New Light on

Āryabhata & Chamravattam

K. CHANDRA HARI K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology 17 Feb 2011

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Dr KV Sarma (1919-2005)

 I am to gratefully acknowledge the love, affection and inspiration given by Dr KV Sarma and Mrs Lakshmi Sharma.

 He was one of the few ‘truly’ qualified people to do work in history of science, especially astronomy and mathematics

legacy we speak of today could survive only because of his arrival at the right time if not late…

 ('Contributions to the study of the Kerala School of Hindu Astronomy and Mathematics' (1977) )

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Kerala legacy of Astronomy & Mathematics

 Pre-historic epochs seen recorded in the alpha-numeric chronograms

 These chronograms combine both mathematical and astronomical information and attests for an antiquity that finds little support in the known historical details. Most of these chronograms specify a day count, i.e. the kalidinam expired and had implicit in them the epoch of ‘Kaliyugadi’ i.e. midnight /sunrise of 17-18 February – 3101 CE when the siddhāntic planetary means had a computational super-conjunction at 0 degree. Such dates can be traced as far back as 29 April -58 CE, 13 November -26 CE etc. th  In the 4 century after Christ, these chronograms make us meet with a legendary astronomer Vararuci and he is succeeded by -I in Kali 3623 (Giritunga)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Epoch of Vararuchi

th  20 March 403 CE apogee conjunction of can be shown to be the epoch of the Girnasreyadi vakyas

 Udayagiri epoch of Indian astronomy can be identified as 20 March 402 CE (K3503 i.e. 120 years prior to K3623)

 Studies today lack an audience that can understand and appreciate the astronomical evidence

 Vararuchi of Kerala known through the Chandravakyas had the epoch of his Vakyas related to the Udayagiri of Chandragupta-II .

 In Kerala, the chronogram yajnasthanamsamrakshyam puts his son’s epoch as 14 Feb 378 CE.

 Aryabhata epoch Kali 3623 (elapsed) is 120 years after the Udayagiri epoch

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Āryabhata

 Āryabhata stands renowned even in modern times for the scientific treatise he presented on Astronomy and Mathematics.

 Knowledge that won him praise in Kusumapura in his own life time continues to win him praise even in the 21st century.

 Apart from his astronomical and mathematical precepts, his advent is looked upon as a turning point in the history of exact sciences in . He not only set forth the right background by drawing the best of the scientific tradition that preceded him but also chose to create a break with the paradigm by enunciating such revolutionary principles like the rotation of earth and a wholesome revision of mathematical astronomy based on observations.

 We are in dark about his observational innovations as the Āryārdhrātrasiddhānta is lost and is known only through brief extracts in texts like those of Bhāskara-I.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Legacy of Āryabhata

 Kerala legacy of Astronomy and Mathematics begins with Aryabhata (522 CE) – ‘practically every astronomical text produced in the land base itself on the teachings of Aryabhata’ Ref: Sarma, KV., Tradition of in Kerala, Revision of Planetary Parameters

 A vast body of astronomical and mathematical literature in such illustrious names as Bhaskara-I, , Madhava, Paramesvara, Nilakantha, Achyuta etc. Modern researchers, Dr. KV Sarma, Kuppanna Sastri, RC Gupta, KS Shukla, CT Rajagopal etc.

 Revision of the older siddhantas was an outcome of the realization of data misfit between prediction and observations of the astronomical phenomena and successive astronomer- mathematicians have been very critical of even the most astute of their predecessors as we see with Brahmagupta and Vatesvara.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Āryabhata, Brahmagupta, Vatesvara

 Brahmagupta minced no words to criticize Aryabhata and Vatesvara followed on the lines of the “lotus-born” (Brahmagupta) :

 “The longitude of a planet obtained from its forged revolution number cannot be the same as that obtained from its real revolution number…. The revolution number for Mars (for example) may be forged by taking the first four figures as 8522, 0635, 7552 or 9292..” (I:20-22)

 and

 “On account of forged revolution numbers, forged civil days and forged positions of apogees and due to ignorance of the epicycles, the longitude of the planets disagree with observations and so they are not true”. (I:27)

 With Vatesvara airing such criticism on Brahmagupta, one can imagine the plight of the lesser folks if anybody were to forge astronomical works without taking into account the data misfit of his times. (Vatesvara siddhanta, translated by Prof. KS Shukla) K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Controversy about place…

 "Aryabhata I or Aryabhata the Elder to distinguish him from a 10th-century Indian mathematician of the same name, he flourished in Kusumapura—near Patalipurta (Patna), then the capital of the Gupta dynasty—where he composed at least two works, Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. Aryabhatasiddhanta circulated mainly in the northwest of India and, through the Sasanian dynasty (224–651) of Iran, had a profound influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight. Aryabhatiya was particularly popular in , where numerous mathematicians over the ensuing millennium wrote commentaries" (sic) (Encyclopedia Brittanica)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Controversy contd.

 "A veritable pioneer of Indian Astronomy, Āryabhata is without doubt one of the most original, significant and prolific scholars in the history of Indian science. He was long known by Arabic Muslim scholars as Arjabhad and later in Europe in the middle Ages by the Latinized name of Ardubarius. He lived at the end of the 5th century and the beginning of the sixth century CE, in the town of Kusumapura..." (Georges Ifra, The Universal History of Numbers)

 "As far as astronomical works are concerned, it seems that the Kerala country was the seat of its development in the South. It is all based on the Āryabhatīya, with or without corrections called the bījas... How Āryabhata came to be connected with the Kerala country is yet to be explained. He is called Aśmaka (i.e. one born in the Āśmaka region) and some say that an early name of the erstwhile princely state of was Āśmaka (Apte's Dictionary). But many say that the region near the Vindhyās was called the Āśmaka country...“ (TS Kuppanna Sastri)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Āryabhatstviha nigadati kusumapurebhyarcitam jnānam, 'Āryabhatstviha nigadati kusumapureƒbhyasitam jnānam‘

 "...scholars have thought for a long time that Āryabhata was either born in Kusumapura or lived and taught in that great city of ancient India. Such a view now appears untenable in the light of recent studies on the works of Bhaskara-I and his commentators and also of the medieval commentators of Āryabhata. In these works, Āryabhata is frequently referred to as an aśmaka, that is one belonging to the Aśmaka country which is the name of a country in the south, possibly Kerala....the fact that commentaries of and works based on Āryabhatīya have come largely from South India, from Kerala in particular certainly constitute a strong argument in fvaour of Kerala being the main place of his life and activity"

 ('A Concise History of Science in India‘, INSA)

 SB Dikshit to Dr KV Sarma (1977/2001)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology

(Dr KV Sarma, IJHS, 2001)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Astronomical Evidence

 Conflict of the latitude of Ujjayinī ºlɱÉVɱɨÉvªÉɱ±ÉÆEòÉ ¦ÉÚEòIªÉɪÉÉ ¦É´ÉäSSÉiÉÖ¦ÉÉÇMÉä* =VVÉʪÉxÉÒ ±ÉÆEòɪÉÉ& iÉSSÉiÉÖ®Æú¶Éä ºÉ¨ÉÉäkÉ®úiÉ&*14*

 Verse spells out that on the prime meridian, Ujjayinī is located at one-sixteenth of the earth's circumference North of Laňkā and thus the latitude of Ujjayinī turns out to be 3600/16 = 220N30'. 0  "...This makes the latitude of Ujjayinī equal to 22 30'N. This is in agreement with the teachings of the earlier followers of Āryabhata, such as Bhāskara-I (AD 629), Deva (AD 689) and Lalla and the interpretations of the commentators Someśvara, Sūryadeva (b. AD 1191) and Parameśvara (AD 1431). Even the celebrated Bhāskara-II (AD1150) has chosen to adopt it.

 Brahmagupta (AD628) differed from this view. He takes Ujjayinī at a distance of one-fifteenth of the earth's circumference from Laňkā and the likewise the latitude of Ujjayinī as equal to 240N

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology 0  Āryabhata gave the latitude of Ujjayinī as 360 /16 North of Laňka and it had acceptance among only his followers.

 Brahmagupta and a host of others like Varāhamihira did not agree with Āryabhata and had given rise to an alternate school of thought and tradition.

 Bhāskara-II apparently had agreement with Āryabhata but some followers of Āryabhata like Sūryadeva could not find any rationale underlying the Āryabhata's notion and they did tacitly accept Brahmagupta as correct.

 Apart from what Shukla and Sarma have discussed, we can see that the Sūryasiddhānta also did not agree with Āryabhata in the matter.

 Shukla has quoted Nīlkantha who has tried to explain the conflict by crediting Āryabhata’s reference of 220N30’ to a different Janapada at that latitude. But this is not correct as any reference to Ujjayinī in ancient texts obviously hinted at the location of Mahākāleśvar temple whose latitude according to modern determination is 230N13’.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Kusumapura (25N35, 85E15) vs. Ujjayini

 In any country places falling on the tropic of cancer is well known to astronomers. How could Aryabhata miss it had he been living beyond the tropic of cancer but close by?

 Can Aryabhata at 25N35 be unaware of the intersection of the prime meridian and the tropic of cancer (240)?

 How can Aryabhata at Kusumapura (25.5N) place Ujjayini at 22.5N with Palabha = 5? 0  Kusumapura was 87 yojanas (9.5 ) east of the prime meridian. How can he produce a treatise without the mention of Desantara?

 Can a treatise as accurate as Aryabhatiya be created in Kusumapura (25.5N, 9.5E of Ujjayini) without Desantara and Udayantara?

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Indications of Āryabhatīyam 1. Two traditions of equatorial circumference of

earth (C0) and latitude of Ujjayinī 2. Āryabhata tradition gives the least value of

C0 = 3299 and  of Ujjayinī as 360/16 =22.50.

3. Brahmagupta tradition gives C0  5000Y and  of Ujjayinī as 360/15 =240.

4. Least values of C0 and C suggest that the tradition evolved at low  as C-Co increased with increase of . 5. Ujjayinī was unknown to Āryabhata and thus gave the  = 22.5 at Palabha = 5.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Earth’s Circumference as in Suryasiddhanta

Moon’s horizontal parallax (Angle subtended by earth’s radius of 800Y at the centre of moon) in SS is 53’20”. Sine  =  when  is small and radius of the moon’s orbit is obtained as 51570Y. Moon’s orbit will be 324000 Y. Basic definition is Moon’s orbit of 21600 minutes of arc in Yojanas. Suryasiddhanta takes 1 arc min = 15 Yojanas. Same in Ardharatra paksha.

51570 800

) M M

Moon

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Orbital Dimensions in Āryabhatiya

Moon’s horizontal parallax (Angle subtended by earth’s radius of 525Y at the centre of moon) in SS is 52.5’. Radius of the moon’s orbit is obtained as 34380Y. Moon’s orbit will be 216000 Y. Basic definition is Moon’s orbit of 21600 minutes of arc in Yojanas. Aryabhatiya takes 1 arc min = 10 Yojanas.

34380 525

) M M Earth

Moon

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Impossible Equivalence with Modern

Ancient Modern Ancient Distance/ Orbit value Yojana KM Graha Orbit Earth’s Radius = Distance - for each case in Yojanas radius Distance KM

Moon 216000.000 34377.387 384400 65.48 11.18

Sun 2887666.800 459585.371 149,600,000 875.40 325.51

Mars 5431291.460 864414.862 227900000 1646.50 263.65

Jupiter 34250133.368 5451065.280 778,300,000 10382.98 142.78

Saturn 85114493.163 13546360.638 1,427,000,000 25802.59 105.34

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Place of Āryabhata

Evidence of Spashta-bhūparidhi Spashta bhūparidhi = Bhūmadhyaparidhi* Cos 

C = C0* Cos 

where  is latitude and C is earth’s circumference

C0 = 3299, Interger Yojanas per degree of longitude at  demands

C = 3240 = 360*9 and

 = ACOS(C0/C). i.e.  = 10N51

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Historical Values of C0 & Derivation of 

Astronomer C0 C Yojanas  Place Āryabhata 3299 3240 9*360 10.85 C.Vattam Brahmagupta 5000 4680 13*360 20.61 Bhilmala Varāhamihira 3200 2880 8*360 25.84 Kusumapura Manjula 3600 3240 9*360 25.84 Kusumapura Bhāskara-II 4967 4680 13*360 19.57 Bid Bhāskara-II 3927 3600 10*360 23.55 Ujjayinī Eratosthenes 5040 4320 12*360 31.00 Alexandria

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Camravttam:10N51

Latitude 10 N51 of Kerala marks the place where the Ujjayinī meridian (75E45) intercepts the west coast.

Camravattam having the ancient Jain temple of Bāhubali is close to the ancient port and one-time Cera capital, Ponnāni. It is also very close to Tirunāvāya, place of the Mahāmaghā congregation in ancient times.

Aśmaka was the Jain Country surrounding Śravanabelgola (12N50) and the place received its name from the stone monoliths out of which the great statues got carved in later times.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology CHAMRAVATTAM near PONNANI

Chamravattom, close to the illustrious Ponnani is a village located 11 km away from Tirur. This serene village is on the shores of the river Bharata_puzha

10° 49′ 7.99″ N, 75° 57′ 10.01″ E

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology What is special about this place?

 5 Mistakes of Computational Rules 1. Arkāgrā, verse 31 of Goḷa which stipulates the condition for samamaṇḍala śaňku 2. Earth’s diameter and circumference 3. Erroneous use of Rversed sine (verses 35, 36 and 45 of Āryabhaṭīya) 4. Precept on the visibility of Agastya 5. Modification of the revolutions of Moon’s Nodes K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Chamravattam Attests the Truth…

 It stands scientifically established that the alleged mistakes were in fact observational truths locally at his place of observation – banks of Nila at 10N51, 75E45.

 Bhaskara’s reference to Asmaka as his place originated from the fact that he was a Jain and Chamravattam and the river Bharata_puzha were part of the Jain country in his days.

 Chamravattam is named after the Jain muni Sabara and Bharata_puzha after the Jain King Bharata famous as Bharatesvara.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Vindhya_giri Hill

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology AŚMAKA – Hard Stone 12N51, 76E30

Asmaka received its name from the stone monolith of Sravanabelgola out of which the statue of Gomatesvara got carved out.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Current Science Journal: 25 December 2007

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Evidence of Arkāgra

Verse 30 & 31 of Gola, Āryabhatīyam {É®ú¨ÉÉ{ÉGò¨ÉVÉÒ´ÉÉʨɹ]õVªÉÉvÉÉǽþiÉÉÆ iÉiÉÉä ʴɦÉVÉäiÉÂ* VªÉÉ ±É¨¤ÉEäòxÉ ±É¤vÉÉEòÉÇOÉÉ {ÉÚ´ÉÉÇ{É®äú ÊIÉÊiÉVÉä**30** ºÉÉ Ê´É¹ÉÖ´ÉVVªÉÉäxÉÉ SÉänÂù ʴɹÉÖ´ÉnÖùnùM±É¨¤ÉEäòxÉ ºÉRÂóMÉÖÊhÉiÉÉ* ʴɹÉÖ´ÉVVªÉªÉÉ Ê´É¦ÉHòÉ ±É¤vÉ& {ÉÚ´ÉÉÇ{É®äú ¶ÉRÂóEÖò&**31*

Amplitude of the Rising Sun: Distance of the rising or Setting sun from the East-West line

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Evidence of Arkāgra

Verse 30 of Gola, Āryabhatīyam Rsin λ*Rsin/Rcos  = Agra = , where  is the latitude of the place and  shall be used to denote agra in the following discussion.

Verse 31, Gola: "When that (agra) is less than the Rsine of the latitude in the northern hemisphere, s… Rsine h = *Rcos/Rsin =  / Rtan 

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Criticism of Brahmagupta

"The statement (of Āryabhata) that the sun, in the northern hemisphere, enters prime vertical when the (sun's) agra is less than the Rsine of the latitude is incorrect, because this happens when the Rsine of the sun's declination satisfies this condition (and not the sun's agra)"

In modern terms, the prime vertical altitude 'h' is Sin h = Sin δ / Cos  where δ is the declination of sun and  is the latitude. Declination δ has to be more than  for the altitude h to be positive and the fact could have been obvious to a mathematician who is rightly believed to be the originator of the modern trigonometric functions.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Observation that shaped the precept

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Arkāgra: Cut off  =  =  at  =10N51

At lower latitudes (Rsin agrā - Rsin) and (Agrā-δ) tend to be lower.

 of Kerala 8.5 to 12.5, (δ – agrā) ~ 0.250. At 10.85

When  = 0, Agra or Amplitude is 0 and when  > , sun cannot cross PV. In the precept Agra comes into picture as an observation in south latitudes where  =  = Agrā for low  values like 10N51, the location of Āryabhata

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Carakhandas at Palabhā 1:  = 4.775

How the Āryabhata tradition as seen in Mahābhāskarīya III.8 is able to spell out the Carakhandas at Palabhā =1?  = 4.775 is in the sea and Carakhandas at Palabhā =1 is unlikely to be derived from those of Palabhās = 5 or 6?

Āryabhata’s Ujjayinī of Palabhā = 5 and  = 22.5 is a hypothetical place or there is a latitude error of 1.50 and Palabhā error of 20 Vyaňgulas.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Values of Āryabhata tradition Cara-khandas in asus

Palabhā P l a c e  Mesa Vrsbha Mithuna 1.0 4.76 236 192 80

Mahābhāskarīya III.8 240 192 81

2 . 0 9 . 4 6 476 388 160

M a h ā bhā s k a r ī y a I I I . 8 480 384 162

5 2 2 . 5 0 1184 960 392

M a h ā bhā s k a r ī y a I I I . 8 1200 960 405 K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Evidence of Ancient Shadow Floor at  = 9.46

Cara-khandas in asus/4 Palabhā Place  Mesa Vrsbha Mithuna 1.0 4.76 236 192 80 2.0 9.46 476 388 160 2.3 10N51 10.85 548 444 180 5.0 22N30 22.50 1184 960 392 5.34 Ujjayinī 24.00 1272 1032 424 5.72 Kusumapura 25.50 1360 1108 452 6.0 26.57 1428 1160 476

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Egg Shaped Shadow Floor : Palāňgulam =2

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Chengannur & Camravattam

1 - 2 = (10N51 – 09N30) = 01020’

Palabhā1- Palabhā2 = 0.3 = 18 Vyaňgulas, combined with 1 - 2 = 01020’ gives the value of Palabhā = 5 at  = 22.5. i.e. (22.5/1.33)*18=303  5 Angulam

Carakhandas fixed at this place of Palabhā = 2 formed the basis of the formula that we see in Mahābhāskarīya III.8

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Evidence of Eclipses

Eclipse observations give critical evidence that supports Āryabhata’s drastic modifications of the revolution numbers of Moon’s Apoge and Nodes.

Total Solar Eclipse of 15 Feb 519 CE with totality at 10N51, 75E45 & Annular Solar Eclipse of 11 Aug 519 CE with totality at 8N30, 75E45

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Solar Eclipse of 15 Feb 519 CE

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Eclipse Observed from Sea

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Evidence for these Observations

Modern Mean Longitudes for Mean New moon of 11 Aug 519 AD

Planet Aryabhatīyam Sūryasiddhānta

Ārya Modern

Sun 1400 58' 141000' 1400 58'

Moon 1400 58' 140021' 1400 58'

Apogee 3250 20' 325015' 3200 30'

Rāhu 317035' 317036' 3130 57'

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology One-point fit of Siddhāntic Mean s

‘0’ Mean s at Yugādi introduces errors in the siddhāntic mean motions.

Therefore Siddhāntic Mean s precisely match with observation and modern values only at the epoch of the Siddhāntic treatise.

Mysterious precision we see between Āryabhatīyam and modern Mean s for 519 CE establishes that the mean motions were fixed based on the eclipse observations of 519CE and around.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Node and Apogee of Moon

Node of Moon Apogee of Moon

Kali Modern Āryabhata Diff: Modern Āryabhata Diff: elapsed

3590 185 26 185 43 -17' 168 29 168 51 -22' 3600 352 1 352 12 -11 215 26 215 42 -16 3612 119 54 119 58 -4 343 46 343 55 -9 3616 42 32 42 34 -2 146 32 146 40 -8 3619 344 30 344 31 0 268 37 268 43 -6 3623 267 8 267 6 2 71 24 71 27 -4 3646 182 16 182 0 16 287 21 287 12 9

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Mistaken Use of Versed Sine

Verses 35, 36 and 45 of Golapāda of Āryabhatīya had been a matter of conflict and discussions since the days of Brahmagupta.

Āryabhata precepts on the two components of Valana, viz., Aksa and Ayana which make use of the versed sine had their origin in the observations of the total lunar eclipse of 23 March 517 CE at Camravattam, 10N51, 75E45 Meridian transit of eclipsed Moon close to the equator led to Aksavalana rule – i.e. Versed sine of the hour angle and sine of Aksavalana becoming zero simultaneously and aksavalana attaining maximum on the horizon.

Simultaneously the eclipse also presented Versed sine (Moon+900) equaling 1 and Ayanavalana attaining the maximum value of the obliquity of the earth’s axis and thus leading to the proportion of verse 36 of Golapāda.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Page of the Bhaskariya Commentary

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Rule of Āryabhata for the Latitudinal Correction (Aksa Drk-karma) Verse 45 of Golapāda ¨ÉvªÉɼxÉÉäiGò¨ÉMÉÖÊhÉiÉÉäƒIÉÉä nùÊIÉhÉiÉÉäƒvÉÇʴɺiÉ®¿iÉÉä ÊnùEÂò* κlÉiªÉvÉÉÇSSÉEæòxnùÉäκjÉ®úÉʶɺÉʽþiÉɪÉxÉÉiÉ º{ɶÉæ ** (45)

 Lunar eclipse of 23 March 517 CE had maximum of the eclipse at 23:56 almost coinciding meridian and could be observed on horizon later as hypothetically described by Prof KS Shukla.

 Total eclipse of the moon on the meridian and intersection of the equator where the versine was zero and horizon where it reached a maximum = latitude could be observed by Aryabhata on the meridian of Ujjayini.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Circumstances of the Lunar Eclipse of – 23 March 517 CE

Sun Aksa-Valana0 Moon Mid- H Time λ0 Versine Āryabhat B.Gupt Modern λ0 heaven hours a a 17:00 180.570 78.560 4.450 1.227 6.87 13.350 -16.720 10.530 18:00 4.49 181.18 0.976 92.33 5.91 10.58 -16.60 10.85 19:00 4.53 181.79 0.726 106.16 4.94 7.86 -15.39 10.48 20:00 4.57 182.40 0.494 120.33 3.97 5.34 -13.28 9.44 21:00 4.61 183.01 0.294 135.09 3.01 3.17 -10.47 7.79 22:00 4.66 183.62 0.139 150.55 2.04 1.50 -7.22 5.64 23:00 4.70 184.23 0.039 166.64 1.07 0.42 -3.77 3.14 0:00 4.74 184.84 0.000 183.07 0.11 0.00 -0.36 0.69 1:00 4.78 185.46 0.025 199.41 -0.86 0.27 2.79 2.46 2:00 4.82 186.07 0.112 215.29 -1.83 1.21 5.51 5.03 3:00 4.86 186.68 0.256 230.49 -2.79 2.76 7.64 7.31 4:00 4.90 187.30 0.447 245.01 -3.76 4.82 9.08 9.11 5:00 4.94 187.91 0.673 259.02 -4.73 7.28 9.80 10.32 6:00 4.98 188.53 0.920 272.78 -5.69 9.97 9.79 10.850

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Observation shaped the Precept

 The lunar eclipse presented an occasion when the eclipsed moon had a meridian transit when posited close to the equator. Versine and Sine During Eclipse  At the intersection of the meridian and the equator, the Rversed sine of the hour 1.5 angle H was zero and the Rsine of the aks̟avalana was also zero. 1.0  Rversed sine of the hour angle and the aks̟avalana increased thereafter. 0.5  When the eclipsed body was at the horizon at 18:00 hrs on 23 March or at Sine H 06:00 hrs on 24 March, the Rversed sine 0.0 of H had its maximum value and the 180 184 188 192 Versine Functions H aks̟avalana also had its maximum value -0.5 equal to the latitude of the place.

 East to West horizon observations received the correct representation -1.0 with versine H but the same failed for the intended purpose during the -1.5 progress of the eclipse between first Moon Longitude as Time Advanced contact and last contact

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Golapāda verse 36 Ê´ÉIÉä{ÉÉ{ÉGò¨ÉMÉÖhɨÉÖiGò¨ÉhÉÆ Ê´ÉºiÉ®úÉvÉÇEÞòÊiɦÉHò¨ÉÂ* =nùMÉÞhÉvÉxɨÉÖnùMɪÉxÉä nùÊIÉhÉMÉä vÉxɨÉÞhÉÆ ªÉɨªÉä** (36)

0 “Rversed“Rversed sine sine of theof themoon’s moon’s longitude longitude (λ) increased (λ) increased by 90 by0 multiplied 90 multiplied by the Rsine of the byobliquity the Rsine ( of) theand obliquity the latitude () of and the the moon latitude (β) givesof the themoon Ayana (β) -givesvalana (Ay)….” i.e. the Ayana-valana (Ay)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Ay Results of Āryabhata and Brahmagupta Brahmagupta Āryabhata 0 Moon 0 Versine Sine Ay Time 0 λ+90 0 0 λ λ+90 λ+90 A 0 0 0 0 β*Ay Ay β*Ay y 17:00 180.57 270.57 0.99 -1.00 23.75 6.87 -24.00 -6.95 18:00 181.18 271.18 0.98 -1.00 23.48 5.47 -23.99 -5.59 19:00 181.79 271.79 0.97 -1.00 23.21 4.10 -23.99 -4.24 20:00 182.40 272.40 0.96 -1.00 22.94 2.76 -23.98 -2.88 21:00 183.01 273.01 0.95 -1.00 22.67 1.44 -23.96 -1.53 22:00 183.62 273.62 0.94 -1.00 22.40 0.16 -23.95 -0.17 23:00 184.23 274.23 0.93 -1.00 22.13 -1.10 -23.93 1.19 0:00 184.84 274.84 0.92 -1.00 21.86 -2.32 -23.91 2.54 1:00 185.46 275.46 0.90 -1.00 21.60 -3.52 -23.88 3.89 2:00 186.07 276.07 0.89 -0.99 21.33 -4.68 -23.86 5.24 3:00 186.68 276.68 0.88 -0.99 21.06 -5.82 -23.83 6.58 4:00 187.30 277.30 0.87 -0.99 20.80 -6.93 -23.79 7.92 5:00 187.91 277.91 0.86 -0.99 20.53 -8.00 -23.76 9.26 6:00 188.53 278.53 0.85 -0.99 20.27 -9.05 -23.72 10.59

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Fig-2: Aryabhata and Brahmagupta: Ayana Valana Magnitudes 9

6 Arya

3 Bgupta

0 180 182 184 186 188 190 -3

-6

-9 Ayana-valanaMagnitudes

-12 Moon Longitude in the night of Eclipse

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology

Contrast of Āryabhatīya and Modern Results Total lunar eclipse of , 23 March 517 CE Full Mean True Mean True λ λ Kalidays Moon JD λs λs λm λm node apogee

Āryabhatīya Mean Full 0 0 0 0 0 0 1321508.30 1909974.05 02 22’ 04 27’ 182 22’ 178 46’ 183 44’ 48 17’ Moon 1 1

Modern λ 02024’ 04018’ 182023’ 178016’ 183043’ 48012’

Mod. Mean 1321508.30 1909974.05 0 0 0 0 0 0 Full Moon 2 23 02 24’ 04 18’ 182 24’ 178 18’ 183 43’ 48 12’

Āryabhatīya 1321508.73 1909974.48 True Full 0 0 0 0 0 0 7 66 02 47’ 04 52’ 188 06’ 184 52’ 183 44’ 48 17’ Moon

Modern 1321508.74 1909974.49 True Full 0 0 0 0 0 0 2 22 02 50’ 04 44’ 188 12’ 184 44’ 183 42’ 48 15’ Moon

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Mahābhāskarīya VI. 39-41 Meridian Transit of Eclipsed Moon at 10N51, 75E45

Trial Time Lagna Midheaven Moon Moon +900 1 23:00 254.47 167.03 184.21 274.21 2 23:30 261.27 174.30 184.5 274.50 3 0:00 268.09 181.61 184.79 274.79 4 0:08 270 183.65 184.87 274.87 5 0:14 271.2 184.93 184.92 274.92 6 0:30 274.98 188.98 185.08 275.08 7 1:00 281.98 196.47 185.38 275.38 In the case of the lunar eclipse of 23 March 517 CE discussed above, moon thus determined as equal to meridian ecliptic point also gave the value of sun as the totally eclipsed moon on meridian obviously indicated sun to be moon+1800. Equation of centre and the year length of Aryabhata do carry the signature of the spring equinox on 19 March 517 CE, 03:39 ZT.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Description of Lunar Eclipse V |ÉOɽþhÉÉxiÉä vÉÚ©É& JÉhb÷OɽþhÉä ¶É¶ÉÒ ¦É´ÉÊiÉ EÞò¹hÉ&* ºÉ´ÉÇOÉɺÉä EòÊ{ɱÉ& ºÉEÞò¹hÉiÉÉ©ÉÉä iɨÉÉä ¨ÉvªÉä ** (46)

 “At the beginning and end of lunar eclipse, the obscured disc appears smoky and during partial obscuration it is black. In totality the obscured part is yellowish brown and at maximum obscuration the disc appears bluish with black tinge”

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Aryabhata and Jain Tradition

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Jain Legend on Asmaka

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Precepts on Agastya (Canopus)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Rise and Set of Canopus Agastya sets when λs = 60 - 

20.0

18.0 10N51 16.0 Fig.1 60 - Latitude 14.0

12.0

10.0

8.0 24N

Altitude of Agastya 6.0

4.0

2.0

0.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 Sun's Longitude

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Agastya Rises when λs = 120+ 

25.0

20.0

15.0 10N51

24N

10.0 Altitude of Agastya Altitude 5.0

0.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 Sun's Longitude

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Sum up on Agastya rule

 Precept is location specific and without knowing the place where Agastya was observed, tradition had been alleging mistake on the Aryabhata precept.

 Empirical rule of Aryabhata is very precise at Chamravattam and none could give a better rule at their place or could even guess out that the place of observation may be south latitudes.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Agastya 8 May 517 AD, Sun 60 – 110 (Kerala)

Canopus

Horizon is the white line above

At sunset18:12 ZT, sunset Agastya had 12+ degree altitude, sufficient for observation after twilight over the sea horizon

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Agastya 26 April 517 AD, Sun 60 – 230 (Ujjayini)

Canopus

Horizon is the white line above

At sunset18:25 ZT Agastya had only 4 degree altitude

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology 10 Aug 2007, Current Science

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Origin of the Sine and Cosine

 Kerala legend speaks of Haridatta who became a lunatic in later times had been rolling stones up the Rayiranellur hill and rolling them down to demonstrate that obtaining height is too difficult while reaching the foot is very easy

 Unnati varuthuka = Deriving R*Sin 

 Padam varuthuka = Deriving R*Cos 

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology

Once height is achieved, base is easy

B Sine  = BF/OB = BF/R AYhm BF = R*Sine  = D¶Xn.

 Cos  = OF/OB AYhm A O F OF = OB*Cos  = R*Cos  = ]mZw

R*cos = R*sin(90-)

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology 0  Deriving these 24 Rsine values for  up to 90 was difficult as the slope increased or angle increased.

 Once the 24 Rsines were determined, the cosines could be easily derived.

 Achieving the height was difficult like rolling a stone up hill but once the height or R*Sine  is achieved, the base or R*Cos  was easy by the rule of compliment.

 It is noteworthy that height is achieved by means of the Hypotenuse along which the stone is rolled up to height.

 And thus analogy was drawn for the process of a stone rolled up the hill to achieve the height and then left to roll down with ease to reach the base.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Āryabhata & Jain Tradition

 Verse 9 of Kālakriya giving the Jaina 12 fold division of Yuga

 Verse 5 of Daśagītikā speaks of Bharata, the first Universal emperor of Jains who accessed the throne from the Ādinātha Rsabhadeva at the beginning of Apasarpinī Kaliyuga.

 Āryabhata’s rejection of the 4:3:2:1 cycle of Krtādi yugas based on the Smrtis provide attestation to the new interpretation attempted of the verse.

 Verse 11 of Gola referring to Nandana-vana and Meru represents terminology borrowed from Tiloyapannatti of Jains.

 References to Bramah the primordial deity of Jains in verses 1 of Ganitā and 49, 50 of Gola.

 Use of Kali Era having the distinct signature of Āryabhata for the first time in South India with the Aihole inscription of the Cālukya King Pulikeśi-II. Aihole as Āryapura suggests the possibility that the town may have been the place where he may have attained liberation later in his life and hence named after Āryabhata as Āryapura.

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology Monday, Jun 25, 2007

Aryabhata lived in Ponnani?

“Aryabhata lived in Ponnani in the northern coastal belt of Kerala,” says K. Chandra Hari, …

K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology K. Chandra Hari, 17 Feb 2011, Cochin University of Science & Technology