Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002
An Anecdote on Mādhava School of Mathematics
Athira K Babu Research Scholar, Department of Sanskrit Sahitya, Sree Sankaracharya University of Sanskrit, Kalady,
Abstract
The Sanskrit term ‘Gaṇitaśāstra’, meaning literally the “science of calculation” is used for mathematics. The mathematical tradition of ancient India is an ocean of knowledge that is dealing with many topics such as the Vedic, Jain and Buddhist traditions, the mathematical astronomy, The Bhakshali manuscripts, The Kerala School of mathematics and the like. Thus India has made a valuable contribution to the world of mathematics. The origin and development of Indian mathematics are connected with Jyotiśāstra1.
This paper tries to deconstructing the concept of mathematical tradition of Kerala with respect to Niḷā valley civilization especially under the background of medieval Kerala and also tries to look into the Mādhava School of mathematics through the life and works of great mathematician Mādhava of Saṅgamagrāma and his pupils who lived in and around the river Niḷā.
Keywords: Niḷā, Literature review, Mathematical Tradition of medieval Kerala, Mādhava of Saṅgamagrāma, Great lineage of Mādhava.
Introduction
Niḷā, the Nile of Kerala is famous for the great ‘Māmāṅkam’ festival. The word ‘Niḷā’point out a culture more than just a river. It has a great role in the formation of the cultural life of south Malabar part of Kerala. It could be seen that the word ‘Peraar’ indicating the same river in ancient scripts and documents. The Niḷā is the life line of many places such as Chittur, Ottappalam, Shornur, Cheruthuruthy, Pattambi, Thrithala, Thiruvegappura, Kudallur, Pallippuram, Kumbidi,
1 The Sanskrit word used for Astronomy is Jyotiśāstra. In ancient times Jyotiśāstra was treated as a part of Jyotiṣa which deals with all aspects of Astronomy. Śāstra is the word used to denote science in India. But the meaning of science in western context is quite different from that of śāstra. The word śāstra derived from the root śās which means some rules about a particular knowledge. Jyotiśāstra is the study of the sun moons, stars, planets, comets, galaxies, dust and other non- earthly bodies and phenomena, or may be defined as the scientific study of celestial bodies. It is one of the oldest sciences. It is one of the oldest sciences. Indian Astronomy continues to play an integral role from prehistoric to modern times. Indian astronomers and astronomy credited with interesting theories and discoveries.
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Thirunavaya, Alathiyur, Trikkandiyur and Kottakkal which has a great cultural and scientific tradition. Literature Review
In 1832, a paper ‘On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras, the Tantra Sangraham, Yucti Bhasha, Carana Padhati, and Sadratnamala’ was presented by Charles M. Whish at the Royal Asiatic Society in London. 2 It was a pioneer work in highlighting the mathematical arena of Kerala. The eminent scholars Rama Varma Maru Tampuran, Akhilesvara Ayyar, C. T. Rajagopal, T. A. Sarasvati Amma, R. C. Gupta, K. V. Sarma, S. Madhavan, George Ghevarghese Joseph and the like have done several noteworthy efforts to spread out a light in the mathematical tradition of Niḷā Basin.
This noteworthy efforts and works help to reveal the vital role and contributions of mathematicians Saṅgamagrāma Mādhava, Vaṭaśśeri Parameśvara, Nīlakaṇṭha Somayāji, Jyeṣṭhadeva and the like who lived in and around the Niḷā valley. More precisely it can be said that the mathematical tradition of Kerala flourished in and around Niḷā basin. Among these mathematicians Saṅgamagrāma Mādhava has a significant place.
Due to the presence of the great mathematician Āryabhaṭa in the writings of Kerala mathematicians, Sarasvati Amma refers to that mathematical tradition as Āryabhaṭa School. Some scholars say it as Kerala School; without consider both history and geography. Kim Plofker, in her work ‘Mathematics in India’ names this as ‘School of Mādhava’. Before this, in ‘Mathematics of India: Concepts, Methods, Connections’, P. P. Divakaran prefers to call this the Niḷā School.
Mathematical tradition of Medieval Kerala
In the beginning of the seventh chapter of the book ‘Indian Mathematics’, Kim Plofker says:
“Probably the most famous school in Indian mathematics, and the one that produced many of its most remarkable discoveries, is the guru-paraṃparā or “chain of teachers” originating with Mādhava in the late fourteenth century and continuing at least into the beginning of the seventeenth. These scholars lived in the region known as Kerala on the southwestern coast of India, in its central part between modern Kochi (or Cochin) and
2 Charles M Whish (1794-1833), who was appointed as a civil servant by the Madras Establishment of the East India Company in Kochi, near to the old capital of Mahodayapuram. In 1832, he presented a paper at the Royal Asiatic Society in London on four manuscripts, which are collected by him in Malabar. The manuscripts were Nīlakaṇṭha's Tantrasaṅgraha , Jyeṣṭhadeva's Yuktibhāṣa (a Malayalam text), Nīlakaṇṭha Somayājī's Karaṇapaddhati, Śaṅkara Varmā's Sadratnamālā . The first two of them are the key text of the Niḷā corpus.
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Kozhikode (or Calicut). What survives of their work includes writings in Sanskrit and in the local Dravidian vernacular called Malayalam. In astronomy they are generally considered to be followers of the Ārya-pakṣa, but they also wrote on texts in other pakṣas, as well as on astronomical systems unique to Kerala.
A narrow strip of land between the Western Ghats Mountains and the Arabian Sea, Kerala in the mid-second millennium maintained a distinct regional culture without being entirely isolated from the neighboring parts of southern India. Moreover, its pepper production and geographical location had made it a major international hub, with trading connections stretching back for many centuries. ” (Plofker 2009, 218)
This quote hints the fact about that Kerala, where the śāstras like Astronomy, Mathematics and Ayurveda have been flourished in a traditional lineage of guruśiṣyaparaṃparā and its possible knowledge transmission to other parts of the world; influence of Āryabhaṭa and other Indian scholars and the like.
Kerala School has credited with a number of commentaries and original works related to Mathematics and Astronomy; which were based on Āryabhaṭīya, Laghubhāskarīya, Mahābhāskarīya, Bṛhajjātaka and the like. Especially a number of astronomical works are produced from Kerala School by the influence of Āryabhaṭa. Most of the available commentaries on the Āryabhaṭīya have been written by the scholars such as Vaṭaśśeri Parameśvara, Nīlakaṇṭha Somayājī and the like and they have generally made the correction, supplementation and revision of Āryabhaṭa system in order to get accuracy in results. Parahita system is an example for the correction to the Āryabhaṭa system. The system put forward the correction called bhaṭābdasaṃskāra or śakābdasamskāra which presents all the data, directions and sine tables necessary for the computation of the planets. The system is introduced by Haridatta through his astronomical works Grahacāranibandhana and Mahāmārganibandhana at the occasion of the great Māmāṅkam festival. Saṅkara Varma in his Sadratnamālā says about this Māmāṅkam event as follows:
आचार्ाार्ाभटप्रणीतगणणतं प्रार्ः स्फुटं तत् खलु
गोत्रोत्तु敍गणिताब्दके व्यणभचरन् ब्रह्माददणिद्धाꅍतके ।
दृ嵍वैषम्र्वशाद् िहास्थलणिते क쥍र्ब्धके णनणितः
िंस्कारो णवबुधैर्ातः परणहत配वं तेषु वीनेष्वर्ि् ।।
Later this system gave way to the Dṛggaṇita system introduced by Parameśvara who revised the Parahita system.
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Āryabhaṭa introduced a numeral system based on the Sanskrit alphabets and the schemes of representation of values of this system are shown below:
Ña Ka=1 Kha=2 Ga=3 Gha=4 Ṅa =5 Ca=6 Cha=7 Ja=8 Jha=9 =10 Ṭa Ṭha Ḍha Ṇa Ḍa =13 Ta=16 Tha=17 Da=18 Dha=19 Na=20 =11 =12 =14 =15
Pa=21 Pha=22 Ba=23 Bha=24 Ma=25 Ya=30 Ra=40 La=50 Va=60 Śa=70
Ṣa Sa=90 Ha=100 =80
The twenty-five Vargīyavyañjanas (stops both oral and nasal) from Ka to Ma are consisting of the five Vargas which represent numbers from one to twenty-five. The numbers from 26 to 29 and the numbers in between the other multiples of ten such as 31 to 39, 41 to 49, 51 to 59, etc. are represented by sequences of two letters. In this system, the numbers are to be read from backward, i.e. from right to left (aṅgānām vāmato gatiḥ). But it is difficult for pronunciation.3
Even though the Kerala astronomers had been the strong influence of the Āryabhaṭa School of astronomy, they didn’t follow the Āryabhaṭa system of numerals. Instead of that system, Kaṭapayādi system was used by them. This is an easy method of expressing numbers through letters. This is also a numeral system based on Sanskrit alphabets somewhere similar to Kacaṭapayādi system or Āryabhaṭa system. The system was more popular in south India, especially in Kerala. This is also termed as ‘Paralpperu’ or ‘Akṣarasaṃkhyā’. The legendary figure Vararuci is credited with this innovation. The value of letters in this system is as follows:
1 2 3 4 5 6 7 8 9 0
ka kha ga gha ṅa ca cha ja jha ña
ṭa ṭha ḍa ḍha ṇa ta tha da dha na
pa pha ba bha ma
ya ra la va śa ṣa sa ha ḷa
3 Example: cakha =26 (ca=6, kha =2), gaja=83 (ga =3 ja =8)
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The chronograms using the above systems are widely used to determine the date of the manuscripts and also of the inscriptions. In these systems, the numbers are to be read from backward, i.e. from right to left (aṅgānām vāmato gatiḥ).4 The Tantrasaṅgraha of Nīlakaṇṭha, the Yuktibhāṣā of Jyeṣṭhadeva, Kriyākramakarī of Saṅkaravāriyar and Nārāyaṇa, Karaṇapaddhati of Putumana Somayājin, Sadratnamālā of Saṅkaravarman are the important works in the field of astronomy in Kerala. The important feature of the last four authors is that they all are descendants from Mādhava and Nīlakaṇṭha who are referred to as Ācāryas by them. This continuity of tradition is one of the main features of the Kerala School of Astronomy. The transmission of knowledge from teacher to disciple or father to son was very popular.
The great lineage of Mādhava
The Mādhava School of Mathematics origins with Mādhava of Saṅgamagrāma, who is considered to be the greatest mathematician-astronomer of medieval India. The date of Mādhava is given by 1340-1425 C.E. by K.V. Sarma in the introduction of his critical edition of the works ‘Veṇvāroha’ and ‘Sphuṭacandrāpti’ of Mādhava and most of the scholars have accepted this date. Earlier historians says that his birth place may be at Saṅgamagrāma5, the village of Saṅgameśvara, a deity of Kūṭalmāṇikyaṃ temple and also the information regarding his birthplace, Mādhava himself gives in his work ‘Veṇvāroha’ by the following verses:
बकुलाणधणित配वेन णवहारो र्ो णवशेष्र्ते । गृहनािणन िोर्ं िेर्ाणिजनािणन िाधवः ।। (Veṇvāroha 13) K.V. Sarma has identified him to be the author of the following works. Goḷavāda, Madhyamānayanaprakāra, Lagnaprakaraṇa, Veṇvāroha, Sphuṭacandrāpti and Agaṇita- grahacāra. Among these Goḷavāda and Madhyamānayanaprakāra are known only through references in other works and only Veṇvāroha,and Sphuṭacandrāpti are available in print , both are critically edited and published by K.V.Sarma6. The rest is available in the manuscript form. He
4 Determining the date of the work: The last line of the verse in Nārāyaṇīyam of Melputtūr Nārāyaṇa Bhaṭṭatiri reads as Āyurārogyasaukhyam. By calculating the numerical value of this vākya using Kaṭapayādi system, the number obtained is 0122171 as given in the following column. It is to be read from right to left and thus the sequence indicates 1712210 which is considered as the kalidina. The corresponding Kalivarṣa is 4688 which is 1587 A.D. this is the date of composition of Nārāyaṇīyam.
5 The Village Saṅgamagrāma, the modern Iriñjālakkuṭa, situated besides the joining of the rivers Kunti and Niḷā, in cenral Kerala. 6 The Sphuṭacandrāpti, titled by ‘Computation of True Moon by Mādhava of Saṅgamagrāma’ which is critically edited with Introduction, Translation and Notes, by K.V. Sarma and published by Hoshiarpur Vishveshvaranand Institute in 1973. Veṇvāroha of Mādhava of Saṅgamagrāma with the commentary of Acyutapiṣāraṭī of Trikkaṇṭiyūr has been critically edited by K.V. Sarma and published by Govt. Sanskrit College, Thrippunithura in 1956.
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was referred to by later scholars as ‘Goḷavid’7 (Master of spherics), who knows about planets. The most famous mathematical contribution of Mādhava to the world of Mathematics are Mādhava- Newton’s power series for sine and cosine trigonometric functions, Mādhava-Gregory and Leibnitz’s series for the inverse tangent, Taylor’s series for sine and cosine functions, Mādhava- Leibnitz’s power series for 휋/4 and the like. But all these mathematical treasure survive only through a few verses recorded by his pupils. i.e., Today , all informations regarding to Mādhava comes from the works of later scholars, primarly Nīlakaṇṭha and Jyeṣṭhadeva; a lott of references to his original theories on mathematics appear in later works, including the Tantrasaṅgraha of Nīlakaṇṭha Somayājī's and the Yuktibhāṣa of Jyeṣṭhadeva, and Kriyākramakarī commentary on Līlāvatī by Saṅkara and Nārāyaṇa.
For example, there are some quotations of Mādhava in Tantrasaṅgraha of Nīlakaṇṭha Somayājī. The verse of Mādhava is quoted which gives the trigonometric identities sin(퐴 ± 퐵) = sin 퐴 cos 퐵 ± cos 퐴 sin 퐵, which is known as Jīveparasparanyāya. Mādhava’s expression of this trigonometric identity is stated by the following verses:
जीवे परस्परणनजेतरिौ셍वाकाभ्र्ा-
िभ्र्स्र् णवस्तृणतदलेन णवभा煍र्िाने ।
अꅍर्ोꅍर्र्ोगणवरहानुगुणे भवेताि् ।। - इणत िाधवः।8
The following verses depicted Mādhava’s expression for the power series of trigonometric functions sine and cosine.
णनह配र् चापवगेण चापं तत्त配फलाणन च।
हरेत् ििूर्ु嵍वगैणि煍र्ावगााहतैः क्रिात् ।।
चापं फलाणन चाधोधोꅍर्स्र्ोपर्ुापरर 配र्जेत् ।
जीवाप्त्र्ै, िंगेरहोऽस्र्ैव णवद्वाणन配र्ा ददनाकृतः ।।
7 Mādhava was the teacher of Parameśvara,(A.D.1360-1455), the promulgator of the Dṛggaṇita school of Kerala astronomy and is frequently quoted in the medieval astronomical literature of Kerala with the appellation of Goḷavid ('Adept in Spherics'). Thus Nīlakaṇṭha Somayāji (1444-1545 A.D.), while referring to Parameśvara in his Āryabhaṭīyabhāṣya says: Parameśvaras tu ...Mādhavādibhyo 'Goḷavidbhyo' Gaṇita-goḷa-yuktir api bālya eva samyag gṛhītvā ....Acyuta Piṣāraṭī uses the same appeiiation for Mādhava in the introductory verse in his Sphuṭanirṇaya : vande 'goḷavidaś' ca Mādhavamukhān etc. (K. V. Sarma, Inroduction to his edn. of Sphuṭacandrāpti.p.12)
8 According to the author of Yuktibhāṣā, this is the quotation of Madhava assigned to Tantrasaṅgraha. Hence all the works are interconnected. They belong to the same tradition.
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णनह配र् चापवगेण 셂पं तत्त配फलाणन च ।
हरेणद्विूलर्ु嵍वगैणि煍र्ावगाहतैः क्रिात् ।।
ककंतु व्यािदलैिैव णद्वघ्नेनाद्यं णवभ煍र्ताि् ।
फलाꅍर्धोधः क्रिशोꅍर्स्र्ुपर्ुापरर 配र्जेत् ।
शराप्तर्ै, िंग्रहोऽस्र्ैव स्तेनणि配र्ाददनाकृतः ।।
(तꅍत्रिंग्रहि्)
The power series of sine and cosine are stated as follows:
푥3 푥5 sin 푥 = 푥 − + − ⋯ 3! 5! 푥2 푥4 cos 푥 = 1 − + − ⋯ 2! 4! This is generally known as Newton Power Series and now it also known in the name of Mādhava, as Mādhava Newton Power Series. It was discovered by Mādhava and re discovered 200 years later by Newton. It can be seen that the elaborate explanations of rationales behind those above mentioned verses in the work Yuktibhāṣa of Jyeṣṭhadeva.
Similarly it could be seen that the depiction of Taylor series for sine and cosine functions by Mādhava, quoted by Keḷallūr Nīlakaṇṭha Somayājin in his commentary on the Āryabhaṭīya Gaṇita, with the statement: tatrāha Mādhavaḥ ; Mādhava’s expression of the Gregory and Leibnitz’s series for the inverse tangent, quoted in the commentary Kriyākramakarī on Bhāskarācārya's Līlāvatī by Śaṅkara and Nārāyaṇa , and also quoted in Yuktibhāṣa; Leibnitz’s power series for 휋 and approximations to the value of 휋 by Mādhava, quoted in the Kriyākramakarī and the like. That is, the references to all these above mentioned Mādhava’s statements and their traditional elucidation on mathematics appear only in his later works such as Nīlakaṇṭha Somayāji's Bhāṣya on the Āryabhaṭīya, Gaṇita;Yuktibhābhāṣā; Kriyākramakarī on Līlāvaṭī ;Karaṇapaddhati and the like.
In the great lineage of the Mādhava School of mathematics, Parameśvara of Vaṭaśśeri, the only known direct pupil of Mādhava, has a prominent place. He was born in the village called Aśvatthagrāma in 15th century A.D. The village Aśvatthagrāma , modern Ālattiyūr is located at the coastal on Gāyatripuzha, a tributary of the Niḷā. He wrote several works on astronomy which include Dṛggaṇita, Goladīpikā(3 volumes), Grahaṇāṣṭaka, Grahaṇamaṇḍana, Grahaṇanyāyadīpikā, Candracchāyāgaṇita, Vākyakaraṇa and Commentaries on Āryabhaṭīya, Mahābhāskarīya, Laghubhāskarīya, Sūryasiddhānta, Laghumānasa, Līlāvatī, and the like.
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Parameśvara founded the Dṛggaṇita system which is one of the main traits of Mādhava School. It is a revision of parahita system and more accurate; this system was used for horoscopy (jātaka), astrological inquiry (praśna), and computation of eclipses (grahaṇa), parahita used for fixing auspicious times for rituals and ceremonies (muhūrta).
Keḷallūr Nīlakaṇṭha Somayājin is one of the great savants in this lineage. Nīlakaṇṭha Somayājin was the pupil of Vaṭaśśeri Dāmodara, who was the son and pupil of Parameśvara of Vaṭaśśeri. Nīlakaṇṭha was born in Keḷallūr illaṃ (Keraḷasadgrāma in Sanskrit) at Tṛkkaṇṭiyūr (Kuṇḍapura in Sanskrit) near Tirur. Nīlakaṇṭha’s Tantrasaṅgraha is a famous work.Grahaṇanirṇaya, Candracchāyāgaṇita, Goḷasāra, Siddhāntadarpaṇa, and Sundararājapraśnottara are his other works. There are two commentaries available on Tantrasaṅgraha, Yuktidīpikā and Laghuvivṛti by Śṅkara Vāriyar.
Jyeṣṭhadeva is the author of the popular Yuktibhāṣā which is a prose in Malayalam language, contains detailed proof of Tantrasaṅgraha . He was the teacher of Acyutapiṣāraṭi and the pupil of Vaṭaśśeri Dāmodara. He belonged to the Paraṅṅoṭṭu family of the Ālattiyūr village in South Malabar.
Śaṅkara Vāriyar has a significant place in this lineage. Śaṅkara Vāriyar was the disciple of Netranārāyaṇa, Āzhvāñceri Tamprākkaḷ; and also Nīlakaṇṭha Somayāji and Jyeṣṭhadeva were the teachers of Śaṅkara Vāriyar. Śaṅkara Vāriyar (1500-1560 AD) was born in Tṛkkuṭaveli(Sanskrit: Śrīhuta) near Ottappalam on Niḷā. Śaṅkara Vāriyar wrote several works on mathematics and astronomy. Kriyākramakarī commentary of Līlāvatī is his most celebrated work, but unfinished one. Laghuvivṛti and Yuktidīpikā are the two commentaries on Tantrasaṅgraha of Nīlakaṇṭha Somayāji; written by Śaṅkara Vāriyar. Yuktidīpikā is based on Jyeṣṭhadeva's Yuktibhāṣā and is written in verses. Nārāyaṇa of Mahiṣamaṅgala completed Śaṅkara’s unfinished Kriyākramakarī commentary of Līlāvatī. Several anonymous pupils and commentators include in this great lineage. Tṛkkaṇṭiyūr Acyutapiṣāraṭi is one of the last but not the least identifiable member of this great mathematical lineage. He was a pupil of Jyeṣṭhadeva and the instructor of Melputtūr Nārāyaṇa Bhaṭtatiri who was a famous poet and a grammarian too. He has written a commentary on the Veṇvāroha of Saṅgamagrāma Mādhava. This great lineage shows that the Mādhava School has a great history of mathematical and astronomical tradition and there were a number of scholars in this continuous tradition on astronomy and mathematics.9This continuity of tradition is one of the main features of the Mādhava school of Mathematics.
Conclusion
The śāstras like astronomy, mathematics and Ayurveda have been flourished in a traditional lineage of guruśiṣyaparamparā in and around Niḷā valley. The period between the 13th
9 K. V. Sarma in his A History of The Kerala School of Hindu Astronomy mentioned about above 70 Astronomers.
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to 18th C.E was the golden age of astronomy and mathematics in Mādhava School. Mādhava School of astronomy and mathematics is now regarded as one of the major areas to be explored. Mādhava’s contributions to the field of astronomy and mathematics are unique compared to other mathematicians in India. Continuity of tradition is one of the main features of the Mādhava School of mathematics. It can be said that Mādhava School has a great heritage of scientific literature, especially in mathematics and astronomy. Many of the scholars and their texts have to know where some others still exist untouched. It is our duty to find out them and make them available to the world.
References
Divakaran, P P 2018. The Mathematics of India : Concepts, Methods, Connections.New Delhi: Singapore, Springer . George Gheverghese Joseph. 2009. A passage to infinity : medieval indian mathematics from kerala and its impact. Delhi ; London: Sage. George Gheverghese Joseph. 2011. Ananthathayilekku Oru Patha (R.Padmaraj, Trans.). Kottayam 686001, Kerala: DC Books. George Gheverghese Joseph. 2011. Kerala Mathematics : History and Its Possible Transmission to Europe. Br Publishing Corporation. George Gheverghese Joseph. 2013. Mayurasikha (Second Edition; P. R. Menon, Trans.). Nalanda, Thiruvananthapuram, Kerala: State Institutes of Languages. George Gheverghese Joseph. 2016. Indian Mathematics : Engaging with the World, from Ancient to Modern Times. New Jersey: World Scientific. Plofker, Kim 2009. Mathematics in India. Princeton: Princeton University Press. Sarma, Krishna V (Ed.)1956. Veṇvāroha of Mādava of Saṅgamagrāma with the commentary of Acyutapiṣāraṭī of Trikkaṇṭiyūr. Kerala;Govt. Sanskrit College, Tripunithura. Sarma, Krishna V 1972. A History of the Kerala School of Hindu Astronomy (in Perspective). Hoshiarpur ; Vishveshvaranand Inst.
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