The only surviving work usually attributed to Aristarchus, On the Sizes and Distances of the Sun and Moon, is based on a geocentric world view. It has historically been read as stating that the angle subtended by the Sun's diameter is 2 degrees, but Archimedes states in The Sand Reckoner that Aristarchus had a value of ½ degree, which is much closer to the actual average value of 32' or 0.53 degrees. The discrepancy may come from a misinterpretation of what unit of measure was meant by a certain Greek term in Aristarchus' text.

Though the original text has been lost, a reference in Archimedes' book The Sand Reckoner (Archimedis Syracusani Arenarius & Dimensio Circuli) describes another work by Aristarchus in which he advanced the heliocentric model as an alternative hypothesis. Archimedes wrote:

You (King Gelon) are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.

—The Sand Reckoner

Aristarchus thus believed the stars to be very far away, and that in consequence there is no observable parallax, that is, a movement of the stars relative to each other as the Earth moves around the Sun. The stars are much farther away than was generally assumed in ancient times; and since stellar parallax is only detectable with telescopes, his speculation although accurate was unprovable at the time.

The geocentric model was consistent with planetary parallax and was assumed to be the reason why no stellar parallax was observed. Rejection of the heliocentric view was common, as the following passage from Plutarch suggests (On the Apparent Face in the Orb of the Moon):

Cleanthes (a contemporary of Aristarchus and head of the Stoics) thought it was the duty of the Greeks to indict Aristarchus on the charge of impiety for putting in motion the hearth of the universe ... supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis.

—Tassoul, Concise History of Solar and Stellar Physics

The only other astronomer of antiquity who is known by name and who is known to have supported Aristarchus' heliocentric model was Seleucus of Seleucia, a Hellenistic astronomer who lived a century after Aristarchus.

Aristarchus claimed that at half moon (first or last quarter moon), the angle between Sun and Moon was 87°. Possibly he proposed 87° as a lower bound since gauging the lunar terminator's deviation from linearity to 1° accuracy is beyond the unaided human ocular limit (that limit being about 3° accuracy). Aristarchus is known to have also studied light and vision.

Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 and 20 times farther away than the Moon. (The true value of this angle is close to 89° 50', and the Sun's distance is actually about 400 times the Moon's.) The implicit false solar parallax of slightly under 3° was used by astronomers up to and including Tycho Brahe, ca. AD 1600. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the diameter of the Sun was between 18 and 20 times larger than the diameter of the Moon; which, although wrong, follows logically from his data. It also leads to the conclusion that the Sun's diameter is almost seven times greater than the Earth's; the volume of Aristarchus's Sun would be almost 300 times greater than the Earth. This difference in sizes may have inspired the heliocentric model.

Mentioned by Archimedes and by modern scientists as the first to propose a heliocentric "universe", Aristarchus also proposed an ancient Greek time period, his "Great Year" of 4868 solar years, equaling exactly 270 saroi, each of 18 Callippic years plus 10⅔ degrees. (Syntaxis) Its empirical foundation was the 4267 month eclipse cycle, cited by Ptolemy as source of the "Babylonian" month, which was good to a fraction of a second (1 part in several million). It is found on cuneiform tablets from shortly before 200 BC, though Ptolemy did not attribute its origin to Babylon. (Due to near integral returns in lunar and solar anomaly, eclipses 4267 months apart exceptionally never deviated by more than an hour from a mean of 126007 days plus 1 hour, the value given by Ptolemy at op. cit. Thus, estimation of the length of the month was ensured to have relative accuracy of 1 part in millions.) Embedded in the Great Year was a length of the month agreeing with the Babylonian value to 1 part in tens of millions, decades before Babylon is known to have used it. There are indications that Babylon's month was exactly that of Aristarchus, which would suggest that one party obtained it from the other or from a common source. Aristarchus's lunar conception represents an advance of science in several respects. Previous estimates of the length of the month were in error by 114 seconds (Meton, 432 BC) and 22 seconds (Callippus, 330 BC). The attribution of a mean motion to such a motion as the moon's was possibly new.

The Vatican library has preserved two ancient manuscripts with estimates of the length of the year. The only ancient scientist listed for two different values is Aristarchus. It is now suspected that these are among the earliest surviving examples of continued fraction expressions. The most obvious interpretations can be computed from the manuscript numbers.

The results are years of 365 + 1/152 days, and 365 − 15/4868 days, representing the sidereal year and the civil, supposedly tropical year.

Both denominators can be related to Aristarchus, whose summer solstice was 152 years after Meton's and whose Great Year was 4868 years. The difference between the sidereal and tropical years is identical to precession. The former value is accurate within a few seconds. The latter is erroneous by several minutes.

Both are close to the values later used by Hipparchus and Ptolemy, and the precession indicated is almost precisely 1 degree per century, a value which is too low. 1 degree per century precession was used by all later astronomers until the Arabs. The correct value in Aristarchus's time was about 1.38 degrees per century.

Aryabhata mentions in the Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.

Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." The Asmaka were one of the 16 of Ancient Indian Mahajanapadas, and the only one situated south of the Vindhyas. Since Aryabhata lived in and around 480, he resided in The Gupta's, near the end of the Golden Age.

It is widely attested that during the mid first millennium BCE, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India, and it is possible Aryabhata was born there. However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander.

Many are of the view that he was born in the south of India in Kerala and lived in Magadha at the time of the Gupta rulers.

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapati) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

Some archeological evidence suggests that Aryabhata could have originated from the present day Kodungallur in Tamilakam, modern Kerala. For instance, one hypothesis was that aśmaka (Sanskrit for "stone") may be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Tamilakam was used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Tamilakam, and the Aryasiddhanta was completely unknown in Kerala.

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums - of - power series, and a table of sines.

The Arya-siddhanta, a work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight - day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku - yantra), a shadow instrument (chhAyA - yantra), possibly angle - measuring devices, semicircular and circular (dhanur - yantra / chakra - yantra), a cylindrical stick yasti - yantra, an umbrella - shaped device called the chhatra - yantra, and water clocks of at least two types, bow - shaped and cylindrical.

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

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. 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. 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 pādas or chapters:

1. 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.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. 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-tithis, and a seven - day week with names for the days of week.
4. 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. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

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).

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 explains that knowledge of zero was implicit in Aryabhata's place - value system as a place holder for the powers of ten with null coefficients.

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.

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam 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."

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

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 was proved in Europe only in 1761 by Lambert.

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half - side is the area."

Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it 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". And after that, the sinus became sine in English.

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as diophantine equations. 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 is called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first - order diophantine equations and is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras.

In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:

and

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 as elliptical rather than circular.

Aryabhata correctly insisted that the earth 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 in other parts of the world, that the sky rotated. This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter:

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. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast). 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."

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. 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.

Solar and lunar eclipses were scientifically explained by Aryabhata. Aryabhata states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo - planetary nodes Rahu and Ketu, 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. He discusses at length the size and extent of the Earth's shadow 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.

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; 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) is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).

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, though this has been rebutted. 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, though the evidence is scant. 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.

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighboring 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, modern names "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).

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 c.) 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, 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. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Gregorian calendar.

India's first satellite Aryabhata and the lunar crater Aryabhata are named in his honor. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIOS) near Nainital, India. The inter - school Aryabhata Maths Competition is also named after him, as is Bacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.