3Dr KV Sarma ( )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 mathematicsKerala 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) )
5Kerala legacy of Astronomy & Mathematics Pre-historic epochs seen recorded in the alpha-numeric chronogramsThese 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 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.In the 4th century after Christ, these chronograms make us meet with a legendary astronomer Vararuci and he is succeeded by Aryabhata-I in Kali 3623 (Giritunga)
6Epoch of Vararuchi20th March 403 CE apogee conjunction of moon can be shown to be the epoch of the Girnasreyadi vakyasUdayagiri 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 evidenceVararuchi of Kerala known through the Chandravakyas had the epoch of his Vakyas related to the Udayagiri of Chandragupta-II Vikramaditya.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
7ĀryabhataĀryabhata 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 India. 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.
8Legacy of ĀryabhataKerala 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 Aryabhatiya in Kerala, Revision of Planetary ParametersA vast body of astronomical and mathematical literature in such illustrious names as Bhaskara-I, Haridatta, 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.
9Āryabhata, 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 ” (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)
10Controversy 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 Varahamihira (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 South India, where numerous mathematicians over the ensuing millennium wrote commentaries" (sic)(Encyclopedia Brittanica)
11Controversy 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 Travancore was Āśmaka (Apte's Dictionary). But many say that the region near the Vindhyās was called the Āśmaka country...“ (TS Kuppanna Sastri)
12Āryabhatstviha 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)
14Astronomical Evidence Conflict of the latitude of UjjayinīVerse 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'."...This makes the latitude of Ujjayinī equal to 22030'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
15Āryabhata gave the latitude of Ujjayinī as 3600/16 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’.
16Kusumapura (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?Kusumapura was 87 yojanas (9.50) 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?
18Indications of Āryabhatīyam Two traditions of equatorial circumference of earth (C0) and latitude of Ujjayinī2. Āryabhata tradition gives the least value ofC0 = 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.
19Earth’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 Y. Basic definition is Moon’s orbit of minutes of arc in Yojanas. Suryasiddhanta takes 1 arc min = 15 Yojanas. Same in Ardharatra paksha.
20Orbital 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 Y. Basic definition is Moon’s orbit of minutes of arc in Yojanas. Aryabhatiya takes 1 arc min = 10 Yojanas.
21Impossible Equivalence with Modern GrahaAncient Orbitin YojanasRadius = DistanceModern valueDistance -KMDistance/Earth’s radiusYojana KMfor each caseMoon38440065.4811.18Sun149,600,000875.40325.51Mars263.65Jupiter778,300,000142.78Saturn1,427,000,000105.34
22Place of Āryabhata Evidence of Spashta-bhūparidhi Spashta bhūparidhi = Bhūmadhyaparidhi* Cos C = C0* Cos where is latitude and C is earth’s circumferenceC0 = 3299,Interger Yojanas per degree of longitude at demandsC = 3240 = 360* and = ACOS(C0/C). i.e. = 10N51
23Historical Values of C0 & Derivation of AstronomerC0CYojanasPlaceĀryabhata329932409*36010.85C.VattamBrahmagupta5000468013*36020.61BhilmalaVarāhamihira320028808*36025.84KusumapuraManjula3600Bhāskara-II496719.57Bid392710*36023.55UjjayinīEratosthenes5040432012*36031.00Alexandria
24Camravttam:10N51Latitude 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.
25CHAMRAVATTAM 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_puzha10° 49′ 7.99″ N, 75° 57′ 10.01″ E
27What is special about this place? 5 Mistakes of Computational RulesArkāgrā, verse 31 of Goḷa which stipulates the condition for samamaṇḍala śaňkuEarth’s diameter and circumferenceErroneous use of Rversed sine (verses 35, 36 and 45 of Āryabhaṭīya)Precept on the visibility of AgastyaModification of the revolutions of Moon’s Nodes
28Chamravattam 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.
32Evidence of Arkāgra Verse 30 & 31 of Gola, Āryabhatīyam Amplitude of the Rising Sun: Distance of the rising orSetting sun from the East-West line
33Evidence 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
34Criticism 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' isSin 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.
37Arkāgra: 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) ~ At 10.85When = 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
38Carakhandas 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? = 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.
40Evidence of Ancient Shadow Floor at = 9.46 PalabhāPlaceCara-khandas in asus/4MesaVrsbhaMithuna1.04.76236192802.09.464763881602.310N5110.855484441805.022N3022.5011849603925.34Ujjayinī24.00127210324245.72Kusumapura25.50136011084526.026.5714281160
42Chengannur & Camravattam Palabhā1- Palabhā2 = 0.3 = 18 Vyaňgulas, combined with 1 - 2 = 01020’ gives the value of Palabhā = 5 at = i.e. (22.5/1.33)*18=303 5 AngulamCarakhandas fixed at this place of Palabhā = 2 formed the basis of the formula that we see in Mahābhāskarīya III.8
44Evidence of EclipsesEclipse 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
47Evidence for these Observations Modern Mean Longitudes for Mean New moon of 11 Aug 519 AD57'3130317036'317035'Rāhu30'3200325015''Apogee58'1400140021''Moon141000'SunModernĀryaSūryasiddhāntaAryabhatīyamPlanet
48One-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.
49Node and Apogee of Moon Node of Moon Apogee of Moon Modern Āryabhata KalielapsedModernĀryabhataDiff:35901852643-17'1682951-22'3600352112-1121542-1636121195458-43434655-936163234-214640-83619344303126837-63623267862712427364618216287219
50Mistaken 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, 75E45Meridian 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.
52Rule of Āryabhata for the Latitudinal Correction (Aksa Drk-karma) 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.
53Circumstances of the Lunar Eclipse of – 23 March 517 CE TimeSunλ0MoonVersineMid-heavenHhoursAksa-Valana0ĀryabhataB.GuptaModern17:004.4501.22778.5606.8713.35010.53018:004.49181.180.97692.335.9110.58-16.6010.8519:004.53181.790.726106.164.947.86-15.3910.4820:004.57182.400.494120.333.975.34-13.289.4421:004.61183.010.294135.093.013.17-10.477.7922:004.66183.620.139150.552.041.50-7.225.6423:004.70184.230.039166.641.070.42-3.773.140:004.74184.840.000183.070.110.00-0.360.691:004.78185.460.025199.41-0.860.272.792.462:004.82186.070.112215.29-1.831.215.515.033:004.86186.680.256230.49-2.792.767.647.314:004.90187.300.447245.01-3.769.089.115:00187.910.673259.02-4.737.289.8010.326:004.98188.530.920272.78-5.699.979.7910.850
54Observation shaped the Precept The lunar eclipse presented an occasion when the eclipsed moon had a meridian transit when posited close to the equator.At the intersection of the meridian and the equator, the Rversed sine of the hour angle H was zero and the Rsine of the aks̟avalana was also zero.Rversed sine of the hour angle and the aks̟avalana increased thereafter.When the eclipsed body was at the horizon at 18:00 hrs on 23 March or at 06:00 hrs on 24 March, the Rversed sine of H had its maximum value and the aks̟avalana also had its maximum value equal to the latitude of the place.East to West horizon observations received the correct representation with versine H but the same failed for the intended purpose during the progress of the eclipse between first contact and last contact
55“Rversed sine of the moon’s longitude (λ) increased by 900 multiplied by the Rsine of the obliquity () and the latitude of the moon (β) gives the Ayana-valana (Ay)
56Ay Results of Āryabhata and Brahmagupta TimeMoonλ0λ+900VersineSineĀryabhataBrahmaguptaAy0β*Ay017:00180.57270.570.99-1.0023.756.87-24.00-6.9518:00181.18271.180.9823.485.47-23.99-5.5919:00181.79271.790.9723.214.10-4.2420:00182.40272.400.9622.942.76-23.98-2.8821:00183.01273.010.9522.671.44-23.96-1.5322:00183.62273.620.9422.400.16-23.95-0.1723:00184.23274.230.9322.13-1.10-23.931.190:00184.84274.840.9221.86-2.32-23.912.541:00185.46275.460.9021.60-3.52-23.883.892:00186.07276.070.89-0.9921.33-4.68-23.865.243:00186.68276.680.8821.06-5.82-23.836.584:00187.30277.300.8720.80-6.93-23.797.925:00187.91277.910.8620.53-8.00-23.769.266:00188.53278.530.8520.27-9.05-23.7210.59
58Contrast of Āryabhatīya and Modern Results Total lunar eclipse of , 23 March 517 CE KalidaysFull MoonJDMeanλsTrueλmλnodeapogeeĀryabhatīyaMean Full Moon02022’04027’182022’178046’183044’48017’Modern λ02024’04018’182023’178016’183043’48012’Mod. Mean182024’178018’True Full Moon02047’04052’188006’184052’Modern02050’04044’188012’184044’183042’48015’
59Mahābhāskarīya VI. 39-41 Meridian Transit of Eclipsed Moon at 10N51, 75E45 TrialTimeLagnaMidheavenMoonMoon +900123:00254.47167.03184.21274.21223:30261.27174.30184.5274.5030:00268.09181.61184.79274.7940:08270183.65184.87274.8750:14271.2184.93184.92274.9260:30274.98188.98185.08275.0871:00281.98196.47185.38275.38In 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 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.
60Description of Lunar Eclipse “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”
67Sum up on Agastya rulePrecept 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.
68Agastya 8 May 517 AD, Sun 60 – 110 (Kerala) CanopusHorizon is the white line aboveAt sunset18:12 ZT, sunset Agastya had 12+ degree altitude, sufficient for observation after twilight over the sea horizon
69Agastya 26 April 517 AD, Sun 60 – 230 (Ujjayini) CanopusHorizon is the white line aboveAt sunset18:25 ZT Agastya had only 4 degree altitude
72Origin 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 easyUnnati varuthuka = Deriving R*Sin Padam varuthuka = Deriving R*Cos
74Deriving these 24 Rsine values for up to 900 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.
75Āryabhata & Jain Tradition Verse 9 of Kālakriya giving the Jaina 12 fold division of YugaVerse 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.
76Aryabhata lived in Ponnani? Monday, Jun 25, 2007Aryabhata lived in Ponnani?“Aryabhata lived in Ponnani in the northern coastal belt of Kerala,” says K. Chandra Hari, …