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cholar and chronicler of india abū rayhān al bīrūnī 10 aryabhatiya main article aryabhatiya direct details of aryabhata s work are known only from the aryabhatiya the name aryabhatiya is due to later commentators aryabhata himself may not have given it a name 7 his disciple bhaskara i calls it ashmakatantra or the treatise from the ashmaka it is also occasionally referred to as arya shatas ashta literally aryabhata s 108 because there are 108 verses in the text 20 7 it is written in the very terse style typical of sutra literature in which each line is an aid to memory for a complex system thus the explication of meaning is due to commentators the text consists of the 108 verses and 13 introductory verses and is divided into four pāda s or chapters gitikapada 13 verses large units of time kalpa manvantra and yuga which present a cosmology different from earlier texts such as lagadha s vedanga jyotisha c 1st century bce there is also a table of sines jya given in a single verse the duration of the planetary revolutions during a mahayuga is given as 4 32 million years ganitapada 33 verses covering measurement kṣetra vyāvahāra arithmetic and geometric progressions gnomon shadows shanku chhaya simple quadratic simultaneous and indeterminate equations kuṭṭaka 19 kalakriyapada 25 verses different units of time and a method for determining the positions of planets for a given day calculations concerning the intercalary month adhikamasa kshaya tithi s and a seven day week with names for the days of week 19 golapada 50 verses geometric trigonometric aspects of the celestial sphere features of the ecliptic celestial equator node shape of the earth cause of day and night rising of zodiacal signs on horizon etc 19 in addition some versions cite a few colophons added at the end extolling the virtues of the work etc 19 the aryabhatiya presented a number of innovations in mathematics and astronomy in verse form which were influential for many centuries the extreme brevity of the text was elaborated in commentaries by his disciple bhaskara i bhashya c 600 ce and by nilakantha somayaji in his aryabhatiya bhasya 1465 ce 20 19 aryabhatiya is also well known for his description of relativity of motion he expressed this relativity thus just as a man in a boat moving forward sees the stationary objects on the shore as moving backward just so are the stationary stars seen by the people on earth as moving exactly towards the west 7 mathematics place value system and zero the place value system first seen in the 3rd century bakhshali manuscript was clearly in place in his work while he did not use a symbol for zero the french mathematician georges ifrah argues that knowledge of zero was implicit in aryabhata s place value system as a place holder for the powers of ten with null coefficients 21 however aryabhata did not use the brahmi numerals continuing the sanskritic tradition from vedic times he used letters of the alphabet to denote numbers expressing quantities such as the table of sines in a mnemonic form 22 approximation of π part of a series of articles on the mathematical constant π 3 14159 26535 89793 23846 26433 uses area of a circle circumference use in other formulae properties irrationality transcendence value less than 22 7 approximations milü madhava s correction term memorization people archimedes liu hui zu chongzhi aryabhata madhava jamshid al kashi ludolph van ceulen françois viète seki takakazu takebe kenkō william jones john machin william shanks srinivasa ramanujan john wrench chudnovsky brothers yasumasa kanada history chronology a history of pi in culture indiana pi bill pi day related topics squaring the circle basel problem six nines in π other topics related to π tau v t e aryabhata worked on the approximation for pi π and may have come to the conclusion that π is an irrational number in the second part of the aryabhatiyam gaṇitapāda 10 he writes caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ add four to 100 multiply by eight and then add 62 000 by this rule the circumference of a circle with a diameter of 20 000 can be approached 23 this implies that for a circle whose diameter is 20 000 the circumference will be 6 832 i e π displaystyle pi 62 832 20 000 displaystyle 62 832 over 20 000 3 1416 displaystyle 3 1416 which is accurate to two parts in one million 24 it is speculated that aryabhata used the word āsanna approaching to mean that not only is this an approximation but that the value is incommensurable or irrational if this is correct it is quite a sophisticated insight because the irrationality of pi π was proved in europe only in 1761 by lambert 25 after aryabhatiya was translated into arabic c 820 ce this approximation was mentioned in al khwarizmi s book on algebra 10 trigonometry in ganitapada 6 aryabhata gives the area of a triangle as tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ that translates to for a triangle the result of a perpendicular with the half side is the area 26 aryabhata discussed the concept of sine in his work by the name of ardha jya which literally means half chord for simplicity people started calling it jya when arabic writers translated his works from sanskrit into arabic they referred it as jiba however in arabic writings vowels are omitted and it was abbreviated as jb later writers substituted it with jaib meaning pocket or fold in a garment in arabic jiba is a meaningless word later in the 12th century when gherardo of cremona translated these writings from arabic into latin he replaced the arabic jaib with its latin counterpart sinus which means cove or bay thence comes the english word sine 27 āryabhata uses the following formulae relating sin functions in his works sin π 2 x sinπ 2 versinx sin π x sinπ 2 versinπ 2 sinxsl sin 3π 2 x sinπ 2 versinπ 2 sinπ 2 versinx 28 these formulae are no longer used as the function versin has become obsolete citation needed indeterminate equations a problem of great interest to indian mathematicians since ancient times has been to find integer solutions to diophantine equations that have the form ax by c 29 this problem was also studied in ancient chinese mathematics and its solution is usually referred to as the chinese remainder theorem this is an example from bhāskara s commentary on aryabhatiya find the number which gives 5 as the remainder when divided by 8 4 as the remainder when divided by 9 and 1 as the remainder when divided by 7 that is find n 8x 5 9y 4 7z 1 it turns out that the smallest value for n is 85 in general diophantine equations such as this can be notoriously difficult they were discussed extensively in ancient vedic text sulba sutras whose more ancient parts might date to 800 bce aryabhata s method of solving such problems elaborated by bhaskara in 621 ce is called the kuṭṭaka कुट्टक method kuṭṭaka means pulverizing or breaking into small pieces and the method involves a recursive algorithm for writing the original factors in smaller numbers this algorithm became the standard method for solving first order diophantine equations in indian mathematics and initially the whole subject of algebra was called kuṭṭaka gaṇita or simply kuṭṭaka 30 algebra in aryabhatiya aryabhata provided elegant results for the summation of series of squares and cubes 31 1 2 2 2 n 2 n n 1 2 n 1 6 displaystyle 1 2 2 2 cdots n 2 n n 1 2n 1 over 6 and 1 3 2 3 n 3 1 2 n 2 displaystyle 1 3 2 3 cdots n 3 1 2 cdots n 2 see squared triangular number astronomy aryabhata s system of astronomy was called the audayaka system in which days are reckoned from uday dawn at lanka or equator some of his later writings on astronomy which apparently proposed a second model or ardha ratrika midnight are lost but can be partly reconstructed from the discussion in brahmagupta s khandakhadyaka in some texts he seems to ascribe the apparent motions of the heavens to the earth s rotation he may have believed that the planet s orbits are elliptical rather than circular 32 33 motions of the solar system aryabhata correctly stated in aryabhatiyam that the earth is round and it rotates about its axis daily and that the apparent movement of the stars is a relative motion caused by the rotation of the earth contrary to the then prevailing view that the sky rotated 24 34 30 this is indicated in the first chapter of the aryabhatiya where he gives the number of rotations of the earth in a yuga 35 and made more explicit in his gola chapter 36 in the same way that someone in a boat going forward sees an unmoving object going backward so someone on the equator sees the unmoving stars going uniformly westward the cause of rising and setting is that the sphere of the stars together with the planets apparently turns due west at the equator constantly pushed by the cosmic wind aryabhata described a geocentric model of the solar system in which the sun and moon are each carried by epicycles they in turn revolve around the earth in this model which is also found in the paitāmahasiddhānta c 425 ce the motions of the planets are each governed by two epicycles a smaller manda slow and a larger śīghra fast 37 the order of the planets in terms of distance from earth is taken as the moon mercury venus the sun mars jupiter saturn and the asterisms 10 the positions and periods of the planets was calculated relative to uniformly moving points in the case of mercury and venus they move around the earth at the same mean speed as the sun in the case of mars jupiter and saturn they move around the earth at specific speeds representing each planet s motion through the zodiac most historians of astronomy consider that this two epicycle model reflects elements of pre ptolemaic greek astronomy 38 another element in aryabhata s model the śīghrocca the basic planetary period in relation to the sun is seen by some historians as a sign of an underlying heliocentric model 39 eclipses solar and lunar eclipses were scientifically explained by aryabhata he states that the moon and planets shine by reflected sunlight instead of the prevailing cosmogony in which eclipses were caused by rahu and ketu identified as the pseudo planetary lunar nodes he explains eclipses in terms of shadows cast by and falling on earth thus the lunar eclipse occurs when the moon enters into the earth s shadow verse gola 37 he discusses at length the size and extent of the earth s shadow verses gola 38 48 and then provides the computation and the size of the eclipsed part during an eclipse later indian astronomers improved on the calculations but aryabhata s methods provided the core his computational paradigm was so accurate that 18th century scientist guillaume le gentil during a visit to pondicherry india found the indian computations of the duration of the lunar eclipse of 30 august 1765 to be short by 41 seconds whereas his charts by tobias mayer 1752 were long by 68 seconds 10 sidereal periods considered in modern english units of time aryabhata calculated the sidereal rotation the rotation of the earth referencing the fixed stars as 23 hours 56 minutes and 4 1 seconds 40 the modern value is 23 56 4 091 similarly his value for the length of the sidereal year at 365 days 6 hours 12 minutes and 30 seconds 365 25858 days 41 is an error of 3 minutes and 20 seconds over the length of a year 365 25636 days 42 heliocentrism as mentioned aryabhata advocated an astronomical model in which the earth turns on its own axis his model also gave corrections the śīgra anomaly for the speeds of the planets in the sky in terms of the mean speed of the sun thus it has been suggested that aryabhata s calculations were based on an underlying heliocentric model in which the planets orbit the sun 43 44 45 though this has been rebutted 46 it has also been suggested that aspects of aryabhata s system may have been derived from an earlier likely pre ptolemaic greek heliocentric model of which indian astronomers were unaware 47 though the evidence is scant 48 the general consensus is that a synodic anomaly depending on the position of the sun does not imply a physically heliocentric orbit such corrections being also present in late babylonian astronomical texts and that aryabhata s system was not explicitly heliocentric 49 50 legacy this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed march 2017 learn how and when to remove this message india s first satellite named after aryabhata citation needed illustration of aryabhatta aryabhata s work was of great influence in the indian astronomical tradition and influenced several neighbouring cultures through translations the arabic translation during the islamic golden age c 820 ce was particularly influential some of his results are cited by al khwarizmi and in the 10th century al biruni stated that aryabhata s followers believed that the earth rotated on its axis his definitions of sine jya cosine kojya versine utkrama jya and inverse sine otkram jya influenced the birth of trigonometry he was also the first to specify sine and versine 1 cos x tables in 3 75 intervals from 0 to 90 to an accuracy of 4 decimal places in fact the modern terms sine and cosine are mistranscriptions of the words jya and kojya as introduced by aryabhata as mentioned they were translated as jiba and kojiba in arabic and then misunderstood by gerard of cremona while translating an arabic geometry text to latin he assumed that jiba was the arabic word jaib which means fold in a garment l sinus c 1150 51 aryabhata s astronomical calculation methods were also very influential along with the trigonometric tables they came to be widely used in the islamic world and used to compute many arabic astronomical tables zijes in particular the astronomical tables in the work of the arabic spain scientist al zarqali 11th century were translated into latin as the tables of toledo 12th century and remained the most accurate ephemeris used in europe for centuries calendric calculations devised by aryabhata and his followers have been in continuous use in india for the practical purposes of fixing the panchangam the hindu calendar in the islamic world they formed the basis of the jalali calendar introduced in 1073 ce by a group of astronomers including omar khayyam 52 versions of which modified in 1925 are the national calendars in use in iran and afghanistan today the dates of the jalali calendar are based on actual solar transit as in aryabhata and earlier siddhanta calendars this type of calendar requires an ephemeris for calculating dates aryabhatta knowledge university aku patna has been established by government of bihar for the development and management of educational infrastructure related to technical medical management and allied professional education in his honour the university is governed by bihar state university act 2008 india s first satellite aryabhata and the lunar crat...