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Summary

Sketches the basic astronomical ideas as developed in Indian in ancient times. Calculation of astronomic events require precise conceptualization of time. Particularly, it calls for computing compute the total elapsed time or ahargana अहर्गण from a reference event (start of epoch) - i.e. one must refer to the calendar.

Time is defined from certain celestial events, defined in chapters 2 and 3. sUryasiddhAnta [Late] सूर्यसिद्धान्त onwards, Indian calendars were generally lunisolar - i.e. the pattern of lunar phases control the month boundary, but the transition of the sun determines what name the month will have. This enables the lunar month to be roughly in the same part of the solar year.

The ecliptic is divided into twelve equal rAshis (राशि, signs of the zodiac, chapter 3), and there is a lunar month for each episode when the sun enters a particular rAshi. The month vaishAkha वैशाख is so named because the full moon on this month occurs near the asterism vishAkha (विशाखा, in Libra).

This is ensured by calling the month when the sun is in meSha (मेष, Aries) as vaishAkha वैशाख (in new-moon based systems of South India, this would be the month of chaitra चैत्र). Note that the mesha rAshi, covering the nakshatras asvini अश्विनी नक्षत्र, bharaNI भरणी, and kr^ittika कृत्तिका, is roughly opposite vishAkha on the ecliptic.

The Indian lunisolar calendar

Sometimes, the sun may remain in the same rAshi for more than a lunar
month; in such situations, one has an extra month (adhikamAsa, chapters 5 & 6)
Occasionally, the sun may transition into two rAshis during the same month,
here one has a lost month (kshaya mAsa).  This makes it difficult to
compute the total elapsed time.

In the pure solar calendar which is also used in most regions, the months are
defined only by the solar transitions.  This method of solar year reckoning
is similar to the Besselian year used in astronomy today.  Thus the year
usually begins with the same Mesha transition (around April 14-15), but
because the sun at sunrise may remain in a rAshi for 29 to 32 days, the
length of months are not fixed; but the overall year length is within a day
or two of 365.

All this means that the Indian calendars are always fixed to the sun, there
is no need for leap years or other corrections.  However, this also means
that the number of days elapsed are a bit variable.  Thus, it is a problem to
determine the astronomical time, the period elapsed from a given epoch.
This is the computation of ahargana, the total elapsed time, given in
Chapter 7.

Corrections to the "mean" position

Once we have the total elapsed time, one may compute a "mean" position, which
assumes uniform velocity across the sky (chapter 7).  

Since planets don't move uniformly across the sky, a manda (slower speed,
retrograde motion) or shighra (faster speed) correction has to be applied
(chapter 8).  This of course, correlates to a geocentric view of the
heavens, and the process, of obtaining a mean position, and then the
corrections, is pretty much the same as the Ptolemaic system (about 2nd
c. AD).   Whether the Indian system is likely to be derived from it (though
the surya siddhAnta is most likely older), is not commented on by
Balachandra Rao.

Thus, finally, we have a prediction for the "true positions" of each
celestial object at any given time.  This can also be used together with
measurements on a gnomon (sanku) to compute latitude.  Also, one may
compute the time of sunrise and sunset (chapter 10).  Ultimately, one may
also predict solar and lunar eclipses, (ch 11/12).

Excerpts

		 	यथा शिखा मयूराणां नागानां मणयो यथा। 
		 	तथा वेदाङ्गशास्त्राणां गणितं मूर्ध्नि स्थितम्॥

	 	yatha sikha mayuranam naganam manayo yatha | 
	 	tadvad vedanga sAstrANAm jyotiSam jyotisam murdhani sthitam ||
						vedAnga jyotiSa 4

	Like the crests on the heads of peacocks, like the gems on the hoods
	of the cobras, jyotiSa (astronomy) is at the top of the vedAnga
	shAstras.  [opening quote, p.1; translation edited AM]

ch1 : Six vedangas and authors

* shikShA शिक्षा - phonetics - gargeya [phonetics was systematized for
	sounds - so that vedic texts could be pronounced properly.  the word
	shiksha gained the current meaning - training / education - because
	this was how it was pronounced]
* vyakAraNa व्याकरण - grammar (lit. analysis) - ashTAdhyayI panini
* nirukta निरुक्त - etymology - yaska
* chhandas च्हन्दस - prosody -  pingala
* jyotiSa  ज्योतिष - astronomy / astrology - lAgadha
* kalpa कल्प (ashvalayAna-shrautA) - ritual / procedure - kautsa

[AM: 
for many centuries, these consittuted the "syllabus" for traditional
brahminic study.  the idea of a syllabus - also prevalent in china from
confucian times - could it idea have influenced the  Arabs, and thence the
"Seven Liberal Arts" of medieval europe:  Grammar, rhetoric,
logic/dialectic, arithmetic, geometry, music, and astronomy/astrology.]

vedAngajyotiSa : c. 1300BC

vedAMgajyotiSha वेदांगज्ज्योतिष : appears in two rescensions: Rigveda
	jyotiSha and yajurveda jyotisha.  One verse says: "I shall write on
	the lore of time, as enuciated by the sage Lagadha."  Based on this,
	the authorship of the vedAMgajyotiSha is attributed to Lagadha. p.2

At the time of its composition, the winter solstice was at the
beginning of the ShrAviShThA श्रविष्ठा (Delphini) constellation and summer
solstice in the middle of the AshleShA आश्लेषा (Hydrae).  VarAhamihira stated
that in his own time the summer solstice was at the end of three quarters
of punarvasu पुनर्वसु and the winter solstice at the end of the first qtr of
uttarAShARhA उत्तराषाढ, there had been a precession by 1 and 3/4 of a
nakshatra (7 pAdas, each pAda = 3 deg 1/3), or about 23deg 20' 

This difference, x rate of precession = 72 years per degree --> 1680 years
or about 1150 BC.  

Generally agreed period for vedAngajyotiSa  is between 14th c BC and 12th.

Astronomical knowledge

r^gveda maNDalas --> knowledge of moon phases, newmoon, etc.

nakShatra नक्षत्र: very old system, used for days (27 and 1/4); moon covers one
	   nakshatra each day (lunar month = 27.3217 days)
		nakSatras : each occupy 13 deg 20'

agrahAyaNa अग्रहायण - old name for mr^gashira - means beginning of the year -
	     corresponds to about 4000 BC. p.3


TIME MEASUREMENT:

in the vedAngajyotiSa, time measurement is mentioned by having a specified
quantity of water flow through an opening (a clepsydra); a small unit is
one nARikA, i.e. 1/60th of the day. 

shortest and longest days

vedAMgajyotiSha mentions that the longest and shortest days on the two
solstices as 36 and 24 nARikas नाडिक (1/60th of day, as measured by a
certain quantity of water flowing through a small hole [clepsydra]).

Year was known to be 360 days plus 5 (4 is too less, and 6 too much).
Alternately, 254 days (lunar years) + 11 days = 11 days of sacriice.

shortest day : 24/60 x 24 hrs = 9h 36m :  dinArdha = 4h 48m
longest day : 36/60 x 24 hours = 14h 24m: dinArdha = 7h 12m
	Thus diff from 6hrs = +/- 1h 12m = ASCENSIONAL DIFFERENCE

sin (asc diff) = tan(phi) tan (delta)

where phi = latitude, delta = declension of sun - about 23 deg 53 min
- VJ takes it as 24 deg --> leads to a latitude about 35 deg

  --> VJ was written from a lat about 35deg [may be gAndhAra 5]

1.4 siddhAntic astronomy

"siddhAnta" : the word has a connotation of "established theory" -
	    several arose around 100 BC to 100 AD

- introduced the twelve signs of the zodiac
- more precise value for the year
- computations of planet motions, solar and lunar ecpipses
- idea of parallax

principally 18 siddhAntas :
   	* surya, [most important]
   	* paitAmaha
	* vyAsa
	* vAsiShTha
	* atri
	* parAshara
	* kAshyapa,
	* nArada
	* gArgya
	* mArIchi
	* manu
	* AngIra
	* romaka (or lomasha),
	* paulisha (paul of alexandria?)
	* chyavana
	* yavana
	* bhr^gu
	* shaunaka.
most are lost.  

even during varAhamihira (b.505 AD), only five were extant: surya or saura,
   paitAmaha (or brahma), vAshiShTha, romaka and paulisha.

   --> compiled by varAhamihira as panchasiddhAntaka


AryabhaTa: 
system for measuring days  [two systems, both based on aryabhaTa]
 	- audAyikA : from sunrise to sunrise 
	- ArdharAtrikA : midnight to midnight  
[AryabhaTa text for the latter is lost; however, description given in
brahmagupta] p.8 

1.6 Historical Texts in Indian Astronomy

 1 AryabhaTa I          499 AD      AryabhaTiyam, AryasiddhAnta
 2 varAhamihira         b.505       pañchasiddhAntika, br.hatsaMhitA
 3 bhAskara I           c. 600      bhAShya on AryabhaTiyam, mahAbhAskarIyam
                                      laghubhAskarIyam
 4 brahmagupta          b. 591      bhamashpuTasiddhAnta, khaNDakhAdhyaka
 5 vateshvara           880         vateshvarasiddhAnta
 6 mañjula              932         laghumAnasam
 7 AryabhaTa II         50          mahAsiddhAnta
 8 bhAskara II          b. 1114     sidhhAntashiromaNi, karaNakutUhala
 9 parameshvara         c.1400      dr.ggaNitam, sUryasiddhAntavivaraNam,
                                      bhaTadIpikA
10 nilakanTha somayaji  1465        tantrasaMgraha, AryabhaTabhAShya
11 gaNesha daivajn~a    1520        grahalAghava
12 jyeShTadeva          1540        yuktibhAShA
13 chandrashekhara      b. 1835     siddhAntadarpaNa
14 shankara varman      19th c.     sadratnamAlA
15 venkatesa ketkar     1898        jyotirgaNitam, grahagaNitam
						(p.11-12)

1.7 siddhantas : four chapters (adhikAras)

calculating the position of planets (tArAgraha) तरग्रह + sun/moon
	[VJ had only sun/moon]
in addition, some special points - moon's apogee (mandocca) and also its
	ascending node (Rahu) would be computed. 

grahas : budha, shukra, kuja, guru, shani


four chapters gave the various parts of the computation: 

Average angular velocity

1. madhyamAdihkAra - expected mean position - based on mean angular velocity
	(rotation speed)
	[these would later be corrected]

mean angular velocity : specified as number of rotations in one mahAyuga (=
	43,20,000 years; 
[but some small differences between AryabhaTa and bhAskara)

	expected motion from base (epoch) in degrees
		= (No of revolutions * ahargaNa x 360 deg)
			/ (no of civil days in mahayuga)  MOD 360

[Note:
ahargaNa - count of the number of civil days from the epoch start 
	(meShAdi - differs between systems; Lahiri almanac, starting from
	Spica, the most preferred)

computing ahargaNa : complicated because year was solar, but calendar was
	lunar [purNamanta or amAnta]
the astronomical calculations were done on a lunar calendar -
--> convert solar years since [beginning of era =epoch] to lunar months
+ months in current years + number of tithis
--> convert total number of tithis
	--> convert to civil days  = ahargaNa

sun's position : moves back by 1 deg every day [covers 360 deg in 365 days]
moon's position : moves by 12 deg x ' [360/29.5] 
]

Corrections to madhyam (mean) position

2. spaSTAdhikAra - corrections to the madhyam position
	manda - slower - applic to all seven celestial objects
		[needed because planets orbits are elliptical (not circular)]
	shighra - faster - only for planets, 
		[needed because  rotation is about the sun, not earth]

3. triprashNAdikAra त्रिप्रश्नाधिकार

fix the three questions: 
	- direction (dik)
	- place (desha)
	- time (kAla)

how to find the latitude of a place, the times of sunrise and sunset,
variations of sunset and sunrise along eastern and western horion (solstice
to solstice), gnomon problems and calculation of lagna_. 

4. chandra and sUrya - grahaNAdhikAra  - lunar / solar eclipses

this was the true test of an astronomer's abilities.  

bIja saMskAra : changes suggested to account for deviation between
prediction and observation.  p.13


[ch.2 deals with basic issues in positional astronomy - the ecliptic vs the
celestial horion, meridians, pole star, equinoxes and solstices, etc.]

ch.3 : Coordinate Frames

There are three systems for assigning coordinates to a star, 
	- using the ecliptic, 
	- the celestial equator, and 
	- the horizon as reference.  

The ecliptic is stable, the equator wobbles slowly, the horizon changes all
the time but is good for immediate reference.

Ecliptic system

celestial latitude of star S = angle from ecliptic up to S
longitude : In Indian system, longitude is measured from meshAdi (start of
	mesha) - this is a fixed point, and hence this sidereal or
	nirayana longitude (ls) is measured w.r.t. the stars .  

	Because of the precession of the equinoxes, this point varies a
	little and immediate measurements of longitude w.r.t. the equinox,
	called the tropical longitude, (lt), varies from the nirAyana by an
	amount called ayanAMsha (degrees of precession of equinox).

	some debate as to the first point of Aries (meShAdi) - which is
	moving back steadily (abt 1 deg every 70 years) due to precession. 
	Indian Calendar reform committee : take 285AD as year of zero
	precession.  p.31


* Right ascension and declination system (w.r.t celestial equator)
        angle along celestial equator from first point of Aries is the right
        ascension (R.A., alpha), and the elevation of the object from the
        equator is the declination

* Azimuth and altitude (w.r.t. celestial horizon)
	Angle along horizon from north is azimuth; angle from horizon is
	elevation.

	these change rapidly. 

Seasons

Rigveda:  four-red eyed dog --
	alpha canis majoris / alpha canis minoris
	
joining them --> points to the south pole.  (not visible from N hemisphere)

[seasons: early RV: 3 seasons
       	   then 5, 
	   late RV: 6 ]

RV names for seasons

vasanta : madhu
grISma 	: sukra / shuchi
barShA  : nabhas
hemanta : sahas
sarat	: Isha / Urja
sisira 	: tApas  
	p.(50)


[
RV divisions of the day: 

period from sunrise to sunset - were divided differently
2, 3, 4, 5 or 15 divisions

4 divisions: prahara: 
	purvAhNa, madhya, aparAhNa, sAyahNa
]