒÊxɤÉxvÉÉä %ªÉ¨ÉEòÉÊ®ú Ê´ÉnÖù¹ÉÉÆ ¨ÉÖnäù*

¸ÉÒ´ÉÉSɺ{ÉÊiÉʨɸÉÉähÉ ´Éº´ÉRÂóEò´ÉºÉÖ (898) ´ÉiºÉ®úÉä**

He has written the commentary [Bhamati] on ['Sankara's] [Brahmasutra-bhasya]. Thus, this also sets the limit.

A S T R O N O M Y

I. A Brief Historical Sketch*

The study of astronomy was purusued by the Chinese from very ancient times. Even from the third millennium B.C. we hear of professional astronomers in China, who enjoyed the patronage of the Chinese emperors and who broguht the art of astronomical observation to a state of perfection the art of astronomical observation to a state of perfection surpassed only by the achievements of modern times. They discovered the sidereal periods of the sun, moon and the planets, and could predict their positions with tolerable accuracy. They discovered the metonic cycle and could even predict the occurrence of eclipses.

The ancient Egyptians were highly interested in astronomy and this interest was partly due to their being 'astrolators.' Palto says that they learnet their astronomy from the inhabitants of the submerged continent of Atlantis, believed by some to have been situated in and beyond the [Sahara] Desert position.

The Babylonians, who occupied the valley of the Tigris-Euphrates, observed independently all that the Chinese had done, and not stopping with that, they reduced their knowledge to a system. The practive of dividing the zodiac into twelve signs comes from them and it is supposed that they were the first people to give names for the days of the week. It was they who taught astronomy to the Greeks in the seventh and sixth centuries B.C.

To the Greeks should be given the credit for sublimation experience into theory and from their time begins the scientific study of the subject, however crude the basis of that science might be. Aristarchus of Samos who lived in the third century B.C. even attempted the determination of the distance of the sun form the earth by a method theoretically quite sound but having no practical value inasmuch as the errors of observation would be too many to yield any result in the neighbourhood of accuracy. In spite of the fact that he found the distance of the sun from the earth to be twenty times the distance of the moon-which is only a twentieth of the real distance -his very attempt is noreworthy in the history of astronomy. (The Hindu Siddhanta belief was only 13 times).

Next came Hipparchus who did more than anybody else in those days for astronomy. He computed the sidereal, tropical and synodic periods of the moon, the sun and the five planets, Mercury, Venus, Mars, Jupiter and Saturn. The value he found for the obliquity of the ecliptic was only five minutes in error or what it was then. He determined the moon's horizontal parallax. He located the sun's apogee, the position at which it is at its greatest distance from the earth. He discovered the phenomenon of precession by comparing the results of his observation with those of his predecessors, for which he gave the value of 36 seconds per annum, which is not very far from the true value when we consider the shortness of the period of authentic observation preceding him. He found with tolerable accuracy the values for the equation of the centre of the various heavenly bodies. He imagined with his contemporaries that uniform circular motion was the only form of perfect motion. He had to reconcile this idea with the apparent motion of the heavely bodies, and discovered how their apparent motion can be represented by a series of eccentric circles; and this paved the way for the introduction of epicycles by his successors.

Ptolemy who came about 250 years after Hipparchus wrote the [Almagest] in which he placed the geocentric hypothesis on a firm footing; and the popularity of this work was responsible more than anything else for the lack of progress in astronomical science for several centuries to come. Those who came after him were wither his expounders or his commentators.

Meanwhile the Hindus were not behind hand in the pursuit of this branch of knowledge. If in other countries it was curiosity and the desire for knowledge for it sown sake, coupled with its use for finding time and guiding ships at sea, that led to a study of this subject, in India there was the additional impetus of the need of this knowledge for fixing the dates of the Vedic ceremonies and sacrifices; and astronomy became one of the 'Vedangas.' As there was cultural contact between the Greeks and the Hindus, there should certainly have been some give and take. But it is wrong to imagine, as some do, that the Hindus borrowed their science of astronomy from the Greeks. The [Vedangajyotisa], teh earliest astronomical work we know of, bears no traces of such borrowings. The great scholar and historian, P.T. Srinivasa Iyengar, has pointed out in his essay, 'Our Hellenic debt,' that it was the science of astrology that we borrowed from thee Greeks, as the list of the names of the [Purvacaryas] or pioneers and the numerous techincal terms found in out astrological works show.

Among the Hindu astronomers, [Aryabhata] who lived at the end of the fifth century A.D. deserves special mention as one who gave the diurnal rotation of the earth as teh explanation of the apparent diurnal rotation of the stellar sphere. He should have been a bold man to have said so, for almost everybody would have characterised it as being opposed to 'reason.' [Varahamihira] who came after him refuted it in his works, by an ingenious but fallacious reasoning based on an ignorance of the laws of kinematics; and so did others.

The Hindu astronomers had a knowledge of the theory of the lunar parallax, and by an application of that theory they found that the distance of the moon from us is about 65 times the radius ofthe earth. If this is the correct value in excess by 9 ./. it is not because their process of reasoning was faulty, but because their value for the horizontal parallax of the moon was erroneous, combined as it was with the effects of atmospheric refraction. Taking this result together with the geocentric hypothesis and their theory that the speed of the heavenly bodies in their orbits is constant, they tried to determine the distance of these bodies from the earth. Of course, the results were wrong, for the velocity is inversely as the root of the distance. If they had not been wedded to the geocentric hypothesis, they might easily have arrived at the correct values without having recouse to any physical laws; for the sees of these values are embedded in the ['sighraparidhi] of the inferior planets and [madhyaparidhi] of the superior planets given by them, and the problem is one of geometry.

In general the method employed by the Hindus was superior to that of the Greeks, for they had a better grasp than the Greeks, fo the mathematical 'instruments' like algebra, geometry and trigonometry.

The 'Ptolemaic' system reigned supreme in the West for about fourteen centuries, and then came Copernicus, an original thinker, who broke away from the trammels of the old system. He taught that the sun is the centre of the solar system and the earth with the five planets Mercury, Venus, Mars, Jupiter and Saturn, moves round the sun while the moon moves round the earth. He did not advance any proof for his new hypothesis other than showing that the geometrical explanation of the movements of the bodies would be simpler on this assumption, a smaller number of epicycles being sufficient to represent the movements. This simplicity itself may be taken as a proof,for all natural laws are characteristically simple, and any explanation based on a large number of ad hoc hypotheses is viewed by scientists with suspicion. But the substantial proof based on the phenomenon of 'aberration' was a long time coming, and Copernicus could only add to what he said an appeal to the emotion of the people by saying how beautiful it would be to have the glorious sun as the centre of the solar system.

His theory met with a good deal of opposition, especially from the church. To save himself from the wrath of the church he declared that his theory was given only as a geometic devise for ease in computation. But slowly laymen as well as scientists began tacitly to recognise its truth. The telescope was invented, better astronomical instruments were made, and better observations taken by Galileo etc. It was on the results of the careful observations of the astronomer, [Tycho Brahe], that Keper, his successor, based his famous laws of planetary motion.

By an instinct which is the privilege of the genius Kepler perceived that if the theory of uniform circular motion was abondoned, matters would become extremely simple. By examining the values given by [Tycho Brahe] for the position of Mars at different times, and trying hypothesis after hypothesis, Kepler discovered and formulated his laws of planetary motion:

(1) The orbit of a planet round the sun is an eclipse with the sun at one of the foci.

(2) Tue radius verctor, i.e., the line joining the centres of the sun and the planet, sweeps equal areas in equal intervals of time. (It is interesting to note that he arrived at this truth by a reasoning based on two wrong assumptions the effects of which fortunately cancelled each other).

(3) The square of the periodic time of a planet is proporationate to the cube of its distance from the sun.

Further advance was not possible in the then state of the science of dynamics, the laws of motion not having been clearly formulated as yet. Bold guesses were made that the gravity which manifests itself on the surface of the earth is responsible for retaining the moon in its orbit; and from the earth -moon system it was extended to the solar system. some even guessed the inverse square law of the variation of gravitational force; but it was left to Newton, one of the greatest scientists the world has produced, for formualting the law of gravitation and giving a formal proof if it from the motion of the heavenly bodies.

Newton stated clearly the three laws of motion known after him, and, basing his reasoning on these laws, he investigated the import of Kepler's Laws. He showed that if Kepler's second and third laws must be true, as indeed they were, the gravitational force retaining the planets in their orbits must be directed towards the sun, and the gravitational force of the sun alone was responsible for this, no tangential force being necessary or admissible. He showed that if Kepler's first law must be true, then the gravitational force should vary inversely as the square of the distance; for it is only the inverse square law that can make bodies describe conic sections with the attracting body at one of the foci, (and an ellipse is a conic section.) He formualted his Law of Gravitation : F=K mm/d, where F is the gravitational force, m and m, the masses of the attracting and the attracted bodies, d, the distance between them and K, a constant depending ont he units chosen.

When he tried to verify this law with reference to the motion of the moon, his results were at first not satisfactory, for he did not know the correct radius or the earth, the received value of it being too small by 12 per cent. Later on he heard of a fresh determination of the size of the earth made by Picard in Paris. He made the necessary corrections and found close agreement. Then and then only did he publish the results of his investigations.

Prejudice dies hard even in the scientific world, and his opponents tried to disprove his Law of Gravitation by referring to the inequalities in the motion of the planets and the satellites, to the motion of the apsides and the nodes and to the phenomenon of precession of the equinoxes. Newton showed that the very existence of these is a proof of his law, and if they did not exist his law was not true. Slowely opposition from a study of the inequalities in the motion of Unanus, was a triumph for this Law. Astronomers who came after Newton showed this law to be universal and not merely restricted to teh solar system. The method of investigation was perfected by the use of calculus and otehr branches of higher mathematics, and every observed inequality, except one or two, was tallied with the results of theoretical investigation with complete success.

BUt there were two disconcerting factors about this law which annoyed the scientists not a little. However much they tried they could not explain the residue of about 42 seconds a century in the angular motion of the apsides of Mercury. Secondly, while all other known phenomena of nature admitted of a mechanical explanation, the Law of Gravitation did not admit of such an explanation without necessitation the inadmissible assumption of 'action at a distance;' and it remained an isolated fact.

It was left to A.Einstein to prove, on teh strength of his 'Principle of Equivalence' and the postulates of the Theory of Relativity, that gravitation itself may be dispensed with and the complicated motion of the heavenly bodies may be due to their inertia and the curved nature of the 'space-time continuum' in the neighbourhood of matter.

From the above we should not conclude that Newton was superseded by Einstein and that his Law of Gravitation is of no use. The results obtained by an application of the law are still a very close approximation to the truth, Relativity giving us the additional security of a knowledge of the limits of error; and , in the words of J.Rice, Einstein stands on the shoulders of the greatest man of science ever born.

THE STARS *

On a clear moonless night every one of us has looked up at the stars and wondered what they are. Excepting a few that are called planets, (literally 'wanderers'), we see they occupy the same relative positions, so much so that we even fancy they form unchanging pictures of animals like the ram, the bull, the crab, the lion, the dog, the scorpion etc., and we name the groips according to the likeness they recall. We also know the stars by their peculiar twinkle -though this is not a sure way of distinguishing a star from a planet because some planets twinle when they are near the horizon. The stars are small dots, we think, and the use of the most powerful telescope still shows them as mere points.

But really they are huge bodies about the great size of which we can takl but cannot form any idea. Most of them are as big as out sun-indeed they are suns-and many, bigger. But how big is the sun? It will take an express train, travelling sixty miles an hour day and night, seventeen days to go once round the earth along the equato. Such is the sixe of the earth. But a million earths can be packed into the sun, fancy that! And many stars are vastly bigger than even the sun. Some are so great in sixe that the whole orbit of the earth ound the sun can be placed into them, an orbit whose radius is 93 million miles.

Like the sun too the stars emit their own light and heat and are called 'self luminous.' If we do not get any appreciable quantity of light or heat from them, their distance is responsible for that; for some of them are so intensely hot that if they are placed at teh distance the sun is from us, the whole of the earth will be evaporated away in a trice. Thank God, they are so far away.

The nearest star is at a distance of 25,500,000,000,000, miles, again something meaningless. Let us try to understand this. Vast as the size of the earth is, the speed of light is so great that it can travel more than seven times round the earth in a second. Travelling at this rate it takes about eight minutes for the light of the sun to reach us. That means if the sun goes out at this moment-which God forbid-we shall continue to see the sun for eight minutes more and then alone see it disappear. Now, it takes light about four years (=4X365X3X60 eight-minutes) to reach us from the nearest star; which means it is at a distance fo about 270 thousand times the distance of the sun. There are stars whose light reaches us after travelling for several hundered years through space. Though we think we see them as they are now, we only see them as they were when the light started from them, i.e. as they were several hundred years ago. If at this momemt we transport ourselves to a star at a distance fo 2500 light years from us, (i.e. a distance which light travels in 2500 years) and if we provide ourselves with some means of observing what goes on, say in India, from there, we can see India at it was at the time of the Buddha; and if we continue the observation for 2500 years we can 'see' the complete History of India from the time of the Buddha.

Situated at these inconceivably great distnaces, it is no woner they look like mere pin-heads, with all their heat and light reduced to a glow-worm's twinkle. It is their distance also that makes us think they are fixed. They are no more fixed then we are, or the earth or the moon is . Many of them are moving about at a rate of several miles per [second], but yet we see them fixed, theyt are so far away. If they are observed at intervals of several years with the help of accurate measuring instruments we can see that they chage their positions relative to one another. Even if they have not done so, we cannot be sure they are motionless, for they may be moving along the line of sight. There are ways of measuring even this kiind of motion by the application of whatis known as the 'Doppler's principle,' which is the same in its nature as the sudden fall int he pitch of the sound of the horn as teh railway engine passes us. Scientists have measured the rate fo the 'proper' motion of the stars as this is called, and given us the results for several stars. Even the sun, which we know is fixed with relation to the planets has been discovered to be moving at the rate of thirteen miles per second towards the star Vega (---) with all its family of planets and moons.

It is known that the stars are not fixed though they appear to be so, that they have their own or 'proper' motion and that the rate of the proper motion has been calculated for several stars. This calculation is possible when their distances are known. Astronomers have found out the distances of several stars and given them in so many millions of miles or in so many radii of the earth's orbit round the sun (this is the most natural unit, as we shall see presently) or in so many light-years.

The method is easy. Thus, to find the height of a tower, or its distance from you are, without going to the tower or climbing it, the following is the familiar method:-"A man finds the angle of elevation of the top of a tower on level ground, he finds the angle of elevation 60o. To find the height of the tower and its distance from the first place of observation." You draw a horizontal line, take a point on it and draw a straight line through it making an angle of 30o. You know the top of the tower is somewhere on that line. Then you say to yourself. "Let me represent the hundred yards by 4 inches on the plan" and write "scale: 1"=25 yards." You mark a point at a distance of 4" from the original point on the side where the tower is. Through that point you draw a straight line making an angle of 60o on the side of the tower. You know the top of the tower is on this line too. As it is on both the 30o line and the 60o line, it must be at the point of their intersection. From this point you drop a perpendicular on to the horizontal line. That is the tower. You measure the length of the perpendicular and find it to be 3.464 inches which is the equivalent on the plan of 3.464X25= 86.6 yards. The length from the first point to the foot of the perpendicular represents the distance required, which you find to be 150 yards.

You see it is possible now. Not only so, this is the only methods available for measuring the heights of inaccessible peaks like the Everest; for the barometer or the hypsometer can be used only when we can get at teh places of which we want to find the heights. As the method is capable of very great accuracy, it is used even in accessible places. For instance, countries have been surveyed for purposes of map-making not by going about dragging a chain all along, but by measuring a certain distance called 'the base line' with great accuracy and then measuring angles alone. This is known as the Great Trigonometrical Survey.

By observing the sun from two places on the earth its distance can be found out. It is 93 million miles which is the radius of the orbit of the earth round the sun. To find the distance of a star, the star, S, is observed and its position accurately found out with the help of a telescope when the earth is, say, at E1 (see figure 1). After about six months the earth is at E2, the other end of the diameter. From there the same star is observed again and its position found out. E1 E2 is knwon to be 2X39 million miles (The radius=93 million miles) the distance of the star can be calculated. We see here that the diameter of the earth's orbit which is the same thing as saying the radius (==1/2d) is the natural unit in which the distance of a star is measured; and even if the length of the radius is not known, there is nothing to prevent us from finding out the distance of the star in terms of the radius.

Once the distance is know, it is early to find the velocity of the star. Look at fig.2. A star is observed, say, when it is at A. After some years the same star is observed to be at B. The point from which these observations are made is C. (C may be moving, but we can allow for that and assume it to be stationary.) Agnle C is measured. C A, the distance, is already known. If A B, the distance travelled by the star during the interval, is known, the velocity can be found by dividing it by the interval. The motion along A B can be 'resolved' into a motion along A D, perpendicular to the line of sight and a motion along D B along the line of sight. Angle C is small and is measured in seconds of arc, say, (3,600 seconds ==1 degree). The AD= CA X < C

-----------

206, 265.

The next step is to find D B, the motion along the line of soght. It can be found out, as stated, by using the 'Doppler's priciple.' Let us try to understand how this can be done. We are stading near a railway-line and a railway -engine is whistling past us. The sound of its whistle has a constant pitch. But as the engine approaches us, more waves of sound reach our ear than when the engine is stationary. So the frequency of the sound is increased and the pitch is apparently heightened. When the engine is moving away from us the number of vibrations per second decreases and there is an apparent lowering of the pitch. Now, if we can measure by exactly what amount the pitch has chaged, we can find the speed of the engine. If this method is not used for finding the speed, it is because other easier and more exact methods are available. But in the case of the star this is very useful. The increase or the decrease from the normal of the frequency of a particular kind of radiation can be measured with the help of an instrument called 'Spectrometer,' and from the amount, the velocity along the line of sight calculated. From A D and D B, A B can be found, for D is a right angle (practically) and A B ==A D + D B.

From the distance also we can find the mass of the star, if it happens to be a real double star. These double stars revolve about their common centre of mass. The distance of the bodies from the centre of mass can be calculated from their known distance from us; and the period of revolution observed. Then their combined mass can be found out by using the formula,

K (M1 + M2)== d3

-----

T2

where M1 and M2 are the masses, d is the distance from the centre of mass, T is the period and K is a constant depending on the units chosen. If the radius of the earth is chosen. If the radius of the earth is chosen as the unit of distance, a year as the unit of time and the mass of the earth as the unit of mass,

K= 1

---- and the formula can be written in the form,

330,000

M1 +M2=330,000d

--------

T2 . Then the individual masses can be calculated from their distances from the centre of mass.

THE STRUCTURE OF THE ATOM*

There are two things in the Universe, matter and radiation. Gamma rays, x-rays, ultraviolet rays, light rays, infra-red rays, heat rays and wireless rays are all radiation. Alpha rays, beta rays, cathode rays and the 93 elements with their aggregates are all matter.

What is the distinction between matter and radiation? It was once believed by scientists that matter consisted of particles, and radiation or waves. It is now known that matter too behaves like waves and produces effects like diffraction and interference, characteristic of waves. Radiation, which has longbeen known to behave like waves, also behaves like discrete particles (photons). The possession or the non-possession of mass cannot be a distinguishing feature, for both matter and radiation have mass. But there is one thing by which we can easily distinguish matter from radiation. While matter can never acquire the velocity of light, radiation has always the velocity of light.

All substances are composed of matter. The ultimate constituents of substances as such, are called molecules. They are the smallest particales of a substance, having the properties of the substance. For example the water molecule is the smallest particle of water having the properties of water. It is possible to break the molecule still further; but we no longer get particles having the properties of water, we get two particles of hydrogen and one of oxygen, quite different substances from water.

In general, molecules are built up of such smaller particles. They are called 'atoms,' whichh means, 'indivisible', for it was believed the atoms could not be broken. 92 different kinds of atoms are known to us at present, and at least one awaits discovery. Each of these represents an element and 92 elements are known to us at present.

Are the atoms indivisible, as they were believed to be? Are they the ultimate building bricks of the universe ? Some fifty years ago scientists put this question to themselves. They felt that the atoms could not be the ultimate particles. 93 is too large a number, in comparison with nature's simplicity. Further, a host of fundamental phenomena like the emission and absorption of light, the electrical and magnetic properties of matter, chemical affinity, valency etc. cannot be explained if the atoms are the ultimate particles; for the things to be explained are extremely complex, while the atoms can differ from one another only in their masses, sizes and elastic properties.

Investigation into the electrical properties of rarefied gases led to the discovery that very small particles in the atom are responsible for the conduction of electriity through matter. These were found to be particles of negative electricity and were named electrons. We now see that the atoms are made up of still smaller particles.

The electrons all carry the same amount of electric charge, 4.8X10-1o electrostatic units. On account of their charge, they can be attracted and repulsed by electric and magnetic forces. By subjecting a stream of electrons to these forces we can find the mass, the velocity and (indirectly) the radius of the electrons. All electrons have the same mas, [9X10-28] grams, and the same size, a sphere of radius [10-13cm].

The electron is only one sort of the building -bricks of the universe. Why one sort? Why should not the whole atom be built up of electrons? Taht cannot be, neutral, i.e. it is neither attracted nor repulsed by electric charge of electricity, there must be in the atom some other particle or particles having an equal positive charge to neutralize the negative charge. Secondly, as the mass of the electron is only a very small fraction of the mass of the whole atom, and as no atom contains more than 93 electrons, there must be something in the atom in which almost all the mass is concentrated.

This something is known as the nucleus. It forms the core of the atom. It too has been found to be a compoite structure. The nucleus of the radium atom explodes spontaneously and shoots out helium ions. It can also be broken up by artificial means. From these it has been found that the nucleus is built up of particles called protons and neutrons. They are similar to the electrons in shape and size. But they are far heavier. Each proton and neutron has about 1840 times the mass of the electron. The proton carries a positive electric charge equal in quantity to the charge of the electron. It can be called a particle of the positive electricity. The neutron is electrically neutral, hence the name. It is now believed that these three kinds of particles, the electron, the proton and neutron are the bricks out of which all the matter in the universe is built up.

We shall now see how the different kinds of atoms are built up. Let us take them one by one in the order in which they occur in the Periodic Table of the Fig.1. Hydrogen atom. Fig.2. Helium atom.

elements. The simplest is the hydrogen atom, fig. 1. It has a nucleus of one proton. To neutralize its single positive charge it has one electron moving round it. The next element is helium, Fig. 2. It has two electrons revolving round its nucleus. So it must have two protons in the nucleus to neutralize the charges. But the two protons can give the atom only an atomic weight 2. The atomic weight of helium is 4. So there must be 2 neutrons in the nucleus. Thus the nucleus has the positive charge 2, and the weight 4. The next element has 3 electrons revolving round the nucleus. So its nucleus contains 3 protons which can give it an atomic weight 3. The rest of the atomic weight is supplied by the neutrons. We find each successive element in the Periodic Table has one electron and one proton more than the previous one. If we take the N th element, and if its atomic weight is Z, we see it has N electrons revolving round a nucleus containing N protons and (Z-N) neutrons, The number of protons (or electrons) each atom has is the atomic number. The atomic number of the elements increases one byone in the order of their arrangement in the Periodic Table. As the arrangement is according to the chemical and other properties of the elements, we see at once that the atomci number is the fundamental thing and the outer electrons may in some way be responsible for the chemical behaviour of the elements.

There was a time when scientists thought that the atomic weights were fundamental, on account of the almost perfect agreement between the arrangement according to the atomic weights and that the Periodic Table. So they were disconcerted when they found disagreement in the case of [Argon], [Cobalt], [Iodine] and [Thorium]. They have now found out that the atomic weights do not matter and it is the atomic numbers that count.

Waht is it that binds the electrons to the nucleus and makes them revolve it in orbits? The electrons possessing negative charges are attracted by the protons in the nucleus, having positive charges. It is this attraction that keeps them in their orbits.

There is a close and striking similarity between the atom and the solar system. The atom is a miniature model of the solar system. In the place of the sun is nucleus. In the place of the planets are the electrons which revolve round the nucleus in orbits as the planets revolve round the sun. The only difference is the nature of the forces keeping the planets on the one hand and the electrons on the other, in their orbits. The one is gravitational and the other, electrical.

In another respect also there is similarity. The solar system is practically empty of matter when we consider its size, the planets forming a few particles of dust, and the sun, a small pellet of stone, on a vast field. It is so with the atom. Let us picture the atom which has a radius of 10-8 cm, as a football. The electrons, protons and neutrons having a raduus of the order of 10-13 cm would each be a thousandth of a millimetre across; and there are not more than about 330 of these invisible specks even in the densest atom. All but a millionth of a millionth of the whole space occupied by the atom is empty. If all the matter composing the body of thousand men is collected together, it would not be sufficient to fill a mustard seed. How small we are ! Yet how great, being able to comprehend the whole universe!

NEWTON AND THE LAW OF GRAVITATION*

Introduction

Newton was the formulator of the Law of Gravitation, on which is based the edifice of modern astronomy. Taht we may have a better apreciation of his genius we shall examine the work of his immediate predecessors, [Tycho Brahe] and [Kepler]; and that we might grasp the true import of his discovery, we sahll examine the uses this law is put to and also how far [Einstein's] Theory of Relativity affects the importance of the law.

[Tycho Brahe] was a Dhanish mathematician and astronomer of the time of Queen Elizabeth. He came after Copernicus who was first man in Europe after the Greeks to assert that the sun is the centre of the Solar System and that the planets revolve round it, with the satellites moving round them. But [Tycho] did not accept what all he said. He propounded, instead, that the earth was fixed and the sun, with the other planets going round it, revolved round the earth. This is known as the Tychonian System.

His services to astronomy consist chiefly in his accurate observations of the positions of the stars and the planets, madee with the telescope just coming into use. But for his observations, Kepler, who was his assistant for sometime and who succeeded him in the field of astronomy, could not have formulated his laws governing the movements of planets. Kepler used his observations of Mars for his purpose, for the orbit of Mars has the greatest ellipticity next to that of Mercury, which itself could not be used owing to observational difficulties.

As we have seen, Kepler was a junior contemporary of Tycho Brahe and was his assistant for sometime. After [Tycho's] death he succeeded him as Royal Astronomer to the German Emperor and completed his work, the construction of a set fo astronomical tables. Before his time the calculation of the position of a planet was a very difficult affair. It was thought that perfect motion meant moving in circles and astronomers believed, on metaphysical grounds, that heavenly bodies must have this kind of motion alone. As their theory did not agree with actual observation they said that the bodies must have this kind of motion alone. As their theory did not agree with actual observation they said that the bodies moved in circles the centres of which moved on the circumference of their circles, the centres of which moved on the circumference of still other circles and so on. The explanation is ingenious, and the beauty of it is that it can yield the desired result. In its nature it is only a geometrical representation of the development of a function in series. But in the case of planets with orbits of great eccentricity a very large number of these circles will be required, which will render the calculations laborious.

Kepler felt-herein lies his genius-taht nature's ways are not so involved and elaborate as this. Nature' laws are characteristically simple, and he wanted to simplify the theory of planetary motion. He tried hypothesis after hypothesis and at last arrived at the truth that the orbits of planets are ellipses with the sun at one of the foci. Of course, the ellipse includes the circle as a particular case, the case where the eccentricity zero. Next he discovered that the radius vector, i.e., the straight line joining the sun and the planet sweeps equal areass in equal time. Long after he discovered this law he discovered his third law that the square of the periodic time of the planet's revolution round the sun is proportionate to the cube of its distance from the sun.

Newton's Work

Kepler arrived at these laws empirically, and incidentally they proved true. He did not know the why of tehm It was left for Newton to formulate the Law of Gravitation, and demonstrate mathematically that Kepler's Laws are a result of the Law of Gravitation, and as such, a proof of the lae. Not that people did not know anything of gravitation before Newton. For the matter of that, from very ancient times people have explained the falling of bodies towards the earth as being due to the pull of the earth on the bodies. [Bhaskaracharya], the famous Indian astronomer, in discussing how the earth stands poised in space, says that the earth pulls bodies on its surface, that 'towards the earth' is 'down' and therefore the question why the earth does not fall down is meaningless. Kepler had a vague idea that gravitation is responsible for maintaining the planets in their orbits and making them follow his three laws. But he did not know what exactly was the amount the effects which he supposed must be attributed to it. Some had even surmised, before Newton's days, that gravitation varies inversely as the sqare of the distance. A friend of Newton's, Halley, actually proposed to him the problem, how a body moving round another under the influence of the inverse square law would behave. Newton had already solved the problem and was ready with teh answer: the moving body would describe an ellipse, a parabola or a hyperbola relative to the central body which itself would be at a focus.

Newton's contribution to science in this connection is the exact statement of the Law of Gravitation, first with reference to the pull of the earth and afterwards generalised to include all matter. Some of his predecessors had guessed that the moon is kept in its orbit by gravity which is the same in its nature as the gravity acting on the surface of the earth. But they did not care to verify if it was so; they were not sure of their ground. Newton saw that without being sure of the laws of motion these astronomical problems could not be solved with any degree of certaintly. Some spade work had been done in this direction by Galileo. Newton continued his work and clearly stated his three laws of motion, familiar to every student of science. Then, on the supposition that the acceleration due to gravity varies inversely as the square of the distance, he proceeded to verify if the moon is kept in its orbit by the pull of the earth. He knew the amount of the pull on the surface of the earth, 32 feet per second per second. He knew the distance of the moon from the earth in terms of the radius of the earth (the radius of the earth is the natural unit in measuring the distance of the moon). At first he did not know the correct value of the radius of the earth, the accepted value being in error by 12./.. This gave an error proportionate to this in the result. Any other man but Newton would have been baffled at this and discarded the thind as hopeless. But Newton felt that there might be an error in the value of the radius of the earth. Happily for him, a fresh determination of the radius was being made in France, and when the result reached him he tried a fresh verification with complete success.

The next step was to extend the law to the Solar System as a whole. Here the sun is the attracting body and the planets are the bodies whose motions are to be investigated. By a process of pure mathematical reasoning he arrived at teh conclusions arrived at empirically by Kepler. Thus he supplied the why of Kepler's Laws. He showed that if there is no other force acting on a planet except the central force due to the attraction of the sun, Kepler's second law must hold good, whatever be the nature of the central force. The second law is, as stated earlier, that the radius vector sweeps equal areas in equal time. Provided the body is accelarated only towards the sun, never mind whether the acceleration is inversely proportionate or directly proportionate or anything else, it must sweep equal areas in equal time.

The first law of Kepler, as we have stated, is:(a) that the orbit is an ellipse, and (b) that the sun is at one of the foci. Newton showed that elliptical orbits can be described under two kinds of foces, a force proportionate to the distance, and a force inversely proportaionate to square of the distance. But if the attracting body is to be at one focus, the force must be inversely Propotionate to the square of the distance. Thus the first law of Kepler is a proof, by verification, of the lawof gravitation formulated by Newton.

What is the significance of Kepler's third law, viz., the square of the periodic time is propostionate to the cube of the distance ? Newton theoretically arrived at the following result: [T2 =42 d3/u]. If as Kepler states [T2 d3], 42/u] must be a constant. [4] is obviously, a constant, and [U] is the mass of the attracting body multiplied by the gravitational constant (if we neglect the mass of the attracted body, as we can do in the case of the planets). I fthis is also to be a constant, the attracting body must be the same for all planets, as indeed it is, viz., the sun. Thus Newton showed that the Third Law of Kepler is a proof that the sun is the parent, as it were, of the planets revolving round it.

If the Law of Gravitation holds good in the earth-moon system and in the Solar System, there is no reason why it should not be true beyond the Solar System, in the region of the stars. Better telescopes were required before this could be verified and Newton could not do it with the crude telescopes (crude in coparison with the modern telescopes) of his days. Later on double stars were discovered which revolve round each other obeying Newton's Law of Gravitation. Thus the law has been found to be universally true and it can be stated as follows:-Every particle of matter attracts every other particle with a force proportionate to the product of the particles and inversely proportionate to the square of their distance from each other [F=Gm1 m2/d2]. Newton thus laid the foundations of modern astronomy.

Prejudiced and jealous scientists of Newton's days tried their best to disprove the law. They pointed to the inequalities in the motion of the moon and said they should not exist if the law was true. Newton said that his law required the existence of the inequalities. They would not exist if there were only two dodies, the attracting and the attrached. But there is a host of other bodies, besides, which must affect the body in motion, however small the effect is. Newton actually worked out the major inequalities in the motion of the moon called 'evection' and 'variation'. Gradually all opposition died out and the law recognised by one and all.

The Law of Gravitation in use

A triumph for this law was soon to follow. The planet Uranus was discovered and the elements of its orbits were calculated. From these elements and from the equations for the preturbations of the planet by others the future positions of the planet were calculated. It was seen to occupy a position slightly different from the predicted position. Either the law has failed or there is some other planet not discovered yet, disturbing the planet. Astronomers calculated where the new planet must be found if it existed. Powerful telescopes were directed towards that spot in the heavens, and lo ! there it was, hitherto mistaken for a star; it was christened Neptune. The discovery of the planet Pluto later was another triumph for the law.

The Law of Gravitation has enabled scientists to determince the mass of the earth. The pull of the earth on a known mass is compared with the pull of another known mass on the first mass and as teh pulls are in proportaion to the masses, the earth's mass can be found out. It is about [6X10] tons. From that the mean density of the earth is calculated to 5.5 grams per c.c.

The determiantion of the exact shape of the earth has been made by using this law. The equation for the time of osciallation of a pendulum is [t2=2 1/g], where t is a half-vibration period, 1 is the length of the pendulum and g is teh acceleration due to gravity at the place. Fixing 1, we find t varies inversely as g, and by measuring t we can find the acceleration at teh place. The acceleration will give the distance of the place from the centre of the earth. Many such determinations will give the shape of the earth.

By using the law we can theoretically determine whether there can be an atmosphere on the moon (or any other body). There is a critical velocity cnnected with every body, exceeding which an object on the body will leave it altogether overcoming the gravitational pull of the body. It is given roughly by the equation (v=7*/ m/r), where v is in miles, m is the mass of the body (taking the earth's mass as unity) and r is the radius of the body (taking the earth's radius as unity). For the earth it is about 7 miles. For the moon it is (7*/1/81./.1/4=1.5) miles, roughly. Now, the Kinetic Theory of Gases says that all gases consist of molecules moving about at speeds depending on the molecular weight of the gas and the temperature. At the temperature that obtains on the moon, all gas molecules must exceed the critical velocity of 1.5 miles and fly off into space.

Conclusion

After the advent of the Theory of Relativity there is a tendency among some people to minimise the significance of Newton's work. The Theory of Relativity has shown that Newtons's Law of Gravitation is only an approximation to the truth. But the approximation is so close that Newton's Law suffices for all puposes of astronomy. Colossal as the masses of the astronomical bodies are; the closer formula of Einstein gives practically no measurable difference. It is only in the case of the motion of the apsides of Mercury that an accumulated difference of 42" per century is detectable. Also, Newton's Law is simple in form adn easy to apply. Thirdly, but for Newton Einstein cannot have propounded and verified in a determinate manner the Theory of Relativity at all, for Newton's work is a necessary step towards the progress achieved by Einstein. The very solution of Eintein's gravitational equation is effected by taking Newton's equation as a first approximation.

We can compare Newton with Euclid in this respect. Geometries rival to Euclid's have sprung up, the Hyperbolic Geometry of Johann Balyai and Lobatschewsky and the Elliptic Geometry of Riemann and Hilbert. It is also probable that the geometry of out space is elliptic. But does it mean we can discard Euclid's geometry ? The geometry of Euclid suffices for practical purposes. It has the advantage of being simple, simple in its nature as a particular case of those geometries, and also because of out mental habits and the kind of intuition we have of space-flat. Euclid will live, as also Newton.

*THE EVOLUTION OF THE UNIVERSE ACCORDING

TO SIR JAMES JEANS**

Before seeing how the universe has been evolved, we shall understand what constitutes the universe. The primitive man though that his immediate surroundings with the things in them formed the universe. Gradually, as his sphere of activity increased and knowledge expanded, the Earth came to be supposed the centre and the most substantial part of the universe. The Earth, with the celestial sphere just reaching beyond it on all sides with the Sun, the Moon, the planets and the stars fixed on to the sphere, was thought to be the entire universe.

As knowledge increased man learnt that the earth is only one of the many planets moving round the sun. First there is Mercury (Budha) nearest to the sun, at a distance of 36 million miles. It is about a sixteenth of the size of the earth. Next comes Venus ('Sukra) at a distance of 67 million miles from the sun. It is almost the same size as the Earth. Next to Venus comes the Earth, which is at a distance of 93 million miles from the sun. Its radius is about 4,000 miles and mass, about 7x10 tons. It has one satellite, the Moon. Beyond the Earth is Mars (Kuja) at a distance of 142 million miles. It is small, compared with the Earth, being about a seventh in size. But it has two moons. Beyond the orbit of Mars a vast number of bodies, only a few miles in diameter, called the asteroids, move round the Sun. Next comes Jupiter, (Guru) the Giant Planet. It is the biggest planet and can contain more than a thousands Earths. It has 9 moons. Its distance is 483 million miles, more than five times that of the Earth. Next there is Saturn ('Sani) almost the size of Jupiter, at a distace of 886 million miles. It too has nine moons. It has a beautiful belt of 'rings,' really small stones close together moving round in a belt. Next comes Uranus and Neptune, of size about 60 times and distance about 19 and 30 times that of the Earth. Uranus has 4 moons and Neptune, one. About ten years back a new planet was discovered to which the name Pluto was given. It is at a distance of 40 times that of the Earth adn has a period of revolution of 250 years. Besides these planets there are a large number of comets and swarms of meteors, many of them mere pieces of stones, moving round the Sun. The Sun is the parent of stones, moving round the Sun. The Sun is the parent, as it were, of all the planets and the grand-parent of the satellites. All these consitute the Solar system.

Yet the Sun is only one of the millions of stars that form the star-city-a galaxy as it is called-to which we belong. The number of stars in out galaxy is a little over a hundred thousand million. This is the approximate number of stars in each galaxy. From this we should not think that the galaxy is crowded. Even the Solar system is practically empty of matter when we consider its size, the planets forming a few particles of dust, and the Sun a small pellet of stone, on a vast field. The galaxy is emptier still. It takes 8 minutes for light to reach us from the Sun, but 4 years to reach us from the nearest star. I fthis gives the mean distance between star and star, we can have an idea of the emptiness of a star-city, in spite of its hundred thousand million stars. Such great emptiness can be compared only with the emptiness of what we call matter, which, a great physicist says, is all holes with nothing between.

If we leave our star-city extending over a space of two hundred thousand light-years (a light-year is the distance which light travels in one year), we must travel four times the extent of our city to reach the nearest star-city; and there are millions of star-cities in the universe. This does not mean they are infinite in number. Space, which is co-extensive with teh universe, is finite. There is no meaning for scientists to ask what is beyond, for the question does not arise. [If we think it is infinite, we have erred in our judgement as regards the shape of the Earth and conceived it flat. Experience only tells us that space must be unbounded, and the assumption that it is also infinite is unwarranted. Infinitude belongs to measure relations, while unboundedness belongs to extent relations; and experience confirms the latter and not the former.] The radius of the universe (space) is 10 times the radius of the Earths' orbit.

We shall try to picture the universe. If the Earth represents the universe, the towns on the Earth may represent the star-cities. The stars that are the citizens of these star-cities are the size of a mustard seed, about a hundred thousand million in each town. If the star is a Solar system-the chance for this is very little-the planets, on our model, are specks of dust, a few in number, placed near the seed at a distance of a hair's breadth. And where are we men on one of the smallest of the specks of dust? If we can visualise this, we have obtained an excellent cure for out megalomania.

Let us now see how this universe, as it is at present, has been evolved. Was it ever so, or was there a time when the different orbs in it did not exist ? There was. We have travelled through space and seen its extent. Let us now travel through time, backwards, and see what the universe was millions and millions of years ago.

It was a vast globe of incandescent gas. What it was before that, we cannot even guess. The particles of the gas were moving to and from with enormous speeds, now attracted, now repelled, by one another, but in general attracted by by gravitation. Ages rolled on, and the gas, at first evenly distributed throughout space, condensed and formed smaller globes of gas. Each globe contained the material for a star-city, the material for about a hundred thousand million Suns. Further condensation took place. At the same time the globes began to rotate on account of gravitational attraction between the particles. Each globe became flattened like an organge. The flattening increased, at the same time condensation taking place in the globe; and individual stars began to appear. Thedegree of flattening represents the age of the gas-globe, the nebula, as it is called. The more flattened the nebula is the more condensed it is, and the more the number of stars in it.

One such flattened nebula is our galaxy. The sun is at a distance of a third of the radius of the galaxy from the centre, and takes part in the rotation on the galaxy. It iis this rotation that keeps the stars from coming together at the centre by mutual attraction. Incidentally, it is this rotation that enables us to calculate the mass of a galaxy.

Then millions of years passed, and a strange thing happened about 3,000 million years ago. Our Sun then was a young maiden, as it were, brighter and more glorious than it is to-day. It had a companion, another star, a maid in attendance. They had no work to do, and spent their time in chasing each other round. At the period I referred to, a new star, during his wanderings happened to goe near them. The attraction of the stranger was too strong for out Sun maiden to resist; and the ultimate result was that the planets were born.

This is how it happened. As the new star approached the Sun, great tidal forces were set up in it and it became elongated towards and away from the new star. But the elongated mass could not fall back when the new star receded, because of the velocity of the mass, and it ultimately condensed into the planets. The star moved away after generating the planets, the Sun's old companion eloping with him.

This is the story of the birth of a planet. But such celestial matrimony is rare, in spite of the great number of stars in the universe and the long, long period of time elapsed, because spece is vaster still. "Leave ten bees on the continent of Asia at different places. Waht chance is there of their meeting each other? There is the same chance for stars to visit one another."

Let us continue the story. The planets were globes of gas at first. As they cooled they became liquid and then a solid crust was formed on the surface. On one of these life originated; and by gradual evolution man came upon the earth.

Rare as a Solar system is, a planet with life on it is rarer still, for special conditions are necessary for life to exist. In the Solar system, it is now evident that the Earth alone has life on it, life as we know it. It is just at the range of temperature within which life can exist. The other planets are either too cold or too hot. Mars perhaps has very low forms of life.

One day the Earth too will become a cold, desert planet like many others, unable to support life. In an insignificant corner of the great universe man will have existed for an infinitesmal fraction of time and vanished.

THE DURATION OF ECLIPSES*

Section 1: Lunar Eclipses

The total duration of a lunar eclipse is given in hours, by the formula 2 [D2 -P2{1-P2/(P2+m2)}]1/2/ (P2+m2)1/2 (1), where D is the distance between the centres of the moon and the shadow at first or last contact, P is the latitude of the moon at the time of opposition of the sun and the moon in longitude, P is the increase in P per hour and m is the motion per hour in longitude of the moon, relative to the sun.

This is clearly 0 when P2 etc.=D2

i.e., when [+P{1-P2/(P2+m2)} 1/2=D, i.e., when P is numerically greater than D by D [P2/2(P+m2) approximately. This comes to about 14" on the average. Thus it is wrong to say that there is no eclipse when P is greater than D, for even when P is within about 14" greater than D at opposition we have an eclipse. When P=D, the duration fo the eclipse is not 0, but 2Pp/ (P2+m2), which is about 22 minutes, the eclipse commencing or ending at opposition.

When is the duration a maximum?

Clearly when P, the latitude of the moon at opposition is 0, i.e., the maximum occurs when at opposition the sun and the moon are also at the nodes. It is equal to 2D/(P2+m2)1/2 hours.

But D, m and P are functions of 1, and 1', the moon and the sun's mean anomalies, respectively. Therefore the maximum duration itself varies between limits, whose maximum and minimum we shall evaluate now.

In the neighbourhood of the sun and the moon's conjucntion or opposition in longitude near a node, we have the following equations retaining functions of 1, 1' alone, where 1 and 1' are the moon and sun's anomolies at [sthula-parva]. (This is the time of a fictitious conjunction or opposition with the true moon=mean moon+315' sin 1, and the true sun =mean sun+127' sin 1')

The moon's Equatorial Horizontal parallax

["=3447".9 +224".4cos1].

The sun's Do. Do. parallax ['=8.8"+2"cos 1']

The moon's semi-diameter r=939".6+61".1 cos 1]

The sun's semi-diameter r'=961".2+16."1cos1'

The radius of the shadow s=2545".4+228."9 cos 1

--16."2 cos 1' (m2 +P2)1/2=1875."6+260."1 cos 1

--5."0 cos 1'.

Now the distance between the centres of the moon and the shadow at first or last contact=D=s+r=3485.0 +290.0 cos 1-16.1 cos1'. Therefore 2D/(P2+m2)1/2=2 (3485.0+290.0 cos 1 -16.1 cos 1') /(1875.6+260.1cos1-5.0 cos 1'). This is a maximum when 1=1'=180o and not when 1=0, though it makes the semi-diameters of the shadow and the moon a maximum, for the increase in the numerator is outweighted by that in the denominator. Thus the max. is 2 (3485-290+16.1)/(1875.6-260.1+5)= about 238 minutes. The lower limit occurs when 1=1'=0 and it is, 2 (3485+ 290-16.1)/(1875.6+260.1-5) hours=about 212 minutes,. If we do not neglect functions of 21 the maximum is 237.4 minutes.

The maximum duration of the total phase of a lunar eclipse can be found by making D=s-r, and finding the maximum value. This too occurs when the sun and the moon at oppostion are at teh nodes, and when 1=1'=180; and it is 2 (1605.8-167.8+16.4)/ (1875.6-260 1+5) hours=about 108 minutes.

Section 2 : Solar Eclipses

The formula for the duration of a solar eclipse in general on the earth (as opposed to the duration at any particular place) is the same as for the duration of a lunar eclipse, with this difference, that here [D= ---- +r+r',] and P is the latitude of the moon at conjunction of the sun and moon in longitude. Here too, as in the case of the lunar eclipse, we can see that the duration is 0, not when = p = D, but when P is numerically greater than D by about 20". When =P=D the duration is about 33 minutes.

The maximum duration of a general solar eclipse occurs when P=0, i.e., when the conjunction in longitude is at a node. It is given by 2D/(P2+m2)1/2, in hours= (5339.9+285.5 cos 1 +15.9 cos 1')/(1875.6+260.1 cos1-5 cos1') hours. This is a maximum when 1=180o and 1'= 0 and not when 1=0, though that makes both the parallax and the semi-diameter of the moon a maximum, because the increase in the numerator is outweighed by the increase in the denominator. It is 2x(5339.9-285.5 +15.9)/(1875.6-260.1-5) hours =6 hours 18 minutes nearly. It is easily seen that under this condition the eclipse is annular. Therefore the maximum duration on the earth as a whole happens at an annular eclipse. (If we do not neglect 2 1, the maximum is 6 hours 16 minutes).

The duration of a solar eclipse at a given place on the earth is given by (r+r')/ (P2+m2)1/2 corrected for parallax, which chages rapidly and varies from place to place. So the work fo finding it is a bit difficult. But the maximum duration is easy to determine. This occurs when the central eclipse is at apparent noon. At this time the apparent semi-diameter of the moon is r+about 16". Also, the nearer to noon, the greater is the retardation in the relative hourly motion of the moon, owing to parallax, with the result that the greater is the increase in the duration of the eclipse. For an hourangle of 34o on both sides of noon the average retardation is 850".3+55".4cos 1 per hour.

The total duration is given by 2 (r+16"=r')/{(P2 + m2)}-(the hourly retardation due to parallax). So the maximum duration is given by 2 (1917+61 cos 1+16 cos 1')/ {(1875.6+260cos 1-5 cos1')-(850.3+55.4 cos 1)}=2 (1917+61 cos 1+16 cos1')/(1025.3+204.7 cos 1-5 cos1'), when 1=180o and 1'=0. The maximum, we can easily calculate is about 4 hours 35 minutes, and it occurs when there is a combination of the most favourable circumstances, viz. the conjunction occurring at a Node, the central eclipse falling at noon, and 1 being equal to 180o and 1', 0.

The maximum duration of the [Annular] or Total phase at a given place is also at apparent noon for the same reasond. As the period is very short we shall take the motions per minute for purposes of calculation. The duration of an Annular eclipse near noon is given by 2 (r'-r-16)/ {(31".3+4".3 cos1)-(15"+1"cos1)}=2 (5.7-61 cos 1 +16 cos 1') (16.3+3.3 cos1). This is a maximum when1=180o, 1'=0. Calculating, we can find it to be about 13 minutes. The minimum, of course, is 0, as is patent from the formula.

The Total phase is given by 2 (r+16"-r')/{(31".3+4.3 cos1)-(15'+1"cos1)}=2 (61.1cos 1-16.1cos1'=5.7)/ (16.3+3.3 cos 1). This is a max when 1=0and 1'=180o; whose value is 2X71.5/19.6=about seven minutes. Teh minimum is clearly 0. Comparing the results we find that, other things being equal, an annular eclipse has in general a greater duration than a total eclipse.

THE LUNAR ECLIPSE IN HINDU ASTRONOMY*

1. Introduction

It is a well-known fact that a lunar eclipse occurs when, in the neighbourhood of a moon's node, the sun and moon are in oppostition, i.e., when the moon and the earth's shadow are in conjucntion. At the time of such a conjunction, say T, let P be the latitude of the moon (north latitude being considered +), P', the hourly increase in latitude (increase towards the north being considered +), m', the excess of the hourly increase in longitude of the moon over that of the sun, M, the angular radius of the moon, and S the angular radious of the shadow, at the moon. Then, at any time t hours after the time of conjunction, T, the distance between the shadow and the moon, in longitude, is m't and the latitude of the moon is P+P't; and so the distance between the centres of the shadow and the moon is {m'2+(P+P't)2} 1/2. The eclipse begins or ends when the moon's rim just touches the rim of the shadow in entering it or leaving it.

The distance between them at such a time i sS+M=D, say. Then {m'2t+(p+p't)2}1/2==D gives the time of the beginning or end of the elcipse. Solving this for t, we get. t=-pp'/(m'2+p'2/+{P2P2/(m'2+P'2)2+(D2-P'2)}1/2; in which, obviously, the upper sign gives the beginning, and the lower, the end of the eclipse. The total phase of the eclipse begins or ends when the rims touch the moon being inside the shadow, i.e., when the distance between them is S-M. If we substitute this instead of S+M for D, in the above solution, we get the times of the beginning and ending of the total phase of the eclipse.

Thus we arrive at the following results:

(1) The eclipse begins at T-PP'/(m'2+p'2)-{P2P'2/(m'2+p'2)+(D2-P2)/(m'2+P'2)}1/2, hours.

(2) The eclipse ends at T-pp'/(m'2+p'2) +{p2p2/(m'2+p'2)2+(D2-p2)/(m'2+P'2)}1/2, hours.

(3) The middle of the eclipse falls at T-pp'/(m'2+p'2), hours. From this we see: (a) that if p and p' are both postive or both negative, the middle of the eclipse is before the time of conjucntion , and if one is positive and the other negative, the middle is after conjunction; (b) only when the latitude at conjucntion, p, is 0, the middle falls at T, the time of conjunction, (for p' cannot be 0 near a node).

(4) If D=P, (1) and (2)reduce to T-pp'/(m'2+p'2)+pp'/(m'2 +p'2), from which we see: (a) that the eclipse begins or ends at conjunction, and (b) that the duration is 2 pp'/(m'2+p'2), which may amount to about 22 minutes.

(5) The duration is 0 when the expression between the double brackets is 0, i.e., when P is greater than D by Dp'2/2 (m'2+p'2), (neglecting fourth powers of p'/m'); which may amount to about 14" in the mean.

(6) If t is not real, there is no eclipse or total eclipse according as D is taken to be S+M or S-M. will be in conjunction in longitude at 5.58 a.m. (I.S.T.) P=+3083" p'=-164", m'=1641", S=2339", M=888". Using these we get t=+11minutes +36 minutes, i.e., the middle of the eclipse falls at 6-9 a.m. The eclipse begins at 5-33 and ends at 6-45. Putting D=S-M=1451", t is not real, and therefore there is not total phase.

2. Calculation of the Lunar Eclipse in Hindu Astronomy

So far as we know, the [Vedangajyotisa] (about 1220 B.C.) is the most ancient Hindu astronomical work. Two recensions of it are extant, the Rigvedic and the [Yajurvedic], neither of which makes any mention of eclipses; as also the Jain and Buddhistic works the [Suryaprajnapti], the [Jyotiskaranda] and the [Kalaloka praka'sa] and also the Brahminical works like the [Pitamaha-Siddhanta], which all came a few centuries later.

The next period forms a tansition from this to the time of the advanced [siddhantas] of later times. Three of the [siddhantas] condensed by [Varahamihira] in his [Pancasiddhantika], viz., the [Vasistha], the [Pauli'sa] and the Romaka belong to this period. In them we find the beginnings of an attempt at finding the true longitudes of the sun and the moon and calculating eclipses. [Varahamihira's] Romaka does not give the rules for calculating a lunar eclipse, thought the more difficult calculation of the solar eclipse is given. The methods of the [Vasistha] and the [Pauli'sa] are mixed up. This is what they say in effect. Rule (1). Find the difference n in longitude at the time T of the moon's conjunction with the shadow, between the near node and the moon. If n is within 13o, there is a lunar eclipse. (2). [T + 3/20 */169-n2] hours gives the beginning and end of the eclipse. (3). Find the latitude of the moon P at that time, in minutes. Then, if m' is the difference in the hourly motion in longitude of the sun and the moon, [T+*/552 p2/m'] hours gives the beginning and the ending of the eclipse.(4). [T+21/5m,*/25-n2] hours gives the beginning and end of the total phase.

From (1) and (3) we see that 13o difference between the longitudes of the node and the moon is equivalent to 55' of latitude: i.e., near a node the latitude chages at 55/13 minutes per degree and the maximum latitude is 4o. From (3) we see S+M=55'; from (4), S-M=21'; and so according to these [siddhantas], S=38'and M=17' which are taken to be constant. (2), (3) and (4) neglect p', so that the middle of the eclipse falls at T, and the duration is 0 when S+M=p. 92) neglects m' as well, so that the duration is only mean , as given by this rule.

Applying these rules to the example given in 1, taking T and m' as given there, and n equal to 10o 2', we have according to (3), T+77 minutes and according to (4), T+74 minutes. In each case the duration is more than double whaat it is .

The [Suryasiddhanta] condensed by [Varahamihira] and the later [Siddhantas] in general give fairly accurate methods and constants for the calculation of the eclipse. Their methods are practically the same, though the constants vary a little from one another's. They furnish the basis for finding ['] and ['], the parallaxes of the sun and the moon, and R and M, their radii. ['] [']R and M in minutes are according to the 'condensed' Sur. Sid. 3.8, 51.4, 16.1 and 16.1; according to the [Suryasiddhanta] 4.0, 53.5, 16.2 and 16.0, and according to the [Siddhanta 'Siromani] of [Bhaskaracharya], 3.9, 52.7, 16.2 and 16.0. All three give 270 minutes for the maximum latitude of the moon.

The following, in effect, are the rules they give for computing the eclipse: (1) Find T. Find p, moon's latitude at T, from : lat=270' sin(Moon-Ascending Node). (3) Find m'. 94) Correct ['] and R by multiplying each by the sun's daily true motion in longitude and dividing by the mean motion, 59.1'. Correct['] and M by multiplying by the true daily motion of the moon and dividing by the mean motion, 790.6'. Using these values, [']+[']-R=S. S+M=D for the first and last contacts, and S-M=D, for the beginning and ending of the total phase. (5) [T+*/D2-P2/m'] hours is a first approximation for the beginning and the ending of the eclipse. Repeat, usingfor P the latitude at the approximate time of beginning. A more correct time of beginningis thus obtained. If the latitude of this time is substituted and computation again made, a still more correct time is got, and so on, till we have the necessary accuracy. The same process is to be followed to find the corrrect time of ending.

It is to be noted that this process of successive approximation is necessary because the chage in P caused by P' is neglected in the formula in (5) above. Perhaps the authors feel that this way of finding the times is easier than finding them once and and for all by solving the quadratic with P' included; for, even [Bhaskaracharya] gies this rule though he knows how to solve a quadratic and interpret the solution. But this methods will fail when P=D, for then [*/D2-P2/m'] is 0, and no time before or after T is got for a second approximation. (We have already seen that in this case the eclipse begins or ends at T and the duration will be about 22 minutes). But one thing must be mentioned here, that for the [Siddhantas] the difficulty will not arise in the matter of calculating the first or last contact, because they do not require us to calculate such eclipses. The [Siddhanta 'Siromani], for instacne, says that if S+M-P is less than a sixteenth part of the moon's diameter, i.e. 2', the eclipse should not be calcuted because its visibility will be marred by the brightness of the moon.

Another matter to be noted is that the parallaxes and radii are mentioned as varying proportionately to the true motions, which is not strictly correct.

Let us now apply the above rules to the example in 1, assuming T and m' there,and also given the moon's true longitude 141o 14', the longitude of the Ascending node 331o 16', the moon's true daily motion 717.3' and the Sun's 60.7'. Taking the constants of [Siddhanta 'Siromani], (1) T=5.58 a.m. (2) P=270' sin 169o 58'=47'. (3) m'= 27.4'.(4) 4'and 16.3', each multiplied by 60.7 and divided by 59.1 equal 4.1' and 16.7. 52.7' and 16' multiplied by 717.3 and divided by 790.6 make 47.8' and 14.5'. Thus S=35.2' and M=14.5'. S+M=D=49.7'. (5) */49.7 2-47 2/ 27.4 hours +35 minutes. So the approximate time of the beginning is 35 minutes before T and of the ending,35 minutes after T. The latitude decreases at 2.32' per hour. So at T-35m the lat. is 48.35' and at T+35m it is 45.65'. Using these we get T-25m for the beginning, and T+43 for the ending, as a second approximation. Repeating the process, we finally find that the eclipse begins at T-28m., and ends at T+45m. (5-30 a.m. and 6-43 a.m.) The agreement with the correct values is close, as we see.

THE THEORY OF RELATIVITY*

PART I

At the beginning of the present century, a new scientific theory, known as the Theory of Relativity was propounded by the great German scientist, Einstein. It consequences in the field of science were far-reaching : it necessiated a chage in the very out look of the scientists. The Theory was at first restricted in its scope to uniform motion in a straight line and therefore called the Restricted Theory. In 1916, ten years after the Restricted Therory, the General Theory of Relativity was given to the world.

The Restricted Theory is based on two hypotheses which are really scientific truisms: (1) the velocity of light in a cacuum is constant, i.e., every observer finds it has the same value, and (2) it is impossible to find the absolute velocity of a body through the ether, relative velocity alone is observable, i.e., of two persons on two frames in relative motion 'to each other, if each asserts that he is rest and the other is in motion, there is no test to find out who is right. (It is from this second hypothesis that the theory gets its name). The General Theory rests on only the second of the above-mentioned hypotheses, generalised to include accelerated motion. Before we proceed to see what circumstances led to the framing of the Theory and how it is developed from the given hypotheses, we shall understand the import of the Theory, i.e., the results that follow from it.

A man stands on a railway track and observes a train moving away from him with a velocity v. A Passenger in the train is running from the guard's van towards teh engine with a velocity, u. With what velocity is the passenger moving away from the man ? v+u, we say. For instance, if the train is moving at 30 miles an hourm, he must be leaving the man behind at 38 miles an hour. But, no, says the Theory of Relativity. The formula according to the Theory is (u+v)/(1+v u/c2), where the new quantity c is the velocity of light. If, as in the example given, u and v are small in comparison with c, (which is about 186,000 miles per second), there is practically no difference which formula we take-the reader may work out the difference if he is curious, 0".017. But if u and v are large, the difference is considerable. Suppose a nebula is receding from us at 18,600 miles a second and an electron is moving on it in the same direction at 93,000 miles a second. The velocity of the electron as observed by us is not 18,600+93,000=111,6000 miles, according to the familiar formula. If is (18,600+93,000)/{1+(18,600X93,000)/186,000}=106,286 miles. There is a difference 5,314 miles, nearly a third of the velocity of the nebula.

There is something very interesting to be said in this connection. Long before the Theory of Relativity was though of , a scientist by name [Fresnel], while experimenting on the behaviour of light, measured the velocity of light through the water flowing in a tube. Allowing for the drag of the flowing water, the result showed not a velcoty [c/u,](u is the refractive index for water), as must be expected. It was [c/u+u](1-1/u2), where u is the velcity of the flowing water. He was puzzled at this result. The experiment was repeated, but the result was the same. At last scientists came to accept the discrepancy u (1-1.u2), as an empirical fact and came to call (1-1/u2) the Convection Coefficient. The Relativity formula for the composition of velocities explains this easily, furnishing an indirect proof of the Theory of Relativity.

Let us now examine the equation, (u+v)/(1+uv/c), for its other implications. We shall take the case where u and v are both positive. If they are each less than c, the velocity of light, the composed velocity is always less than c, i.e., no velocity can be added to the velocity of light. (This result is inherent in the first hypothesis). If at least one of the velocities is greater than c, the resultant is less than at least one of them, which result is absurd, because we have taken both u and v positive. Therefore we have to conclude that no velocity can exist greater than that of light. This result is confirmed by another consideration as we shall see later on.

As another consequence of the Theory of Relativity, we have to give up our old idea of mass. We have been defining mass as the amount of matter. But, Relativity teaches that mass has nothing specially to do with matter. Radiation, as well as matter, ha smass; and even though the amount of matter in a body does not change, its mass may change. The mass M is given by the equation M=m/*/(1-v2/c2), where m is the mass when the body is at rest, the rest mass or proper mass, as it is called; v is the velocity of the body, and c, the velocity of light. Usually v is small in comparison with c, and there fore neglecting v4 and higher powers, we can write M=m+1/2 m v2/c2. Now, we know that the kinetic energy of a moving body is [1/2 Kmv2] where K is a constant depending upon the units chose. So we can conclude that the additional mass may be due to the kinetic energy of the body, i.e., the energy E, has mass equal to [E/c2] grams, if E is given in ergs and c in cm per second. This additional mass riding on energy is usually very small because ordinary velocities are small and also because the divisor c2 is enormous (=9X10). But it increases rapidly with the velocity. For an electron with a velocity of 149,000 miles per second, the increase in mass is 60 per cent. For still higher velocities the mass increases very rapidly, tending to infinity as v approaches c. But as the mass increases, increasingly greater forces will be required to increase the velocity by a given amount, with the result that no body can acquire the velocity of light. Thus we return to the conclusion that the velocity of light is absolute and cannot be exceeded.

At the end of the last century, J.J. Thomson, in his experiments with electons, observed that the mass of the electrons increased with the velocity, and he arrived at the empirical formula for the mass, M=m+1/2 mv2/c2. This result is explained by the Therory of Relativity, which itself is proved thereby.

We saw that the additional mass is due to the kinetic energy. May it not be that the rest mass itself is due to the energy inherent in matter in a potential form? Considerations of convenience recommend this assumption. This is also highly probable for the following reasons. Matter has been known to be annihilated, exhibiting itself in the form of energy, the mass of the matter appearing as the energy mass. Secondly, matter which to all appearances is static and contains no energy, really possesses a large quantity of energy, the kinetic energy of its molecules and atoms in motion, (which exhibits itself in the form of temperature), the energy that binds the atoms together to form the molecule, the energy that keeps the electrons and protons together in the atoms and the energy in the electrons and protons themselves owing to their spin and electric charges, energy being required to charge them, So we conclude that all mass is due to energy, with which it is inseparably associated. Conversely to what we said about the mass of energy, the energy in a given mass is enormous (E=mc2) because the mass has to be multiplied by c2 (=9X10) to get the energy. For example, an ounce of matter is equal to [28.35X 10X9 ergs=2.55X10] ergs which amounts to about 700 million kilowatt- hours or 940 million horse-power-hours. This is greater than the energy generated by a thousand horse-power engine working incessantly for 108 years. Only, there is no known means of making this energy available for us.

The Theory of Relativity has revolutionized our ideas of space and time and at the same time given us an insight into their true nature. We derive our ideas of space and time indirectly through our sense-impressions. This iswhat philosophers mean when they say space and time are modes of perception. But time enters our consciousness also directly, without the intervention of the senses. It is this time that is perceived as flowing from the past to the present and from the present to the future. By identifying this subjective time with the objective time obtained through our senses, we get the idea of the flow of time in general. It is this subjective time that gives the world its dynamic character. We shall revert to this later on.

We know that every event is four-dimensional, i.e., we require four qualities to specify each event. We say an event has happened at a particular place at such and such a time. To specify the place we require three quantities, co-ordinates as they are called: so much in length, so much in breadth and so much in height. In other words, space is three-dimesional. But we always separate the four dimensions into two sets, three of space and one of time, never mixing the space and the time. Between two events there is, we feel, a definite interval in space as well as in time. If the interval in space is zero we say they happen at the same place. If the interval in time is zero we say they take place at the same time, i.e., the events are simultaneous.

According to the classical theory, everyone measures the interval in space as being equal, as also the interval in time (provided an ideal measuring rod and an ideal subective time, i.e., time as sensed by each individual . Subjective time is notoriously different for different persons: a person awating his love, and another, the execution of his death-sentence, do not have the same appreciation of time. True, there is a classical theory of relativity which says that if a person A measures an interval x in space and t in time, aperson B, moving uniformly away from A at u units of length per second, measures the interval in time as t, but the space as x-ut. But the classical theory does not take this difference seriously, for it takes it for granted that it can be known who is actually in motion, an da correction may be made accrodingly; so that there is agreement between A and B in their measurement of space as well.

But Einstein's Theory of Relativity has changed the possibility of all this. It says that neither the interval in space nor that in time is the same for two different persons in motion relative to each other. (As for absolute motion, it is denied by the postulates of Relativity.) If A measures an interval x and t in space and time respectively, B measures a space-interval of (x -ut)/*/1-u] units and a time interval of (t-ux)/[*/1-u] seconds. (For the sake of simplicity the distance which light travels in one seconds is taken as the unit of length and the second is taken as the unit to time).

Let us now proceed to examine the formula for their implications. Suppose A has a clock with him and observes an interval t seconds. As the clock is with him always, x, the interval in space, is zero. Accoding to the formula for time given above, B measures the time as (t-u X0)/[*/1-u2=t/*/1-u2]. B says A 's clock is slow, as there is a correspondence between A's acts and his clock, he says A is a lazy person. But A accuses B of being lazy, the relation being reciprocal. Sometimes each envies the other that the other's [jilebi] lasts longer than his own, but finds a consolation in the fact the other's visit to the dentist lasts longer. If we do not experience this difference in time it is because u, the relative velocity, is very small (remember, the unit of length is 186,330 miles. Even if u is 400 miles an hour, the differences is only 1/10 percent). Another thing to be noted is that */1-u2] becomes imaginary if u is greater than 1. From this we see the velocity of light is the limit to which relative velocity can attain. But there is nothing very remarkable in this, for we have only taken out now what we have put into out postulate as the constancy of the velocity of light.

Let us now go to their measurement of space. While A mesures x, B measures (x-ut/*/1-u2 (according to the classical theory he must measure x-ut) His measuring rod as well as his time has shortened. A thinks that it si natural because B is in motion. But B says A is in motion and his rod has shortened. As we cannot find absolute motion we cannot say who is right. As x cannot be greater than t (to measure a length greater that t in time we should move faster than light) no one can see events happening in an order reverse to what another sees they happen. In y article on the stars I humorously remarked that if, at this moment, we transplant ourselves on to a star distant 2,500 light years, we can see that Buddha preaching [Ahimsa] to the world. But we now see we can never have that joy, poor, mortals, and the Buddha must preach without us as audience! This is evident from the following consideration. You start now moving away from the world, all the while observing it through a telescope. As long as you do not move faster than light you cannot catch up with the light that left before you and perceive the event again. But nothing can make you move faster than light. So, all that you can do is to see the processes of the world slowing down and have the satisfaction of accusing the world slowing down and have the satisfaction of accusing the world of laziness, because you are there to whip it up to activity!! Thus the world is saved from topsyturvy-dom by the constancy of the velocity of light. Time is furnished with its arrow, as it were, by the limit set to attainable velocity, and shown which way it should move.

The Theory of Relativity has thus compelled us to throw overboard our concepts of absolute velocity, absolute space, absolute time and simultaneity, the impressions. But [Minkowski], a great mathematician, has shown how to recover the absolute by going back to the four-dimensional origin of our perceptions of space and time. Though A and B calculate the interval in space and in time to be different, they agree that their interval in space-time is the same. If A finds the four-dimensional interval x2-t2, B finds it (x-ut)2/(1-u2)- (t-ux)2/[1-ux2=x2-t2]. Thus events are points in a four-dimensional world of space-time which is absolute, and their interval apart for every person is the same. But each carves out his own interval of space and interval of time. The accident that we move with a small velocity relative to one another has masked the unity of the physical entity, space-time, because if we had considerable relative velocities, we would have obtained different values for physical space and physical time, and the conclusion would have been forced upon us that space being real. Thus Relativity has given us an insight into the true nature of space and time.

We shall now examine some results of the General Theory of Relativity. The force of gravition as given by Newton is G m m'/d2, where m and m' are the masses of the dodies attracting each other, d is their distance apart, and G is a constant depending on the units of mass and length. From what we have said about space and time it is evident that the law cannot be in this form in the Theory or Relativity; for, mass and distance are variable according to the frame of reference with respect to which they are measured. The Theory of Relativity gives a law of gravitation essentially different from Newton's but practically agreeing with it. One difference is remarkable. According to Newton's law it can be shown that as long as only two bodies are involved (says, the sun and a planet) the apsides of the orbit of revolution lie fixed in space. But in fact the apsides of the orbits of the planets are revolving in the plane of the orbits, though the revolution is very slow (for the earth it is 1,163 seconds of arc per century). This is due to the disturbance, known as perturbation, caused by a third body. Let us take the case of Mercury. If there were no other planets, its orbit would be fixed in space and there would be no revolution of the apsides. But owing to the disturbance caused by the other planets there is a revolution of the apsides. The contribution of each planet to the revolution can be calculated from theory and compared with the results of observation. While observation gave a difference of 43". At first it was thought that some other planet, as yet undiscovered, was responsible for this. The existence of Neptune and Pluto were thus predicted from theory before they were discovered at teh very places predicted. It was thought that there was a planet between Mercury and the Sun affecting the orbit. Astronomers had even a name ready for it, Vulcan. But it was not discovered and the 43" remained unexplained, until the Theory of Relativity accounted for it.

The law of gravitation according to the Theory of Relativity shows that even when there are only two bodies there must be a slow revolution of the apsis equal to 6 ['] m/1 radians per revolution of the body, where m is the gravitational mass of the attracting body, where m is the gravitational mass of the attracting body and 1 is the semi-latusrectum of the elliptical orbit, the unit of length being the [lux] (186,330 miles). For the planets of the solar system it is about [*/16/r] seconds of arc per century, where r is the distance of the planet from the sun, the distance of the earth from the sun being taken as teh unit of length. Substituting the value of r for Mercury, we get 43", the exact amount of discrepancy between the old theory and observation. Thus an experiment performed in Nature's laboratory has vindicated the Theory of Relativity.

According to the Theory of Relativity light rays passing near a body are bent towards the body in the same manner as the path of material bodies. Supposing the light comes from a star, and the ray of light passes very close to the body in question, the amount of bending is 4 m/r radians, where m is the gravitational mass of the body and r is its radius in [lux]. It is not possible to say how waves of light could be attracted by a body, if we use Newton's law, which applies to material particles. Even supposing that the law applies to non-material particles as well, or supposing, as Newton himself did, that light rays consist of material particles, Newtons's lae gives only half the value given by Einstein's law. Here is a crucial test for the Theory of Relativity. Einstein predicted from his theory that light rays from a star passing close to the sun would be verified by observing the position of a star very close to the sun; but such an observation is possible only during a total eclipse of the sun, when the light of the sun cannot obscure the light of the star. The first verification of this prediction was made during the total eclipse of 1919, and this brough Einstein and his Theory of Relativity into the limelight. When Einstein made epoch-making discoveries like the photon (light-quantom) or the two Theories of Relavity, nobody took any notice of him outside scientific circles. But when the above-mentioned verification was made, he caught the imagination of the people and there was a general desire to know him and his discoveries. I distinctly remember how all sorts of things (not excluding nonsense) were talked about the Theory of Relativity, and how meaningless comparisons were made between Einstein and Newton, of course to the disadvantage of the latter.

Einstein made another important prediction form his Theory. He showed that the natural period of vibration of a particle is shorter on a light body than on a heavy body. The relation is given by the equation (1-2m1/r1) [t 2/1=](1-2m2/r2) [t 2/2], where [m1, r1 and t1] refer to one body (its gravitational mass and radius and the time of vibration on it) and [m2, r2 and t2] refer to another. If we suppose the left-hand member to the sun, as [m1/r1] is negligible in comparison with [m2/r2], and as [m2/r2=.000002], we can say t2 (the period on the sun)=[.000002 t1]. So the frequency of vibration is greater for the terrestrial particle than for the solar particle. If the spectral lines of a terrestrial atom, say an atom of hydrogen, are compared with the same lines in the solar spectrum, we must find the lines in the solar spectrum shifted by a small amount towards the red-end. But the shift is so small (for the HB line the increase in wave-length is only .01A) that it cannot be verified as it is masked by errors of observation. There are certain stars called white-dwarfs in which, owing to peculiar of space. One such white-dwarf is the companion of Sirius in the constellation, Canis Major. Examination of the stellar spectrum of this dense body has established the truth of this prediction, thus proving the Theory of Relativity.

In the discussion above we took the equations arising from the Theory of Relativity for granted and compared them with the corresponding equations of classical (also called Newtonian) mechanics. We also tried to understand their implications which gave a shock to our accepted ideas. Now we shall derive the equations used above from the hypotheses of the Theory of Relativity. We shall first take the Restriced Theory. The two hypotheses of that theory are :(1) The velocity of light is constant as measured by all observers, whether they are relatively at rest or in motion with uniform velocity as regards one another. (2) Of two men having uniform relative motion, if one asserts that he is at rest and the other is in motion, and if the other asserts that it is he that is at rest, there is no means of determining who is right.

Let us take two men, A and B, having such relative motion. A thinks, naturally, that he is at rest and B is in motion, say with a velocity u. For the sake of convenience let us take as out unit of length the distance light travels in one second. Supoose B phones to A, "I sent at the same instant two beams of light, one along the line of out relative motion and the other at right angles to it. They were reflected back to the source of light by two mirrors placed at a distance, d, from the source. I found that they reached the source at the same instant, i.e., 2d seconds after they started." A says to himself, "How is it possible? I made the same experiment and got the same result. But B is in motion and so he must get a different result. Evidently there is something wrong with his measurement of time or measurement of length or possibly both. Let me find out by calculation the amount of his error once for all, so that in future I can correct his results for agreement with mine." Let me first take the reflection at right angles to his line of motion.

Fig. 1.

Though he had placed his mirror on a line at right angles to his motion (see fig.1), it was the oblique ray of light, marked by the arrow that was reflected by the mirror, (though he was not aware of it), for by the time it reached the mirror, the mirrror would have moved along with B. As light travels one unit in one second, and as in one second B moves a distance u, the light will approach the mirror a distance [*/1-u2] in one second. As the distance of the mirror is d, the time taken to reach it would be [d*/1-u2] seconds. As the same argument holds for the return journey to the source, the total time must be [2d/*/1-u2]. Evidently his clock is slow and gives a shorter interval of time, 2d. So let me hereafter divide the time given by him by (*/1-u2), so that, his result may agree with my 'correct' result.

"Then, for the beam of light along his line of motion. (See fig.2). In one second light travels one unit and is at P. But the source S has moved a distance u and is at S'. The mirror M has moved a distance u and is at M'. Thus in one second the light nears the mirror by 1-u units of length. So the mirror must have been reached in d/(1-u) seconds. During the return journey the light approaches the source at one unit per second while the source approaches the light at u units per second. So the distance is lessened 1+u units every second. Therefore the distance, d, from the mirror to the source is done in d/(1+u) seconds. Adding the two we have 2d/(1-u2), the time from the source to the mirror and back. But the fellow says it is 2d. As his clock is slow I divide 2d by [*/1-u2], as I have already resolved to do. But it makes [2d /*/1-u2] but not 2d/(1-u2) which I have found to be the correct time. Evidently there is something wrong with his measurement of lengths [along the line of motion]. It must have shortened. I must divide his lengths [along the line of motion by[*/1-u2] so that I may get the correct time 2d/(1-u2)."

Fig. 2 .

So says A, and we appreciate his standpoint. BUt we must remember that the second hypothesis allows B to claim he is stationary and A is moving away from him with a velocity, -u. That is no special privilege of A 's. Arguing exactly like A, but putting -u in the place of u in A's argument, B comes to the conclusion that it is A 's clock that is slow and ... he must divide A's time by [*/1-u2 (for */1-(-u)2=*/1-u2)] to get his own correct measure of time. In the same way, he says it is A's length that has contracted and divides that too by [*/1-u2] to obtain agreement with his own correct measurement of time. These two results are fundamental in the Theory of Relativity.

Beaing these two results in mind let us proceed to derive the formulae used earlier. Suppose between two events B measures a time interval, t', and a space interval along the line of motion, x'. What space interval will A find between the same two events? As marked in fig. 3, let B measure the first event to be distance K from his origin, 0. Let him measure the second

Fig. 3

event to be at a distance L from the second position of his origin )'. (For the origin has shifted according to A). B thinks that his frame of reference is stationary. So he does not distinguish between 0 and 0', and measures x'=L-K. But A says that at the time of the second event B has moved a distance u t or which is the same to him, u t'* [*/1-u2]. B's measurement of length x' itself should be divided by [*/1-u2], according to A, to compensate for the contraction of B's lengths along the line of motion. Making these two correction we find A's measure of the distance between the events,

[X=x'/*/1-u2 + ut'/*/1-u2=x'+ut'/*/1-u2............(1)]

But the argument being reciprocal, as we have seen, if A measures a distance X, and a time interval t, between two events, B will calculate his distacne, x'=

[x-ut/*/1-u2]. (Note: for B, u is negative)...............(2).

Eliminating x between the above two equations, we get an expression for the interval in time between the two events, as measured by A, (expressed in B's

measurements, t' and x'), t=t' +ux'/*/1-u2]............(3).

Eliminating X' between equations (1) and (2) we get B's measurement of time in terms of A's measurement of

time and spece, viz, [t'=t-ux/*/1-u2]...........(4).

We can also derive (4) from (3) directly by substituting-u for u, and t, x, for t',x', which the second hypothesis permits us to do.

If u is small, so that we may neglect u2, equation (1) and (2) reduce to, [x=x'+ut', x'=x-ut, respectively. This is the same as the equations of Classical (or Newtonian) relativity, x=x'+ut, x'=x-ut, if we bear in mind that in Classical Relativity t=t', i.e., there is no difference in the measurement of time of two people in relative motion. If we neglect u2 in equations (3) and (4), they reduce to t=t'+ux', t'=t-ux. There is nothing equivalent to this in Classical Relativity, according to which these equations are absurd. Only in the case when u=0, these equations acquire a meaning for Classical Relativity, viz., t=t'. But then there is no relative motion and the equations are outside the pale of the Theory of Relativity.

The four equations (1) to (4) show that two persons in motion relative to each other do not have the same measurement either of distance or of time The implications of this fact have already been discussed earlier. We have seen there that though space and time are different for each, the space-time interval between two events is the same for all. Let us call thsi interval

'separation' or s. We have seen, [s2=t2-x2=t12-x12].

Suppose B reports a velocity, v1, of a body. With what velocity does the body move as observed by A? Velocity is defined as displacement (i.e. distance travelled) per unit time. For A, v=x/t=(x1+ut1/*/1-u2)/

(t1+ux1/*/1-u2)=(x1+ut1)/(t1+ux1)=(x1/t1+u)/(1+ux1/t1)= (v1+u)/(1+uv1)..........................(5).

It may be remembered this is the equation given earlier for the composition of velocities, c2 having disappeared from the formula, because we have taken c as the unit of length. Thus we see that all teh results mentioned there follow from the Theroy of Relativity.

We can, in the same way, derive the velocity [v1] fo rB if we know v, the velocity for A. Substituting -u for u, on the strength of the Relativity hypothesis, in equation (5) and v for v1 and vice versa, we get v1= (v-u)/(1-uv).................(6)

Oedinarily u and v are very small, for our unit of length is the enormous distance that light travels in vacuum in one second. If we therefore neglect the product of small quantities uv or uv1, the equations (5) and (6) reduce to v=v1+u and v1=v-u, respectively, which are the equations of the compostion of velocities according to Classical Relativity. Also, we have seen earlier how Fresnel's Convection Co-efficient can be derived from equation (5).

There are three conservation laws that are fundamental in Newtonian or Classical mechanics:(a) the Law of conservation of Mass, (b) the Law of Conservation of Momemtum and (c) the Law of Conservation of Energy. Most of our work in Physics and Chemistry is based on these laws. So we should examine how far these laws are affected by the Theory of Relativity, especially by the new equation for the composition of velocities.

The Law of Conservation of Mass states that as long as no matter (which is identified with mass) is added to or taken away from a body or a number of bodies, the total mass remains the same. For example, if two molecules of masses m1 and m2 unite to form a new molecule, the mass of the new molecule is m1+m2.

The Law of Conservation of Momentum states that if a number of bodies are in motion and if no external force, whichh alone is capable of adding new momentum, acts on the system of bodies, the total momentum remains constant, though the momentum of the individual bodies may change by their interaction or collision. Let us take, for example, two masses m1 and m2 with velocities v1 and v2, respectively. Momentum=massxvelocity. The law affirms that m1 v1+m2 v2 is constant, under the given condition, though v1 and v2 may change by their collision.

The Law of Conservation of energy states that energy can neither created nor destroyed. There are several form s of energy, kinetic energy, potential energy, heat energy, chemical energy, electrical energy and energy of radiation. The law says that one form of energy may be transformed into another form, but it cannot be destroyed or created. For example, if there are two masses, of velocities, v1 and v2 respectively, their total kinetic energy is(1/2 mv+1/2mv). By the impact between the two bodies, part of the energy may be converted into heat, but the remaining kinetic energy plus the energy in the form of heat is equal to the original kinetic energy.

Let us take the conservation of momentum, first. It is easy to see that according to Classical Relativity momentum is conserved. If B observes velocities [v1 and v2], respectively, A will observe velocities [v1+u] and [v2+u]. If B findsa total momentum [m1+v1+m2 v2], A finds a total momentum, (m1 v1+ m2v2)+u(m1+m2). As (m1 v1 +m2 v2) is a constant, by the law of conservation of mass and u is a constant, A's total momentum too is conserved, though it may have a different value from B's.

We shall now examine whether momentum is conserved if in the place of u+v we use the Relativity formula for the composition of velocities,[u+v/1+uv].

The total momentum for A is :-

(m1(u+v1)/1+uv1+m2(u+v2)/1+uv2=U(m1/1+uv1+m2/1+uv2)+(m1v1/1+uv1+m2 v2/1+uv2)

In this expression although u1 (m1+m2) and (m1 v1 + m2 v2) are constant; the value of A's expression for total momentum will not in general be constant, because v1 and v2 may vary in all sorts of ways. A numerical example will make this clear. Let m1 and m2 be 2 and 5 units respectively and v1 and v2 be .3 and .4 units. The total momentum of rB is 2x.3 +5x.4=2.6 units. Now let v1 become .8 units after an impact. As the total momentum fo rB should still be 2.6 units, v2 must be (2.6-2x.8)/5=.2 units. The total momentum for A under the first condition is, if u=.6 units,

Under the second condition it is

The total momentum has changed. The Law of conservation of momentum does not hold fpr A. But it must hold, for he too, like B, has discovered the law from the experiments. So there must be a mistake either in the definition of mass or in the definiation of momentum. The fact that in Theory of Relativity space and time have merged to form a single entity, 'separation', suggests the following definition, mass X velocity, i.e., momentum=mXx/s, wheres2=t2-x2. Then s2/t2=(t2-x2)/t2=1-x2=1-v2. Therefore

t/s=*/1/1-v2. Therefore[ mXx.s=mX x/t X t/s =mv/ */1-v2].

This new formula for momentum is tantamount to taking the old formula m v with the mass defined as [m/*/1-v2]. i.e., taking the mass as varying with the velocity in the specified manner.

Let us now see if the new definition of mass saves for A his Law of conservation of momentum. The total momentum for B is [m1 v1/*/1-v+m v 1-v], and this is conserved. The total momentum for A is{m(u+v/1+uv)/1-u+v/1+uv)2+{m(u+v/1+uv)/ 1-(u+v/1+uv)2, which reduces to 1[1-u]2 {m1[1-v]+

(m/1-v)+ u (m v/1-v)+ m v/1-v)}. Here, as u and 1/(1-u)

are constants, as loong as (m/1-v)+m/1-v) and

(m v/1-v1+m2 v2/1-v) are constant, the whole expression

will be constant. Now (m/1-v)+m2/1-v) is the total

mass according to B and (m1v1/1-v2+m2v2/1-v2) is the total

momentum. They are both conserved, and therefore constant. So the total momentum according to A is also conserved. From this discussion we conclude that though two persons in relative motion will not find the momentum equal, yet if it is conserved according to one, it is conserved according to the other as well.

We shall now proceed to examine how the new definition of mass affects the conservation of mass for

A. The total mass for A is m1/(1-u+v/1+uv)2+m2/

(1-u+v/1+uv)2 which reduces to

1/1-u2{U (m1/1-v1+m2/1-v2)+(m1v1/1-v2+m2v2/1-v2}

i.e., u times B's total mass is added to his total mementum and the whole is divided by (1-u2) to get the total mass for A. As B's total mass and momentum are conserved and as u is constant, the total mass accroding to A is conserved, though A does not agree with B about the actual value. It is interestingto note that mass and momentum occur together in both the abvoe formulae showing an underlying unity. Of course this is expected, for when B says he is at rest and measures only mass, A says B is in motion and measure momentum as well. We thus see that a new definition of mass has been forced upon us by the Theory of Relativity. m/(1-v2=m) when v=o, i.e., when the body is at rest, the mass is m. Let us call this the rest mass. As v is usually small, m/(1-v2=m+1/2 mv2], neglecting v4 and higher powers. The additional mass acquired by motion is 1/2 mv. Now the kinetic energy of a body due to its motion is 1/2 mv. This suggests taht the additional mass is due to the kinetic energy. This leads us to conclude that the proper mass, m, itself may be due to the energy inherent in the body. We have seen earlier taht E=mc, (where if m is in grams, and c in centimetres per second, E is the velocity of light). E is in ergs. (For a full discussion see p.417 ff.). Thus as all mass is due to energy, conservation of mass turns out to be merely conservation of energy. In this way, all the three conservation laws are saved for us and in addition to this, our investigation has unmasked the close relation between them.

COMPUTATION OF THE SOLAR ECLIPSE IN HINDU ASTRONOMY*

I. INTRODUCTION

It is a well known fact that a solar eclipse occurs, when in the neighbourhood of a moon's node the sun and the moon are in conjuction. The computation of the solar eclipse for the world in general without reference to any given place is like that of the lunar eclipse. Let T be the Indian Standard Time of conjuenction in longitude. P the latitude of the moon, P the hourly change in latitude, (north latitude, and motion towards the north being considered postive), M the excess of the hourly motion of the motion of the moon in longitude over that of the Sun, L the angular radius of the moon, and S, the angular radius of the Sun. Then at any time t hourly ater conjunction, the distance between the sun and the moon's longitude is Mt and the moon's latitude is P+Pt. So the distance their centres is [{M2t2+(p+Pt)2}1/2] . The eclipse begins or ends when their rims appear to touch. This can happen even if the distnce between them is greater than L+S, for the moon's parallax may push it towards the sun. The maximum of this effect is [@-@'(=@)],(@) being the equatorial horizonal parallax of the moon,(@') that of the sun. Thus the rims can appear to touch when the distance between the centres is (@+L+S(=d) at the most. Then {M2t2+(p+Pt)2=d2} gives the times of the beginning and end of the general eclipse. Solving for t, we get, t+-pP/(M2+P2)+{p2 p2/(M2 +p2) +d2-p2]/(M2+p2)}1/2, in which , obviously, and the lower for the end. T+t is the I.S.T. of the beginning or end.

At any given place the eclipse begins or ends when the rims appear to touch at that place, i.e. when the (apparent) distance between the centres is L+S. Now at any time T (I.S.T.) near the times of conjunction in longitude, let the apparent distance in longitude between the cnetres be m, the apparent excess of the moon's hourly motion in longitude over that of the sun M, the apparent difference in latitude p, the apparent excess of the moon's hourly motion in latitude over that of the sun P, the sum of the apparent angular radii of the sun and the moon d, and its variation per hour D [By, 'apparent]' is meant here 'as affected by parallax.' apparent m=real m+(@)cos A cos Bx(1+(@) cos A sin B)>

Apparent p+(real P+(@)sin A) (1+(@)cos A sin B). Apparent (L+S)=S+L(1+(@) cos A sin B), where A is the zenith distance of the [Nonagesimal], given by sin A=sin w cos 0 sin v- cos w sin0, and B is the Orient ecliptic point minus the longitude of the moon.

The orident ecliptic point =Tan {tan 1/2 (90+v) cons 1/2 (90o+0-w)/cos 1/2 (90o +0+w)}+ Tan {tan 1/2 (90o+v) sin 1/2 90o+0-w)/sin1/2(90o+0+w)}, where 0 is the latitude of the place, w is the obliquity of the ecliptic and v is the sideral time in degrees at teh moment, given by, v+97o 30+ the [Greenwich] East longitude of the place + the mean longitude of the sun + the I.S.T. at that moment in degrees. For strict accuracy, the geocentric latitude and the horizontal parallax at taht latitude should be used.

If T is the time for which we have found m, P and d, the apparent distance between the centres of the sun and the moon at any time t hours after T is {(m+Mt)2 +(p+Pt)2}1/2. When this is equal to d+Dt, the eclipse begins or ends. Thus it begins or ends at T+(dD-mM-pP)/(M2+P2)+{mM+pP-dD)/(M2+P2) +d2-p2-m2}/(M2+P2)]1/2 3. The middle of the eclipse, i.e the maximum eclipse, occurs at T+(dD-nM-pP)/(M2+P2).

The total eclipse begins or ends when the rims apparently touch, the sun being within the moon. the distance between them at such a time is (L-S). So by substituting for d in the above formula another d equal to (L-S), we can find teh times of the beginning and end of the total phase. Another thing should be ntoed here. S may be greater than L, so that the moon may be immersed in the sun, leaving a circle of light all round. This is called an annular eclipse. Obviously, the beginning or end of the annular phase is got by making D=S-L.

For e.g., let us compute the solar eclipse that occurred on 9-5-1948, for the world in general, and for [Pudukkottai](78o4' 17" E L and 10o 18' 51" N. Geocentric Latiotude) given : time of conjunction in longitude T=7-59.9(I.S.T.), True longitude of the moon=apparent long. of the sun =48o 22' 24". P=+1425". (@)=3439". (@)'=8.7", M+937", S=950".

Hourly change is Sun's longitude +145",

" " Moon's longitude +1992"

M=1847", Hourly change in p=P=+183.5".

Hourly change in L=+.4".

From these, for the general eclipse, d=(@-@')+d+dS=5318". Substituting these in the formula for the general eclipse we have 7h 59.9m-4.6m-2 45.6m=5h 9.7m for the beginning, 7h 59.9m-4.6m+2 45.6m +10h 40m 9m for the end.

As at Pudukottai the sun will be seen already eclipsed when rising, we shall compute the middle and end alone for that place.

T.(i.e the time near

middle or end) 6h 19,7n 6h 31.7m 7h 19.5m 7h 31,5m

Real m.

Parallax in m -3082" -2713 -1242" -873"

+3085 +3091 +3027 +2990

Apparent m. +3 +378 +1785 +2117

Real P. +1119 +1155 +1302 +1338

Parallax in P. -1462 -1407 -1172 -1107

Apparent p. -343 -252 -130 +231

Real d +1886.5 +1886.5 +1887 +1887

Parallax correction +1.5 +2.5 +5 +6

apparent d. +1888 +1889 +1892 +1893

Taking 6h 10.7m as T for the middle, m=+3", M=+375", P=-343", P=+91", d=+1888". D=+1', from which we find the middle occurs at 6h 19.7m +2.6m= 6h 22 3m.

Taking 7h 19.5m as T for the end, m =+1785", M ++332" p++130", P=+101", d=+1892", D=+1", from which we find the end is at 7h 19.5m-1h 0.1m+1h 3.7m=7 23.1.

THE COMPUTATION IN HINDU ASTRONOMY

(A) According to the Paulisa Siddhanta

Though there is mention made of the elcipse in Vedic literature, the ancient astronomical works like the [Vedanga Jyotisha], [garga Samhita], [Paitamaha Sid], etc., do not give methods for its calculation. The Paulisa Sid. seems to be the most ancient work dealing with the solar eclipse. We shall here give its method as condensed by [Varahamihira] in his [Panchasiddhantika].

1. Find the time of conjunction in longitude.

2. Find the time of noon.

3. Find the interval between conjunction and noon.

4. Find the hour angle of the Sun at conj., in degrees, calculating at the rate of 6o for one nadika or 24 minutes.

5. Multiply the sine of the hour angle by 4 nadika or 96 minutes.

6. If the conj, is before noon deduct the time got, from the time of conj. If it is afternoon, add. This is the time of the middle of the eclipse.

7. Find the longitude of the moon and the mean longitude of Rahu at this time.

8. Make the following corrections in Rahu.

a) Multiply the degrees of latitude of the place by 5 and divide by 27. Add the resulting degrees to Rahu if it is the ascending node. If not subtract.

b) Find the declination in degrees of the point of the ecliptic 90o from the Sun. Multiply this by the result of (5) in nadikas, and divide by 22. The resulting degrees are to be added to the ascending node if the sun's longitude lies between 270o and 90o, and the time of conj. is forenoon, or if the sun's longitude lies between 90o, and 270o, and the time of conj. is at the descending node, the opposite of this is to be done, i.e., addition should take the place of subtraction and vice versa.

c) Take the time in nadikas of conj. elapsed from sunrise in the morning or to elapse for sunset in the evening. Multiply this by the degrees of declination of the moon and divide by 80. If conj. is at the ascending node the resulting degrees are to be added if the moon is between 180o and 360o, and subtracted if between 0o and 180o. For conj. at the descending node, subtract and add respectively.

9) Find the difference in longitude between this corrected Rahu and the moon. If it is less than 8o, there is a solar eclipse.

10) Square the difference in degrees and subtract from 64. Take the square root. This multiplied by 18 is the duration of the eclipse in minutes.

11). Half the duration subtracted from the middle is the beginning, and added in the end.

Let us compute the eclipse of 9-5-1948 according to these rules, using the elements given already; and also given Rahu (here the ascending node)=44o, time of noon 12h 11m and the time of sunrise 5h 58m.

1. The time of conj., is 2h 2m after sunrise.

2. Noon is 6h 13m from sunrise

3. Conj,is 4h 11m before noon.

4. The hour angle of the sun at conj., is 62o 48'.

5. Sin 62o 48'X96m=86m or 3.57 nadikas.

6. This deducted from 8h gives 6h 34m (I.S.T.) for the middle.

7. The longitude of Rahu is 44o.

8. a) The geographical lat. of Pudukkottai is 10o4. This X 5/27=1o9. This is to be added to Rahu which becomes 45o.9.

b) 15o 7X3. 57/22= 2o.5. As Rahu is the asc. Node, the sun lies between 270o and 90o and conj. is before noon, this is to be added, making Rahu 48o.4.

c) The time conj. from sunrise is 5.1 nadikas. The declination of the moon is 18o.8 This X 5.1/80=1.1. As the conj., is at the asc. node, and the moon is between 0o and 180o, this is subtractive. The corrected Rahu is 47o.3.

9. The difference of this and the moon is 1o1. As it is less than 8o., there is a solar eclipse.

10. [18X*/64-1.1=143m =2h 23m.

11. 6h 34m+1h 11m=5h 23m and 7h 45m are the beginning and end.

(B) According to the Romaka Siddhanta

The method of the Romaka is more like the modern one, and more accurate. The follwing is [Varahamihira's] condensation of it.

1-7. The same as in the Paulisa.

8. At conj., as affected by parallax, (C.A.P.), find the orident ecliptic point. Deduct 90o from this. This is the Nonagesimal.

9. Find the declination of the Nonag. The declination is North if the Nonag. lies between 0o and 180o, and South otherwise.

10. Deduct the longitude of the asc. node from this point and multiply its sine by 280'. This is North if the result of the subtraction is 0o to 180o, and South otherwise.

11. If the result of (9) and (10) are both north, add, and call it North. If they are both South, add, and call it South. If one is North and the other South, deduct one from the other and callit by the direction of the larger.

12. If the result of (11) is North and greater than the latitide of the place, deduct the latitude and call it North. If it is North and less, subtractit from the latitude and call it South. If it is South, add it to the latitude and call it South.

13. Multiply the sine of this by the daily motion of the moon in minutes and divide by 15. The result is the parallx in latitude. It has the same direction as (12).

14. Find the latitude of the moon by multiplying the sine of the difference between the moon and Rahu by 280'. If the moon is greater near the asc. node, or less near the desc. node, it is North. Otherwise it is South. (The moon at C.A.P. is to be used for this ).

15. Add the latitide and the parallax in latitude if they have the same direction. If not, subtract one from the other. This is P in minutes.

16. The mean angular radius of the sun is 15', and that of the moon 17'. The radious of each multiplied by its true motion and divided by its mean motion gives the true radius. Let us call them S and L.

17.[(L+S)2-p2% M gives the half duration of the eclipse in hours if M is the difference between the true motions in minutes of the sun and the moon, perhour.

Applying these rules to the given example, we have:

1-7. The same as for the Paulisa.

8. At C.A.P., 6h 34m, the Orient Ecliptic point for Pudukkottai can be found to be 57o 44'.4. This -90o=327o44'.4=Nonagesimal.

9. The declination of this point is 12o 16'.S.

10. 327o44'.4-44o=283o 44'.4. The sine of this is .971. .971X280'=272'.S.+4o32'.S.

11. 12o 16'.S. +4o 32'. S.=16o 48'. S.

12. The Geographical lat. of Pudukkottai is 10o 24'. 16o 48'+10o24'=27o12'.S.

13. The daily motion of the moon=787'. Sine 27o 12' 797'/15=24'.3.S.=parallax in latitude.

14. The moon at C.A.P. is 47o 35'. The difference between this and Rahu=3o 35'. 280' sin 3o 35'=17'.5.N.

15. 24'.3-17'.5=6'.8=p.

16. 15'X58 /59.2=14'.7.=S.17'X797/790.6= 17'.1=L.

17. M=(797'-58')/24=30'.79.[31.8-6 8/30.79]= Ih=half duration. The middle being 6 h 34m, the eclipse begins at 5h 34m and ends at 7h 34m.

The correct times of the middle and end, as we have seen, are 6h 22m and 7h 23m. The following are the inaccuracies in this method. The maximum mean parallax is assumed to be 53' instead of the correct 57'. The max. mean latitude of the moon is taken to be 280'. The declination of the Nonagesimal minus the latitude of the place is taken to be the zenith distance of the Nonagesimal. The formula for the correction of this declination (10) is wrong. We are not instructed that the times obtained are only approximate, and that for a closer approximation, the claculation should be repeated taking the latitude and parallaxes to the first approximate times.

(c) According to teh Later Siddhantas

The method of the Surya Siddhanta condensed by Varahamihira in the Pandchasiddhantika is typical of that of the later Siddhantas, and fairly correct. Only in their constants do these differ from one another. I shall here give the method, and the constants fo Varaha's Sur. Sid., teh New Sur, sid. and the Siddhanta Siromanai of Bhaskaracharya, in order. The reader will do well to compare this method with the modern one, given in the first part.

1. Directly or indirectly each Siddhanta gives the mean angular diameter of the sun and the moon. For the Sun it is 32'.1, 32'.4 and 32'.5 (=2S), and for the moon 32'.2, 32'.0 and 32'.0(=2L). These are to be made true by being multiplied by the true motion and divided by the mean motion. (The correct mean angular diameters of the sun and the moon respectively are 32'.0 and 31'.3, and it will be better to muliply by half the sum of the true and mean motions).

2. The mean horizontal parallax of the moon i s51'.6, 57'.5 and 52'.9, and of teh sun 3'.8, 4'.3 and 3'.9. These too are to be made true by being multiplied by the true motion and divided by the mean motion. (The correct figures are 57'5" and 0'.1, respectively). Let their difference be written (@).

3. For any time near enough to the event sough the zenith distance of the Nonagesimal, A, is found. The orident ecliptic point or Lagna is found from which the Lagna minus the longitude of the moon at the time, B, is found. (Though different methods are given for this, they ultimately reduce to what we have given in PartI.)

4. (@) cos A cos B is the parallax in longitude. Let M be the difference between the true motions of the Sun and the moon per hout. (@) cos A cos B/M are the hours to be deducted from the time of conj. to get the conj. as affected by parallax (C.A.P.). (Note that for B greater than 90o the hours will be negative and so C.A.P. will be later.)

5. Using the A and the B of the C.A.P got, (4) is to be repeated which will give a more correct C.A.P. If repeated once again, the process will give a still more correct vlaue. (The explanation of this method of successive approximations is this. To find the time of any event the parallax etc. at that time, which themseles depends upon the time, are required. To get over this difficulty successive approx. is resorted to.) This time is the time of the maximum ecli9pse, technically called the Middle. (Really the Middle will occur within a few minutes of this ).

6. Find p, the latitude of the moon at the last but one C.A.P., because it will be found convenient if we take the last but one instead of the last. P=270' sin (Moon's longitude the asc. node).

7. P+(@) sin A, is the latitude as affected by parallax where A pertains to the last but one C.A.P. and is already found.

9. Deduct this from the last C.A.P. This is the provisional beginning. Added, it is the provisional end.

10. Take this time of beginning and find A,B and p. Find P+(@) sin A. Repeat (8) which gives the half duration in hours. Using the A and B here, find the C.A.P. Deduct the half duration from this. This is the first approximate time of beginning.

11. Take the provisional time of ending and do everything mentioned in (10) up to finding the C.A.P. Add the half duration to the C.A.P. This is the first approximate time of ending.

12. If in the place of the provisional times the first approx. times are used and (10) and (11) repeated, the second approx. times are got, and so on. (But going beyond in the constants).

We shall compute the solar eclipse of 9-5-1948, according to thsi method, for Pudukkottai, using the constant of the New Surya Siddhanta.

1. The true diameter of the sun =32'.4X58.59.1

=31'8.

The true diameter of the moon=32'.0 X 797/

790=32.3.

2. The true horizontal parallax of the

sun =4'.3 X 58/59.1=4'.2.

" " moon =57'.5 X 797.790.6=

58'(@) = 58'-4'.2= 53'. 8.

3. Taking the time of conj. v = 343 o7', cos A=.9606

and B=30o 10'.

4. (@) cos A cos B=44'.7. M =30'.79. 44.7/30.79=

I h 27 m deductive. So C. A.P.=6-33 as a first Approx.

5. v for 6-33 is 321o22, cos A =.9126 and B=9.45' (@) cos A cos B=48.4. This divided by M gives I h 35m. Deducting this from conj., the approx. time of C.A.P. is 6-25. This is the approx time of the middle. We stop here.

6. The last but one C.A.P.is 6-33. The lat. of the moon at thsi time is 270' sin (47o 35'-44o)=+16'9.

7. Sin A for this time is -.4088. (@) sin A=-22.0. p+this =-5'.1.

8. L+S=(32'.3+31'.8/2=32'.05.[32'.05-5'.1]/

9. The provisional beginning is 5-23, and teh provisional end is 7-27.

10. We shall compute the time of the end alone. v=334o52', Lagna=70o 40', longitude of the moon=48o4', B=22o 36', log sin A=(-) T. 5191, and log cos A=T.9749. P=+19'.14 (@) sin A =-17'78.p+(@) sin A=-1.36.[32'05-1.36/30.79=1h 2m, half duration. (@) cos A cos B =46.88. 46.88.30 79=1h 31m. The new C.A.P. =8h-1h 31m =-6-29. This + the half duration =7-31, the first Approx. time of the end of the elcipse. The second approx. time should be found in the same way.

The method of the later Karanas or manuals are based onthe Siddhantas, adn they generally sacrifice accuracy for ease of computation. There is nothing new in them.

HINDU ASTRONOMY THROUGH THE AGES- A SHORT SKETCH*

1. The Vedic Period

The history of Hindu Astronomy goes back to a very ancient period. Evidence of the astronomical knowledge of the Brahmans is found in the Vedas. Even in the age of the Mantras it was known that the Moon returns to its position among the stars once in 27 days. Each day it was observed to be in conjunction with a single star or group of stars, and the day was designated by taht asterism. Thus arose the later division of the ecliptic into 27 asterismal segments. They knew that once in about 29 1/2 days the Moon is in conjunction with the Sun adn this period they used as teh measure of their month. (The word mad means (i) the month,(ii) the Moon, and (iii) a measure.) They knew that the solar year marked by the cycle oc the seasons consists of 365 days and that this is in excess of 12 lunar months by about 11 days. The months were named after the asterisms at or near which the moon became full, like [Phalguna], [Chaitra] etc. The shortest days were noted to be whenthe noon-sun was low down in the sky at Winter Solstice (W.S.) and the days were noted to become longer as the Summer Solstice approached. The year began with the first day of the light were fortnight of the month [Phalguna] near the W.S. During the age of the [Brahmanas] it was observed that the Pleiades (Krittikas) rose due east, and this fact was used in the orientation of the sacrificial halls. (From this we can compute the age of the observation to be about 3000 B.C.) Professional star gazers called [nakshatra-dar'sa-s] are mentioned and in the [Chandogya] Upanishad a lore of stars, called [Nakshatra-vidya], is mentioned. The Planets Jupiter and Venus were known. The Atri family was credited with the knowledge of eclipses.

2. The Immediate Post-Vedic Period

The astronomical knowledge of this period is found the [Vedanga Jyotisha](V.J) of Lagadha whcih is the most ancient Hindu astronomical work extant. This deals with the computation of the ending moments of the [Tithi] and the [Nakshatra], the [Sun's] [Nakshatra] etc., are mean,as distinuished from the true [Tithis] etc. given by the modern almanacs. The elements from which the computations were made are the following: In each period of 1830 days called a yuga, it is taken there are 5 solar years, 60 solar months, 62 lunar (synodic) months, 1860 lunar [Tithis], 1809 lunar Nakshatras and 135 solar Nakshatras. Supposing the Sun and the moon have only mean motions, (i.e., move uniformly on the ecliptic), the computations are made. This would give only the approximate [Tithis] etc., but true [Tithis] etc, were fixed by observation. The method of computation is peculiar and very interesting .

The V.J. says that the winter solstice began with the sun at the asterism [Sravishtha], from which we can calculate the date of the work to be about 1200 B.C. Close upon the V.J. followed several other works, the Garga [Samhita], the [Paitamaha] [Siddhanta], the [Surya][Prajnapti], the [Jyotishakaranda], the [Kalaloka] [Prakasa] etc., all of which dealt only with the mean motions of the Sun and the Moon, like the V.J. We do not find anything very much special or noteworthy in them.

3. The Period of Transition to the Regular Siddhantas (Circa 100 B.C. to Circa 300 A.D.)

This period is represented by the [Vasistha], [Pauli'sa] and Romaka [Siddhantas] condensed by [Varahamihira] in his [Pancha-Siddhantika]. In the works of this period the influence of Greek culture on Hindu astronomy, especially astrology, is visible for the first time. The names of week-days like [Ravi-vasara], [Indu-vasara] etc., and the names of the 12 solar signs composing the Zodiac, like Mesha, Rishabha etc., occur for the first time now. Many scholars are of opinion that these originated in Babylonia and reached India via the Greeks .

The [Vasistha] seems to be the oldest of the three. In it we find, for the first time, methods for computing taken by the Sun to traverse each [Rasi] of 30o has been found empirically and given. An ingenious formula for the Moon's Equation of the centre, on the assumption that it increases or decreases uniformly, has been given by using the summation of series, the maximum being 347'. A method for finding the day-time from the shadow is given, and another for computing the lunar eclipse. Herein also are the planets dealt with for the first time.

The [Pauli'sa Siddhanta] closely follows the [Vasishtha]. The name, as also the fact that Alexandria is taken to be the 0o longitude, shows that it must have been affected by Greek influences. In this we find a rough method for the computation of the Solar eclipse.

The Romaka [Siddhanta] uses a tropical year of 365.2466 days in the placeof Sidereal year used by the other [Siddhantas]. This, the name Romaka, and the fact that it gives 143' as the maximum equation of the centre for the Sun, and 296' for the Moon, all show its Greek orgin. For the first time we find the Solar eclipse treated in a scientific manner in this work.

A host of astrological works, many of them claiming Greeks as their [Purvacharyas], seem to have been written in this period, followed later by Hindu authors with zest. It is for this concoction, this bane of our culture, that we are indebted to the Greeks in a large measure. In astronomy proper there seems to have been very little 'borrowing '.

4. The Period of the Regular Siddhantas (300 A.D. to 1200 A.D.)

A succession of astronomical works like the Old [Surya Siddhanta] condensed by [Varahamhira], the [Aryabhatiya] of [Aryabhata], the [Brahma-Sphura-Siddhanta] of [Brahmagupta], the New [Surya-Siddhanta] etc, down to the [Siddhanta-'Siromani] of [Bhaskaracharya] mark this period of the heyday of Hindu astronomy. The following are some of the salient features of these works.

(a)The earth is a sphere of diameter 8000 miles, poised in space with nothing for its support. [Bhaskara], in a beautiful discussion, refutes arguments advanced against this idea. For instance the Puranic cosmogony is that the earth is supported by eight elephants which are supported by the Great Tortoise, which, in turn, stands on the Great Serpent Adi-'Sesha. In order to avoid the fallacy of infinite regress the Puranas say 'Sesha supports himself. "Then why not the earth support itself" asks [Bhaskara]; "If it is also divine, why go so far as ['Sesha] and stop there?" But the layman's question "Will not the earth fall?" Should be answered. He says, "This question arises from our experience that things fall towards the earth, attracted by it. So falling means moving towards the earth and so the question of the earth falling does not arise at all".

(b) The stellar sphere tevolves round the earth once in 24 hours (exactly in 23 hrs. 56 minutes). There were some astronomers like [Aryabhata] who asserted thus. The verse is +xÉÖ±ÉÉä¨É MÉÊiÉxÉêºlÉ& {ɶªÉiªÉSɱÉÆ etc. (MÉÉä±É{ÉÉnù 9.). It is made out by commentators of [Aryabhata] that this is not so. The next 'sloka gives the usual theory. Are teh commentators correct or is the next 'sloka an interpolation? It has been tampered with changing ¦ÉÖ& into ¦É¨É, that this revolution is only apparent, and is due to the rotation of the earth. But the generality of astronomers from [Varahamihira] to [Bhaskara] argued that this rotation theory could not be accepted because "then birds leaving their nests cannot return to their nests again, and flags will fly ever pointing westwards on account of the eastward rotation of the earth". We now know that these arguments exhibit an ignorance of the laws of Kinematics, and that there is a direct proof of the earth's rotation in the behaviour of the 'Focault's Pendulum. But in those days these arguments were considered unanswerable.

(c) The Sun moves round the stellar sphere once in 265 1/4 days along the ecliptic, which cuts the celestial equator at angle of 24o. (The inclunation at the present day is 23o 27' but in those daya it was very nearly 24O).

(d) There is a retrograde movements of the point of intersection mentioned in (c), along the ecliptic called Ayana-chalanam (prcession). [Varahamihira] is the first astronomer who has reported this. According to some it is one degree in 67 years, and others, in 60 years. It is noreworthy that these values are very much nearer the modern value of 72 years, than the value given by the Greeks, 100 years. According to some, [Munjala], the prcession is continuous but others assert that the precession will stop after sometime and change into a processional movement. We know now that [Munjala's] view is correct, but it could only have been a conjecture in those days.

(e) The Moon and the planets move in orbits of their own, each inclined to the ecliptic by a small angle. Their motion is not uniform but varies, being slowest at apogee and quickest at perigee. This varying motion was geometrically represented by eccentric circles or epicycles, and epicycles were also used to represent the equation of conjunction of the planets. The Greeks did the same. But the Hindu constants were generally more accurate.

(f) From the horizontal parallax of the Moon, which was tolerably accurate, teh Hindus calculated the distance of the Moon to be about 65 times the earths's raduis. If this in excess of the correct value by about 9 per cent, it is because their value for the horizontal parallax of the Moon was a little erroneous, combined as it was with the effects of atmospheric refraction. They held the theory that the Sun the Moon and the planets had a uniform speed. From this, and from the distance of the Moon they calculated the distance of the Sun and the planets, but these were bound to be wrong, because their theory of uniform speed was wrong. Really the velocity is inversely as the root of the distance.

(g) A table of sines and another of versines are generally given from which all the other trigonometrical functions are derived wherever they are required. The Greeks gave also a table of chords. The New [Surya Siddhanta] has an interesting method for constructing the table of sines, based on the fact that the second differences are proportaionate to the sines. [Bhaskara charya's] treatment of the trigonometrical functions, addition and subtraction formulaw etc., have a surprizingly modern air.

(h) Generally a separate chapter is set apart called the [Tripra'snadhyaya] (Chapter dealing with the three types of questions, relating to time, place, and direction for problems solved by spherical trigonometry. [Bhaskara] is seen at his best here, giving methods of solving problems that baffle even the modern minds.

(i) Separate chapters are devoted to eclipses, and conjunctions of planets among themselves or with stars. For this purpose a list of the pricipal stars with their coordinates are given.

(j) A chapter called the [Yantradhyaya] is generally devoted to astronomical instruments.

(k) The longiture of Ujjain passing through Rohitika and Kurukshetra on to the N. Pole is taken as the prime meridian, for Ujjain was the Greenwich of India in those days. Mean sunrise or mean midnight at Ujjain is taken as the beginning of the astronomical day. A certain boldness and freedom from prejudice characterises the discussions in these works, especially in the [Siddhanta 'Siromani]. In this matter, the works of this period are in pleasing contrast with several works of the next age, which are authorityridden and in which very little originality is seen.

It seems that during this period two schools of astronomers flourished in this country. Ujjain was the seat of one school and there was a long hierarchy of astronomers there from very ancient times. [Varaha](c.500 A.D.), Brahmagupta (c.600 A.D.) and [Bhaskara] (c.1150 A.D.) all belonged to this school. The other school was on the west coast of South India and the [Aryabhatiya] )c. 500 A.D.) has been followed in this school. Certain peculiariues in his work are found in the system of almanac-making in vogue in the Tamil and Malayalam country.

5. The Period of Decline

After [Bhaskara] there was a decline. This was due to several causes. The Muslim occuption deprived the science of much valuable patronage. With the astronomical instruments which were at their disposal, the astronomers had come almost to the end of their tether, and new discoveries would be possible only with new and more powerful instruments. (This is true for discoveries in all sciences.) Where new discoveries are not made the tendency is to lay stress on ancient authority, which itself lays a strangle-hold on further discovery. The result was that very little that was original came in this period, though here and there some remarkable persons enjoying the patronage of some remarkable princes could be found. Prejudice and subservience to authority reigned supreme and reason was blinded. For example, [Kamalakara], the author of the [Siddhanta-Tattva-Viveka], a very able man, condemns many correct formulae given by [Bhaskaracharya], simply because his rival had taken [Bhaskara] for his authority.

As a result, the chief astronomical works were Karanas or manuals, based upon some ancient [Siddhantas], intended for use by almanac-makers. Bot accuracy but ease of computation was the aim of these works and they were intended to be used only for a few hundred years, not more, because error would accumulate. Not knowing this the present-day almanac-makers still use Karanas that have long ago creased to be serviceable, and the result is a deplorable state of affairs. The so-called ['Siddhanta]' and ['Vakya'] almanacs are examples.

The [Siddhanta] almanac, in vogue throughout India except the Tamil and Malayam parts, are most of them based on a Karana called Grabalaghava by [Ganesa Daivajna] (circa 1500). The [Vakya] almanacs followed in the Tamil country are based on a work called [Vakyakarana] or [Vakya] [Panchadhyayi] commented upon by [Sundararaja]. )about 1300 A.D.) The karanas called [Panchabodha] used in Kerala are prepared on the basis of the Parahita-ganita of Haridatta following [Aryabhatiya]. Is it not time we change our Karana?

Recently, thanks to the study of European astronomy, many works have been written by Indians, using which the correct positions of the Sun, the Moon and the Stars can be computed. Some have taken pains to put them in a form very much serviceable to almanac makers. Even the writer of this article has done some works in this direction. He will he only too glad to help people make correct almanacs.