In 499 CE, Aryabhata Fitted Digit-by-Digit Roots, Series Sums, a 24-Entry Sine Table and an Equation-Solving Method Into About 120 Verses

The Aryabhatiya, finished in 499 CE, fits the arithmetic, algebra and trigonometry of its day into a mathematics chapter of 33 verses. MacTutor, the history-of-mathematics archive at the University of St Andrews, counts 66 rules in those verses, stated without proof. Its author, Aryabhata, gives his own age as 23, which places his birth in 476.

This piece is about the mathematics. The turning Earth and the eclipse explanation are in a separate article, and the satellite that carries his name is in another.

A 23-year-old in Kusumapura

Aryabhata dates himself in the book’s third chapter. In W. E. Clark’s 1930 English translation, published by the University of Chicago Press, the verse counts 3,600 years plus three quarters of a yuga, a vast astronomical cycle, and says twenty-three years of his life had gone by. If Aryabhata started the Kali age in 3102 BCE, as later astronomers did, the verse lands on 499 CE, Clark works out. He adds that Parameshvara, a commentator of the fifteenth century, quotes an earlier commentary that reads 499 as the date for which the book’s planetary tables are exact. MacTutor and Clark both treat 499 as the year of composition.

The book also names the place whose learning it sets out. In Clark’s translation Aryabhata presents “the science which is honored at Kusumapura.” David Pingree, writing in the Dictionary of Scientific Biography, identifies Kusumapura with Pataliputra, modern Patna, the imperial capital of the Guptas for much of the fourth and fifth centuries. The exact site of Kusumapura is uncertain, MacTutor adds. Clark says it “may or may not have been the place of his birth.”

Where he was born is open. A tradition that gives Asmaka, a region whose own location is debated, rests on a comment by the commentator Nilakantha Somasutvan, born in 1443. Pingree reads it as probably a confusion with Bhaskara I, and MacTutor says most historians now think Nilakantha mixed the two up. Others have proposed Kerala, Tamil Nadu, Andhra Pradesh or Bengal. MacTutor quotes S. Parameswaran’s conclusion that no final verdict can be given on either place.

Two more facts from the biographies. The Arab scholar al-Biruni believed there were two mathematicians named Aryabhata, and B. Datta showed in 1926 that they were one person, MacTutor says. And only one of his books survives. Pingree lists a second, now lost, that set out a system of reckoning the day from midnight.

A book built to be memorised

The Aryabhatiya has four sections: a short opening set of verses holding the astronomical constants, the mathematics chapter (Ganitapada, 33 verses), a chapter on the reckoning of time and planetary models (25 verses), and a chapter on the sphere and eclipses (50 verses). How many verses there are depends on how the opening is counted. The count in MacTutor is 118 verses, while Clark’s edition prints 13 opening stanzas, which with the 108 in the other three sections makes 121.

In Clark’s description the work is a summary rather than a full working manual. It states the distinctive parts of Aryabhata’s system and leaves out routine procedures that readers were expected to know. The mathematics are given as rules in compressed verse with no proofs, so a reader needed a teacher or a commentary.

Scholars have asked whether Aryabhata wrote every verse. B. L. van der Waerden suggested, as MacTutor reports, that the ten-verse opening was written later than the other three sections and that a few other verses were added by a member of the school at Kusumapura. Clark found no good reason to reject the opening section, partly because Brahmagupta quoted at least three passages from it in 628 and ascribed it to Aryabhata. Clark judged the work genuine on the whole. He wrote in 1930 that, if its authorship holds, it is the earliest preserved Indian mathematical and astronomical text bearing an individual author’s name.

Numbers written as syllables

Aryabhata wrote large numbers as words. Consonants stand for the values 1 to 25 and then 30 up to 100, and vowels attached to them move the value into higher places. The scheme can reach 10 to the power 18, according to MacTutor. Clark notes that the rule for the vowels can be translated in more than one way, and that the largest number Aryabhata uses himself has only ten places.

His own example is the number of times the Moon circles the Earth in a yuga. One Sanskrit word, read syllable by syllable, gives Clark the values 6, 30, 300, 3,000, 50,000, 700,000, 7,000,000 and 50,000,000. Together they make 57,753,336. Only the opening verses use this notation. Elsewhere the book uses ordinary number words.

A system like this depends on position fixing value, and the mathematics chapter says so directly. Verse 2 names the classes of numbers from one up to a thousand million and says that they run “from place to place each ten times the preceding,” in Clark’s translation. Clark judged that “it is hard to believe that such a descriptive alphabetical notation was not based on a place-value notation.” The wider history of the decimal system, from Brahmagupta’s rules for zero onward, is in the piece on India and zero.

Whether Aryabhata had a symbol for zero is a separate question, and the sources split. The word kha, which he uses for a place, became a common Sanskrit word for zero. In Clark’s words it was still disputed whether a zero symbol was in use in Aryabhata’s time, and he suggested calculation may have been done on a board ruled in columns, with an empty column for zero. MacTutor’s history of zero says he devised a number system that has no zero yet is positional. The historian Georges Ifrah, quoted by MacTutor, holds that “it is extremely likely that Aryabhata knew the sign for zero,” because his root extraction would be impossible without one. Oxford’s carbon dating of the Bakhshali manuscript, which uses a dot for an empty place, puts its earliest material as early as the third or fourth century. MacTutor’s page on Indian numerals says that manuscript’s date is disputed. The first dated zero that all historians accept is an inscription from Gwalior of 876, according to MacTutor.

Roots, sums and interest

The rules for roots come in verses 4 and 5. The square-root rule begins, in Clark’s translation: “One should always divide the avarga by twice the (square) root of the (preceding) varga.” Here varga means a square and also the odd-numbered places counting from the right, and avarga means the even-numbered places. Working through a number place by place, the method takes the digits of the root one at a time. Clark judged the process substantially correct while noting several difficulties in the wording. A cube-root rule follows in the same style.

Verse 22 gives two sums. The sum of the squares of the first n whole numbers is one sixth of the product of n, n plus 1, and twice n plus 1. Their cubes add up to the square of the sum of the numbers themselves. In modern notation, those are n(n+1)(2n+1)/6 and (1 + 2 + … + n) squared. Verse 19 gives the sum of any run of terms in a series that rises by a fixed step, and verse 20 works backward from a total to the number of terms, which involves a square root.

The chapter also holds a rule for a problem in which interest itself earns interest, a rule of three (“multiply the fruit by the desire and divide by the measure,” Clark translates it), and the area of a triangle as the perpendicular times half the base. The triangle and circle areas are correct, MacTutor notes.

The pulveriser

The most technical rules are verses 32 and 33. They find a whole number that leaves given remainders when divided by two different numbers. Whole-number solutions to equations of the form by = ax + c or by = ax minus c are the problem, in MacTutor’s description, and Aryabhata’s is the earliest treatment it knows. Astronomers met the question when working out the periods of the planets.

The method is called kuttaka, “to pulverise”: the problem is broken into new problems with smaller and smaller coefficients. In MacTutor’s account this is essentially the Euclidean algorithm for the highest common factor and is related to continued fractions. A toy case is a number that leaves 2 when divided by 5 and 3 when divided by 7, which is 17. Astronomical cases involve much larger divisors. In a 2006 paper for the RSA security conference, Subhash Kak notes that the algorithm for solving such congruences can replace the Chinese Remainder Theorem in certain situations.

Clark was candid about the two verses. They are compressed, and he found no help for reading them in Mahavira, Bhaskara or the second Aryabhata. The closest parallel is the fuller text of Brahmagupta, chapter 18 of the Brahmasphutasiddhanta (628). Clark’s reading therefore follows Parameshvara’s commentary and Brahmagupta’s wording, and he left it to mathematicians to decide which of the two rules is earlier. Bhaskara I’s commentary of 629 covers only the 33 mathematics verses, MacTutor says, and takes up first-degree indeterminate equations and gives examples illustrating Aryabhata’s rules.

Pi and the table of sines

Verse 10 tells the reader to add 4 to 100, multiply by 8 and add 62,000, which gives 62,832 as the circumference of a circle of diameter 20,000. That makes the ratio 3.1416. The verse calls the result “approximately” the circumference, in Clark’s translation. According to MacTutor, Aryabhata does not explain how he found the figure and used the square root of 10 in practice. A. Ahmad proposed a polygon of 256 sides, and E. M. Bruins showed that doubling the sides cannot produce the value. A fuller account is in the earlier article.

The first section lists 24 sine differences, from 225, 224 and 222 down to 7, for a quarter circle cut into steps of 225 minutes of arc, or 3 degrees 45 minutes. With a radius of 3,438 minutes, the 24 differences add up to exactly that figure. A circle of 21,600 minutes divided by twice pi gives about 3,438 as well. Verse 12 gives a rule for building the whole table from the first entry. Clark noted that the exact method Aryabhata used to compute the sines is not known.

Bhaskara I’s Mahabhaskariya later gives a rational formula for the sine and attributes it to Aryabhata. MacTutor reports a maximum error of under one percent. MacTutor’s page on the trigonometric functions traces Aryabhata’s Sanskrit word for a half-chord, jya, through the Arabic jiba and then jaib to the Latin sinus.

Where the sources differ on the mathematics

Two rules in the chapter are judged wrong. Verses 6 and 7 give the volume of a pyramid and of a sphere incorrectly, Clark says, and MacTutor reports that most historians treat them as wrong. K. Elfering argued that a different translation of two technical terms makes both correct. MacTutor observes that his reading needs those terms in unusual senses, so the question stays open. Clark added that Brahmagupta, who criticised Aryabhata about sixty times, mostly on astronomy and on departures from tradition, said nothing about the mathematics chapter, which Clark thought might only mean he found nothing to object to.

Commentaries from the sixth century to 1854

The book was explained for more than twelve centuries. Pingree lists the commentators: Prabhakara around 525, whose work is lost, then Bhaskara I in 629, Someshvara in about 1040, Suryadeva Yajvan of Kerala (born 1191), Parameshvara (about 1400 to 1450), Nilakantha Somasutvan (born 1443), whose Kerala school is the subject of a separate piece on series and calculus, and on to Kodandarama of Bobbili, whose Telugu commentary dates from about 1854. Pingree notes that scholars in south India studied it especially closely. The book was translated into Arabic around 800 as Zij al-Arjabhar, and Pingree says all of al-Biruni’s quotations come from that translation. H. Kern published the Sanskrit text with Parameshvara’s commentary in Leiden in 1874, and Clark’s English translation followed in 1930.

Bhaskara I, according to MacTutor, called Aryabhata the master who handed down the three sciences of mathematics, kinematics and spherics. India’s first satellite, launched on 19 April 1975, was named after him, ISRO’s page records. In 1979 the International Astronomical Union approved the name Aryabhata for a lunar crater, which the USGS gazetteer lists at 21.89 kilometres across.