Aryabhata

How to use this course

Āryabhaṭa is the first Indian mathematician and astronomer whose own words survive in a work under his name. That work, the Āryabhaṭīya, is short: 121 verses, written to be memorized, stating rules without proofs. Almost everything else about him comes from one autobiographical verse and from what later astronomers, admirers and critics, said about him. The course therefore keeps two kinds of evidence apart: the text itself, and later tradition.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Verse references follow the critical edition of K. S. Shukla and K. V. Sarma (1976); W. E. Clark’s translation (1930) numbers the first section slightly differently. Gītikā, Gaṇita, Kālakriyā and Gola denote the four sections. Angles are often measured in minutes of arc, written ′, as Āryabhaṭa did.

Timeline

All dates AD unless marked. Only the year 499 and the birth year 476 rest on Āryabhaṭa’s own words.

Kusumapura and the sources

Everything we know from Āryabhaṭa himself fits in two verses. In the time-reckoning section (Kālakriyā 10) he says that when sixty times sixty years of the Kali Yuga had passed, he was twenty-three years old. Kali year 3600 corresponds to AD 499, so he was born in 476. The verse fixes his age, not necessarily the year in which he wrote: 499 may be the epoch chosen for the calculations. At the start of the mathematics section (Gaṇita 1) he says he sets out the knowledge honoured at Kusumapura.

Kusumapura

Kusumapura, “city of flowers”, is usually identified with Pāṭaliputra, modern Patna on the Ganges, capital of the Gupta empire. Bhāskara I, his commentator in 629, confirms the link with Kusumapura. The phrase does not say he was born there, only that he wrote in its tradition.

Aśmaka and Kerala

Bhāskara I also calls him Āśmaka, “of the Aśmaka country”, and names his book the Āśmakatantra. Aśmaka was a region in the Deccan between the Godāvarī and Narmadā rivers. Nīlakaṇṭha, writing in Kerala around 1500, also calls him a man of Aśmaka. Since most surviving manuscripts and commentaries of the Āryabhaṭīya come from South India, some scholars have argued for a southern, even Keralan, origin. None of this is settled. A modern claim that he headed the university of Nālandā rests on a single word in a later verse calling him a kulapa, head of an institution, and is speculation.

Indian astronomy before Āryabhaṭa

The oldest Indian astronomical text, the Vedāṅga Jyotiṣa (late first millennium BC), regulated the calendar for Vedic ritual with a five-year cycle and simple arithmetic. In the early centuries AD, Indian astronomers absorbed Greek methods transmitted through texts in Sanskrit: the zodiac of twelve signs, the division of the circle into 360 degrees, epicycles and eccentrics, and the use of chord tables. The Yavanajātaka (“Greek horoscopy”), a Sanskrit versification made in western India in the second or third century, is one witness to this contact.

By Āryabhaṭa’s time several systems called siddhāntas (“established conclusions”) were in use. Varāhamihira of Ujjain, a younger contemporary, summarized five of them in his Pañcasiddhāntikā (“Five Siddhāntas”), including the Romaka, the Pauliśa and an early version of the Sūryasiddhānta. These gave rules for computing planetary positions from large cycles of revolutions.

What was new

Āryabhaṭa’s book was different in three ways. It bears an individual author’s name. It puts mathematics in its own section, as a discipline serving astronomy but stated in general form. And it changed the parameters: new revolution numbers for the planets, a mahāyuga divided into four equal quarters instead of the traditional proportions 4:3:2:1, and a rotating Earth. These innovations created a school, the Āryapakṣa, which competed for centuries with the Brāhmapakṣa that followed Brahmagupta.

The book of 121 verses

The Āryabhaṭīya has four sections called pādas (“quarters”):

• Gītikāpāda, 13 verses in the gīti metre: the alphabetic numeral system, the revolution numbers of the planets in a mahāyuga, the sizes of the Earth and planets, and the sine table in a single verse.
• Gaṇitapāda, 33 verses on mathematics: place value, squares and cubes and their roots, areas and volumes, π, the sine rule, shadows of the gnomon, series, interest, the rule of three, linear equations and the kuṭṭaka.
• Kālakriyāpāda, 25 verses on time: units, the conjunctions of planets, intercalary months, and the motions of planets on eccentrics and epicycles.
• Golapāda, 50 verses on the sphere: the celestial sphere and its circles, the spherical Earth and its rotation, day and night at different latitudes, and eclipses.

Bhāskara I calls the first section the “ten gīti” verses, though it has thirteen; which three are framing or later additions is debated. Another early name, Āryāṣṭaśata, “the 108 āryā verses”, refers to the last three sections.

A text for memory

The verses are compressed to the limit. Numbers are encoded in syllables (Lesson 04), rules are stated without proofs or examples, and a single line can hold a full algorithm. The book was meant to be learned by heart and explained by a teacher. This is why the commentaries matter so much: Bhāskara I in 629, Someśvara, Sūryadeva Yajvan in the 12th–13th centuries and Nīlakaṇṭha around 1500 supply the examples and arguments that the verses leave out. Several key rules are ambiguous, and modern interpretations sometimes depend on which commentator one follows.

The lost second work

Later astronomers quote a second treatise, usually called the Āryabhaṭa-siddhānta, in which the day begins at midnight rather than at sunrise. It described instruments including the gnomon, a water clock and shadow devices. Brahmagupta based his Khaṇḍakhādyaka on it, and Arabic sources knew its parameters.

Numbers in syllables: place value

For the general reader. Astronomy needs huge numbers: the Moon goes round the Earth more than 57 million times in Āryabhaṭa’s great cycle. To fit such numbers into memorizable verse, he invented a code in which every syllable is a number. The code only works because he thought in powers of ten, the decimal place value that India later gave the world.

The places (Gaṇita 2)

Āryabhaṭa names the decimal places: eka (1), daśa (10), śata (100), sahasra (1000), and so on up to vṛnda (\(10^9\)), and says each place is ten times the one before. This is the principle of decimal place value, stated as a rule.

The syllabic code (Gītikā 2)

The Sanskrit consonants are arranged in five groups of five (varga, “classed” letters, from k to m) followed by eight others (avarga: y r l v ś ṣ s h).

Each vowel thus governs two decimal places: a classed consonant fills the lower one (values 1–9 there, or 10–25 spilling into the upper), and an unclassed consonant fills the upper one (30–90, with h = 100 reaching into the next vowel’s range). For example, khyughṛ is \(kh\cdot u + y\cdot u + gh\cdot ṛ = 2\cdot10^4+30\cdot10^4+4\cdot10^6=4{,}320{,}000\), the number of solar revolutions in a mahāyuga.

Zero

The code has no symbol for zero: an empty place is simply omitted. But arithmetic in the Gaṇitapāda, particularly the square-root and cube-root algorithms (Lesson 05), proceeds digit by digit in a way that presupposes a written place-value notation, and Bhāskara I in 629 uses written numerals with zero in his commentary. Historians such as Georges Ifrah argue that Āryabhaṭa must have used such a notation; it is not attested in his own text.

Roots, series and the circle

For the general reader. The Gaṇitapāda is a compact handbook of the mathematics an astronomer needs: how to extract square and cube roots, find areas and volumes, sum series, and relate a circle’s circumference to its diameter. Most rules are right; a few volume rules are wrong. His value of π, correct to four decimal places, became famous.

Square roots (Gaṇita 4)

Split the number into pairs of digits from the right. Find the largest square not exceeding the leading group; its root is the first digit. Then repeatedly bring down the next digit, divide by twice the root found so far, subtract the square of the new quotient from what remains, and continue. The method is the one taught in schools until recently, and it rests on \((10a+b)^2=100a^2+2\cdot10a\cdot b+b^2\).

Series (Gaṇita 19–22)

The first sum counts the balls in a triangular pile (citi). Gaṇita 19–20 give the sum of an arithmetic progression and solve for the number of terms, which requires a quadratic equation.

Areas and volumes (Gaṇita 6–7)

The area of a triangle is half the base times the height, and the area of a circle is half the circumference times half the diameter: both correct. The volume of a “six-edged solid” (a tetrahedron) is given as half the base area times the height, and the volume of a sphere as the area of a great circle times its own square root. Both are wrong: the true factors are \(1/3\) and \(\tfrac43\pi r^3\). Some modern readers have proposed reinterpretations, but the commentators read the rules this way.

The circle (Gaṇita 10)

Āryabhaṭa says: add four to a hundred, multiply by eight, and add sixty-two thousand; this is the approximate circumference of a circle of diameter twenty thousand. That is \(\pi\approx (8\times104+62000)/20000=62832/20000=3.1416\). The word he uses, āsanna, means “near” or “approaching”. Some historians read it as recognizing that no exact ratio exists; that π is irrational was proved by Johann Heinrich Lambert in 1761. How he obtained the value is unknown. A regular polygon of 384 sides inscribed in a circle of diameter 20,000 gives a perimeter close to 62,832.

The sine table

For the general reader. Greek astronomers used tables of chords: the length of the straight line across a circle for each angle. Indian astronomers halved the chord and the angle, which gives the modern sine. Āryabhaṭa’s table of 24 sines in a quarter circle is the earliest that survives, and his rule for computing it uses what we would call a second difference, a first step toward the calculus of the sine.

Half-chords and the radius

If a circle has radius \(R\), the chord subtending an arc \(2\theta\) is \(2R\sin\theta\). The Sanskrit jyā or ardhajyā (“bow-string”, “half bow-string”) is the half-chord \(R\sin\theta\). Āryabhaṭa measures the radius in the same unit as the arc. With the circle divided into \(21600′\), the radius is \(R=21600/2\pi\approx3437.75\), which he rounds to 3438.

The table (Gītikā, one verse)

The quadrant is divided into 24 arcs of \(225′=3^\circ45′\). The verse gives not the sines but their differences \(\Delta_n=J_n-J_{n-1}\), where \(J_n=R\sin(n\cdot225′)\):

Their sum is 3438, the radius, as it must be since \(J_{24}=R\sin90^\circ\). The running sums agree with \(3438\sin\theta\) to within one minute; the largest error is at \(67^\circ30′\), where the table gives 3177 against 3176.3.

Proof. \(\Delta_{n+1}-\Delta_n=J_{n+1}-2J_n+J_{n-1}=R[\sin(n\alpha+\alpha)+\sin(n\alpha-\alpha)]-2R\sin n\alpha=2R\sin n\alpha\cos\alpha-2R\sin n\alpha\).

Āryabhaṭa’s rule (Gaṇita 12)

The verse is terse and its reading is disputed. On the reading of Bhāskara I and most modern scholars, it says that each difference is obtained from the previous one by subtracting the accumulated sum of the sines so far divided by the first sine, 225:

This is the proposition with the factor \(2(1-\cos\alpha)\approx\alpha^2=(225/3438)^2\approx1/233.5\) replaced by \(1/225\), a slightly larger factor that is easy to remember because 225 is the first sine. Applied literally the rule drifts below the table after the first few entries, so the tabulated values were probably computed or corrected by other means, for example from the half-angle formula. The idea that the change in the sine’s increments is proportional to the sine itself is the discrete form of \(\frac{d^2}{d\theta^2}\sin\theta=-\sin\theta\).

The pulverizer: indeterminate equations

For the general reader. Suppose you know that a planet returns to the same place every so many days, and another every so many days, and you want to know when both will be in given positions together. Questions like this lead to equations whose solutions must be whole numbers. Āryabhaṭa gave a general method, the kuṭṭaka or “pulverizer”, which grinds large numbers down to small ones by repeated division.

The problem

Find a number \(N\) that leaves remainder \(r_1\) on division by \(a\) and remainder \(r_2\) on division by \(b\). Writing \(N=ax+r_1=by+r_2\), this asks for non-negative integers \(x,y\) with

The method

Divide the larger divisor by the smaller, the smaller by the remainder, and so on, as in the Euclidean algorithm, recording the quotients \(q_1,q_2,\dots,q_k\). Āryabhaṭa’s verse then tells how to build the solution by working back up through the quotients, multiplying and adding at each stage; Bhāskara I explains it with worked examples. In modern terms the quotients are the partial quotients of the continued fraction of \(a/b\), and the back-substitution computes a convergent \(p/q\) with \(aq-bp=\pm1\). Multiplying by \(c\) and reducing gives the smallest solution.

Why it works. If \(a=q_1b+r\), then \(ax-by=c\) becomes \(b(q_1x-y)+rx=c\); setting \(y'=q_1x-y\) gives an equation in the smaller pair \((b,r)\). Each step reduces the coefficients, and when the remainder reaches 0 or 1 the equation is solved by inspection. Reversing the substitutions recovers \(x,y\). This is the “pulverizing”.

Astronomical use

Indian astronomers used the kuṭṭaka to recover the number of days elapsed since the epoch from observed fractional positions of planets, and to choose revolution numbers that make all planets align at the epoch. Brahmagupta and Bhāskara I extended it to systems of congruences, which is the content of the Chinese remainder theorem.

The yuga and the rotating Earth

For the general reader. Every day the stars rise in the east and set in the west. Almost every ancient astronomer took this at face value: the sky turns. Āryabhaṭa said instead that the Earth turns, and the stars only seem to move, as the shore seems to move to someone on a boat. His later followers mostly abandoned the idea, but his numbers for the length of the day and the year were among the most accurate of antiquity.

The revolution numbers (Gītikā 3–4)

In a mahāyuga of 4,320,000 years the Sun makes 4,320,000 revolutions, the Moon 57,753,336, and the Earth 1,582,237,500 rotations eastward. For the outer planets he gives Mars 2,296,824, Jupiter 364,224 and Saturn 146,564. Mercury and Venus, like the Sun, complete 4,320,000 mean revolutions; their faster cycles appear as the revolutions of their śīghra epicycles. All these are mean motions, which epicycles then correct.

The modern sidereal year is 365.25636 days, so Āryabhaṭa’s is about 3 minutes 20 seconds too long. His sidereal day agrees with the modern value, \(23^{\text h}56^{\text m}4.09^{\text s}\), to a tenth of a second.

The boat (Gola 9–10)

In Gola 9 Āryabhaṭa compares the observer to a man in a moving boat who sees fixed objects on the shore moving backward; in the same way, he says, the fixed stars are seen at Laṅkā moving westward. Laṅkā here is a reference point on the equator at the prime meridian of Ujjain, not the island. The next verse says that the rising and setting are caused by the sphere of stars, with the planets, being driven westward by the cosmic wind. The two verses are hard to reconcile. Some commentators took the second as describing appearances; some later copyists and commentators altered the first to remove the rotating Earth. Brahmagupta and later critics rejected the rotation on physical grounds; al-Bīrūnī later reported that Āryabhaṭa’s followers held the Earth to rotate and judged that the question did not affect the astronomical calculations.

Planets

Āryabhaṭa’s model is geocentric. Each planet’s mean position is corrected by two epicycles: the manda, which accounts for the unequal speed around the zodiac (like Ptolemy’s eccentric), and the śīghra, which accounts for retrograde motion (equivalent to the Earth’s motion around the Sun in a modern description). Unusually, he lets the epicycles change size around the circle. Because the śīghra cycle of an outer planet equals the solar year, some historians, notably Bartel van der Waerden, argued for a hidden heliocentric source; most historians reject this. He also says the Moon and planets shine by reflected sunlight.

Eclipses and shadows

For the general reader. Indian tradition explained eclipses by a demon, Rāhu, swallowing the Sun or Moon. Āryabhaṭa gave the physical explanation: a solar eclipse occurs when the Moon covers the Sun, a lunar eclipse when the Moon enters the Earth’s shadow. He then gave rules for computing the size of the shadow and the duration of an eclipse, using nothing but similar triangles.

The explanation (Golapāda)

The Moon covers the Sun, and the great shadow of the Earth covers the Moon. Eclipses happen only near the nodes, where the Moon’s path crosses the ecliptic, at new Moon (solar) or full Moon (lunar). The verses then give rules for the length of the shadow, its diameter at the Moon’s distance, the half-duration of the eclipse and the part of the disc obscured. Āryabhaṭa’s diameters, given in the Gītikāpāda in yojanas, are: Earth 1050, Sun 4410, Moon 315. The length of a yojana in his system is uncertain, but the ratios are what matter.

Proof. The shadow cone is tangent to both spheres. By similar triangles, the diameters of Sun and Earth are proportional to their distances from the cone’s apex: \(S/(L+d)=E/L\), so \(SL=EL+Ed\) and \(L=Ed/(S-E)\). Along the cone the diameter falls linearly from \(E\) at the Earth to 0 at the apex.

Duration

For a central lunar eclipse, the Moon moves relative to the shadow at the difference of the daily motions of the Moon and the Sun (the shadow’s centre is opposite the Sun). From first contact to mid-eclipse the Moon’s centre travels the sum of the radii of shadow and Moon. Āryabhaṭa’s rule divides this distance by the relative daily motion to obtain the half-duration.

Gnomons, lamps and latitude

For the general reader. The simplest astronomical instrument is a vertical stick, the gnomon, and the length of its shadow tells the time and the latitude. Āryabhaṭa gives a group of rules about shadows cast by a lamp and by the Sun. They rest on two ideas, similar triangles and the theorem of the right triangle, and he states both in general form.

Rule (ii) follows from (i): if the circle has radius \(r\) and the chord meets the diameter at distance \(x\) from the centre, the segments are \(r+x\) and \(r-x\), and \(c^2=r^2-x^2=(r+x)(r-x)\). It is the tool for the versine, the “reversed sine” \(r-r\cos\theta\) that Āryabhaṭa also used.

The lamp and the gnomon (Gaṇita 15–16)

A gnomon of height \(g\) stands at horizontal distance \(d\) from the foot of a lamp of height \(H>g\). Its shadow has length

This is Gaṇita 15. The next rule solves the inverse problem: from the lengths of two shadows of the same gnomon at two positions along a line from the lamp, find the distance and height of the lamp. If the shadows are \(s_1,s_2\) and the tips of the shadows are a distance \(t\) apart, then the distance from the tip of the first shadow to the foot of the lamp is \(s_1t/(s_2-s_1)\), and the lamp’s height is that distance times \(g/s_1\).

Derivation. The lamp’s top, the gnomon’s top and the shadow’s tip are collinear, so by similar triangles \(H/(d+s)=g/s\), which gives the first rule. For the second, let \(D_i\) be the distance from the lamp’s foot to the tip of shadow \(i\). Then \(H/D_i=g/s_i\), so \(D_1/s_1=D_2/s_2\). With \(D_2=D_1+t\), \(D_1(s_2-s_1)=s_1t\).

Latitude

At noon on an equinox the Sun stands on the celestial equator, and its altitude is \(90^\circ-\varphi\), where \(\varphi\) is the latitude. A gnomon of height \(g\) then casts a shadow \(s\) with \(s/g=\tan\varphi\). In the Indian tradition the gnomon was 12 units (aṅgulas) tall, and the “equinoctial shadow” of a place was a standard datum. The Golapāda uses the corresponding sine form: \(R\sin\varphi:R\cos\varphi=s:g\).

Pupils, critics and commentators

Nothing is known of Āryabhaṭa’s later life or death. The date 550 often given for his death is a convention without a source. His influence, however, can be traced in detail.

Pupils

Bhāskara I names Āryabhaṭa’s pupils, including Pāṇḍuraṅgasvāmin, Lāṭadeva, Prabhākara and Niḥśaṅku. Lāṭadeva is usually identified with the Lāṭācārya whom Varāhamihira cites on the older systems. Bhāskara I himself, a century later, called Āryabhaṭa the master who, having reached the far shore of the ocean of mathematics, astronomy and spherics, handed down their jewels. His commentary of 629 is our best guide to how the text was taught, and his own Mahābhāskarīya gave a celebrated rational approximation to the sine.

Brahmagupta

Brahmagupta, writing at Bhillamāla in 628, devoted much of his Brāhmasphuṭasiddhānta to attacking Āryabhaṭa: his equal yugas, his rotating Earth, his eclipse theory in places, and his parameters. Yet in 665 Brahmagupta based his practical handbook, the Khaṇḍakhādyaka, on Āryabhaṭa’s midnight system, which suggests that the attacks were partly about school allegiance. The Āryapakṣa and Brāhmapakṣa remained rival traditions.

The South

The Āryabhaṭīya was taught and copied most intensively in South India. Lalla in the 8th century followed it closely. In Kerala, the school founded by Mādhava of Saṅgamagrāma in the 14th century worked within the Āryabhaṭīya tradition, and Nīlakaṇṭha Somayājī’s commentary around 1500 is one of its major works. The Kerala school went on to find infinite series for the sine, cosine and arctangent, growing directly from the sine-difference ideas of Lesson 06.

From jyā to sine: the long legacy

Baghdad

In the 8th century Indian astronomical works reached the Abbasid court. Āryabhaṭa’s parameters were known in Arabic as the system of the Zīj al-Arjabhar, and Muḥammad ibn Mūsā al-Khwārizmī’s astronomical tables drew on Indian methods. Al-Bīrūnī, around 1030, discussed Āryabhaṭa in his book on India, including the question of the Earth’s rotation.

The word “sine”

Sanskrit jyā was transliterated into Arabic as jība, a word without other meaning. Written without vowels as jyb, it was read as the ordinary word jayb, “pocket” or “fold”. In the 12th century Gerard of Cremona and others translated jayb into Latin as sinus, which also means a fold or bay. The cosine and versine have parallel Indian ancestors (koṭijyā, utkramajyā).

India today

Traditional Indian calendars (pañcāṅgas) still use rules descended from the Āryabhaṭa and Brahmagupta schools. India’s first satellite, launched on 19 April 1975, was named Aryabhata; Aryabhatta Knowledge University in Patna and the Aryabhatta Research Institute of Observational Sciences at Nainital also carry his name, as does a lunar crater.

Assessment

• Mathematics: a systematic decimal arithmetic, the first surviving sine table and a second-difference rule for it, the general solution of linear indeterminate equations, and sum formulas for powers.
• Astronomy: highly accurate mean motions, a rotating Earth, and a physical theory of eclipses with quantitative rules.
• Limits: wrong volume formulas, no proofs in the text, and a geocentric model.

Connections

People linked to Aryabhata across this collection. A name without a link has no page here yet. See Aryabhata in the network of all 337 lives →

Students

• Latadeva
• Nihsanku
• Panduranga-svamin
• Prabhakara

Influenced by

• Hipparchus

Influenced

• al-Biruni
• Madhava of Sangamagrama
• Muhammad ibn Musa al-Khwarizmi
• Bhaskara I 629
• Lalla
• Nilakantha Somayaji

Colleagues and collaborators

• Varahamihira

Rivals

• Brahmagupta

Also mentioned in this life

• Euclid
• Johann Heinrich Lambert
• Claudius Ptolemy
• Bartel van der Waerden

Fields

• astronomy 50 lives
• mathematics 15 lives
• trigonometry 9 lives
• number theory 45 lives

On CoursesHub

• Brahmagupta to Kerala School · History of Mathematics
• Astrophysics for Beginners

Sources and further reading

Texts and translations

1. W. E. Clark, The Āryabhaṭīya of Āryabhaṭa: An Ancient Indian Work on Mathematics and Astronomy, University of Chicago Press, 1930.
2. K. S. Shukla and K. V. Sarma, Āryabhaṭīya of Āryabhaṭa, critical edition with translation and notes, Indian National Science Academy, New Delhi, 1976.
3. K. S. Shukla, Āryabhaṭīya of Āryabhaṭa with the Commentary of Bhāskara I and Someśvara, INSA, 1976.
4. H. Kern, The Aryabhatiya, with the commentary Bhatadipika of Paramadiçvara, Leiden, 1874.

Biography and history

1. Aryabhata I, MacTutor History of Mathematics.
2. Aryabhata and Aryabhatiya, Wikipedia.
3. K. Plofker, Āryabhaṭa I, Biographical Encyclopedia of Astronomers.
4. K. Plofker, Mathematics in India, Princeton University Press, 2009, ch. 4.
5. D. Pingree, Āryabhaṭa I, Dictionary of Scientific Biography, vol. 1, 1970.

Mathematics and astronomy

1. Āryabhaṭa’s sine table, Wikipedia (with the 24 differences and a comparison with modern values).
2. J. Q. Jacobs, The Āryabhaṭīya of Āryabhaṭa: the oldest precise astronomical constant? (revolution numbers and derived periods).
3. G. G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed., Princeton, 2011.
4. B. L. van der Waerden, The heliocentric system in Greek, Persian and Hindu astronomy, Annals of the New York Academy of Sciences 500 (1987).
5. G. Ifrah, The Universal History of Numbers, Wiley, 2000.