Aryabhata

2.1. Name

Aryabhata's name is correctly spelled as आर्यभट_Aryabhata^Sanskrit, not "Aryabhatta," which is a common misspelling by analogy with other names ending in the "भट्ट_bhatta^Sanskrit" suffix. Historical astronomical texts, including over a hundred references by Brahmagupta, consistently spell his name as Aryabhata. Furthermore, the "Aryabhatta" spelling often does not fit the traditional Sanskrit meter. The suffix "भट_bhaṭa^Sanskrit" means "hired person" or "mercenary," while "भट्ट_bhaṭṭa^Sanskrit" denotes an "educated person" or "scholar," which might explain the tendency to use the latter for a revered scholar, though it is historically inaccurate. When it is necessary to distinguish him from a 10th-century mathematician of the same name, he is referred to as Aryabhata I. He was also known as āśmakīya, meaning "one belonging to the Aśmaka country."

2.2. Time and Place of Birth

Aryabhata stated in his seminal work, the Āryabhaṭīya, that he was 23 years old in the year 3600 of the Kali Yuga. This corresponds to 499 CE, implying that he was born in 476 CE. Aryabhata identified himself as a native of Kusumapura, which is widely identified with Pataliputra, modern-day Patna, in the Indian state of Bihar.

2.3. Education and Academic Background

It is widely accepted that Aryabhata traveled to Kusumapura for advanced studies and resided there for a significant period. Both Hindu and Buddhist traditions, along with Bhaskara I's writings, confirm Kusumapura as Pataliputra (modern Patna). A verse indicates that Aryabhata held the position of kulapa (head of an institution) at Kusumapura. Given that the renowned Nalanda University was located in Pataliputra during his time and possessed a large astronomical observatory, it is speculated that Aryabhata might have been the head of Nalanda University. He is also credited with establishing an observatory at the Sun temple in Taregana, Bihar.

2.4. Historical Context

Aryabhata flourished during the late Gupta Empire, a period characterized by significant cultural and scientific revival in India. This era saw renewed cultural contact with the Western world, which influenced the resurgence of Vedic astronomy and mathematics. Key centers for mathematical and astronomical research from the 5th to 10th centuries included Kusumapura, Ujjayini, and Mysore. Aryabhata's activities were primarily centered in Kusumapura.

His period of activity in Kusumapura, specifically around 499 CE when he composed the Āryabhaṭīya, is estimated to have been during the reign of Budhagupta, one of the later Gupta kings. Historical records indicate Budhagupta's rule extended at least until 494 CE, with other Gupta kings appearing in inscriptions until 510 CE. The Gupta Empire, though facing decline and fragmentation during this later period, still maintained significant territorial control, extending from North Bengal in the east to the Malwa region in the west, and as far north as Kannauj. To the south, it bordered the Vakataka dynasty near the Narmada River. Since Ashmaka in the Deccan region was outside the Gupta Empire's direct control at the time, it is plausible that Aryabhata traveled from there, crossing the Godavari and Narmada rivers, to reach Kusumapura. A widely accepted theory suggests that Aryabhata was invited by Budhagupta to become the head of Nalanda University, known for its high intellectual standards. The Gupta and Vakataka dynasties maintained friendly relations through marital alliances.

Before Aryabhata's era, Indian astronomy and mathematics, particularly from the 4th century BCE to the 5th century CE, developed from the religious necessity of conducting sacrifices during the Vedic period. This led to the observation of solar and lunar movements based on 27 or 28 nakshatras (lunar mansions). This knowledge was compiled into a branch of astronomy known as "Ganita Jyotisha," serving as an auxiliary science to the Vedas.

3.1. Āryabhaṭīya

The Āryabhaṭīya is Aryabhata's most renowned and surviving work. The name "Āryabhaṭīya" was given by later commentators, as Aryabhata himself may not have titled it. His disciple, Bhaskara I, referred to it as Ashmakatantra (treatise from Ashmaka). It is also sometimes called Arya-shatas-aShTa (literally, Aryabhata's 108) due to its 108 verses. The text is written in a concise, aphoristic style typical of sutra literature, where each line serves as a mnemonic for a complex system, requiring detailed explanations from commentators. The work consists of 108 main verses and 13 introductory verses, divided into four pādas or chapters:

• Gitikapada (13 verses): This section defines large units of time, such as kalpa, manvantara, and yuga, presenting a cosmology that differs from earlier texts like Lagadha's Vedanga Jyotisha. It also includes a table of sines (jya) in a single verse, and specifies the duration of planetary revolutions during a mahayuga as 4.32 million years.
• Ganitapada (33 verses): This chapter covers mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomons and shadows (shanku-chhAyA), as well as simple, quadratic equations, simultaneous equations, and indeterminate equations (kuṭṭaka).
• Kalakriyapada (25 verses): This section addresses different units of time and a method for determining planetary positions for a given day. It includes calculations related to the intercalary month (adhikamAsa), kShaya-tithis (omitted lunar days), and a seven-day week with names for the days.
• Golapada (50 verses): This chapter delves into the geometric and trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, lunar nodes, the shape of the Earth, the cause of day and night, and the rising of zodiacal signs on the horizon. Some versions also include colophons at the end, praising the work.

The Āryabhaṭīya introduced numerous mathematical and astronomical innovations in verse form, which remained influential for centuries. Its extreme brevity necessitated elaboration through commentaries, most notably by his disciple Bhaskara I in his Bhashya (c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE). The work is also notable for its description of the relativity of motion, where Aryabhata stated: "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."

3.2. Ārya-siddhānta

The Ārya-siddhānta is a lost astronomical treatise attributed to Aryabhata. Its existence and content are known primarily through the writings of his contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. The title translates to "Aryabhata's Astronomy" or "Aryabhata's Siddhanta." This work appears to be based on the older Surya Siddhanta and employed a midnight-day reckoning system, in contrast to the sunrise reckoning used in Āryabhaṭīya.

The Ārya-siddhānta also contained descriptions of several astronomical instruments, including:

• The gnomon (shanku-yantra)
• A shadow instrument (chhAyA-yantra)
• Possibly angle-measuring devices, both semicircular and circular (dhanur-yantra / chakra-yantra)
• A cylindrical stick (yasti-yantra)
• An umbrella-shaped device (chhatra-yantra)
• At least two types of water clocks: bow-shaped and cylindrical.

This text gained significant popularity in North India during the 7th century. Even Aryabhata's critic, Brahmagupta, summarized it in his work Khandakhadyaka, which he ironically titled "sweetened food" to criticize it. The Khandakhadyaka was later translated into Arabic as Al-Kand and widely used in the Islamic world as a simple manual for astronomical calculations. The Persian scholar Abū Rayhān al-Bīrūnī, who spent a decade in India, also re-translated Khandakhadyaka, further disseminating Aryabhata's ideas.

3.3. Other Works

A third text, potentially surviving in an Arabic translation, is known as Al ntf or Al-nanf. This work claims to be a translation by Aryabhata, though its original Sanskrit title remains unknown. Dating probably from the 9th century, it was mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

4.1. Place Value System and Zero

Aryabhata's work clearly demonstrates the use of a place-value system, which was first observed in the 3rd-century Bakhshali Manuscript. Although he did not use a distinct symbol for zero, the French mathematician Georges Ifrah argues that the knowledge of zero was implicitly understood in Aryabhata's place-value system as a placeholder for powers of ten with null coefficients. However, Aryabhata did not employ Brahmi numerals. Instead, 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.

4.2. Approximation of π

Aryabhata worked on the approximation of pi (π) and may have concluded that π is irrational. In the second part of the Āryabhaṭīya (Ganitapada 10), he states in Sanskrit:

चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम्_{caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām}^Sanskrit

अयुतद्वयविष्कम्भस्यासन्नो वृत्तपरिणाहः।_{ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.}^Sanskrit

"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."

4.3. Trigonometry

In Ganitapada 6, Aryabhata provided a rule for the area of a triangle:
: त्रिभुजस्य फलशरीरं समदलकोटी भुजार्धसंर्गः_{tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ}^Sanskrit
This translates to: "for a triangle, the result of a perpendicular with the half-side is the area."

Aryabhata introduced the concept of sine in his work under the name ardha-jya, meaning "half-chord." This term was later simplified to jya. When Arabic scholars translated his works from Sanskrit into Arabic, they transcribed jya as jiba. Due to the omission of vowels in Arabic script, it was often abbreviated as jb. Subsequent writers mistakenly interpreted jb as jaib, an Arabic word meaning "pocket" or "fold (in a garment)," as jiba itself was meaningless in Arabic. In the 12th century, when Gherardo of Cremona translated these Arabic texts into Latin, he replaced jaib with its Latin equivalent, sinus, meaning "cove" or "bay," from which the English word "sine" is derived. Aryabhata was also the first to specify sine and versine (1 - cos x) tables, in 3.75° intervals from 0° to 90°, with an accuracy of four decimal places. His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) were foundational to the birth of trigonometry.

4.4. Indeterminate Equations

A long-standing problem for Indian mathematicians was finding integer solutions to Diophantine equations of the form ax + by = c. This problem was also explored in ancient Chinese mathematics, leading to the Chinese remainder theorem. An example from Bhaskara I's commentary on Āryabhaṭīya illustrates this:
: 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. The smallest integer value for N is 85. Such Diophantine equations can be highly complex. They were extensively discussed in ancient Vedic texts like the Sulba Sutras, some parts of which may date back to 800 BCE. Aryabhata's method for solving these problems, further elaborated by Bhaskara in 621 CE, is called the कुट्टक_kuṭṭaka^Sanskrit method. Kuṭṭaka means "pulverizing" or "breaking into small pieces," and the method involves a recursive algorithm to reduce the original factors into smaller numbers. This algorithm became the standard approach for solving first-order Diophantine equations in Indian mathematics, and initially, the entire subject of algebra was known as kuṭṭaka-gaṇita or simply kuṭṭaka.

4.5. Algebra and Series

In Āryabhaṭīya, Aryabhata provided elegant formulas for the summation of series of squares and cubes:
: $1^2\; +\; 2^2\; +\; \backslash cdots\; +\; n^2\; =\; \{n(n\; +\; 1)(2n\; +\; 1)\; \backslash over\; 6\}$
and
: $1^3\; +\; 2^3\; +\; \backslash cdots\; +\; n^3\; =\; (1\; +\; 2\; +\; \backslash cdots\; +\; n)^2$ (see squared triangular number)

4.6. Mensuration

Aryabhata contributed to the field of mensuration, including the accurate calculation of the area of a triangle, as noted in his Ganitapada (verse 6). While he provided a correct rule for the area of a triangle, he also presented an incorrect rule for the volume of a pyramid, claiming it to be half the height multiplied by the area of the base.

5.1. Earth's Rotation and Relativity of Motion

Aryabhata correctly asserted that the Earth rotates about its axis daily. He proposed that the apparent westward movement of the stars is a relative motion caused by the Earth's rotation, directly contradicting the then-dominant belief that the sky itself rotated. This concept is indicated in the first chapter of the Āryabhaṭīya, where he specifies the number of Earth's rotations within a yuga. He further elaborates this 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.

5.2. Astronomical Models

Aryabhata's primary astronomical system was called the audAyaka system, which reckoned days from uday (dawn) at Lanka (an abstract point on the equator at the same longitude as Ujjayini). Some of his later, lost writings on astronomy, which apparently proposed a second model called ardha-rAtrikA (midnight reckoning), can be partially reconstructed from discussions in Brahmagupta's Khandakhadyaka.

He described a geocentric model of the Solar System, where the Sun and Moon are each carried by epicycles, which in turn revolve around the Earth. This model, also found in the Paitāmahasiddhānta (c. 425 CE), posited that the motions of the planets are governed by two epicycles: a smaller manda (slow) and a larger śīghra (fast). The order of planets in terms of increasing distance from Earth was given as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

The positions and periods of the planets were calculated relative to uniformly moving points. Mercury and Venus were described as moving around the Earth at the same mean speed as the Sun. Mars, Jupiter, and Saturn moved around the Earth at specific speeds, representing their motion through the zodiac. Most historians of astronomy believe that this two-epicycle model incorporates 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 an indication of an underlying heliocentric model, although this interpretation has been debated and largely rebutted. The general consensus is that a synodic anomaly (which depends on the Sun's position) does not necessarily imply a physically heliocentric orbit, and Aryabhata's system was not explicitly heliocentric. He may have also believed that planetary orbits were elliptical rather than perfectly circular.

5.3. Eclipses

Aryabhata provided scientific explanations for solar and lunar eclipses, a significant departure from the prevailing mythological interpretations that attributed eclipses to the pseudo-planetary lunar nodes, Rahu and Ketu. He accurately stated that the Moon and planets shine by reflected sunlight. He explained eclipses in terms of shadows cast by and falling on Earth. Specifically, a lunar eclipse occurs when the Moon enters the Earth's shadow (gola verse 37). He extensively discussed the size and extent of the Earth's shadow (gola verses 38-48) and provided computations for the size of the eclipsed part during an eclipse. While later Indian astronomers refined these calculations, Aryabhata's methods formed the core. His computational accuracy was so remarkable that in the 18th century, the French scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found that Indian calculations for the duration of the lunar eclipse on 30 August 1765 were only 41 seconds short, whereas his own charts (based on Tobias Mayer's 1752 data) were 68 seconds long.

5.4. Sidereal Periods and Calculations

Aryabhata's calculations of sidereal periods demonstrated remarkable accuracy for his time. In modern English units of time, he calculated the sidereal rotation (the Earth's rotation relative to fixed stars) as 23 hours, 56 minutes, and 4.1 seconds. The modern accepted value is 23 hours, 56 minutes, and 4.091 seconds. Similarly, his value for the length of the sidereal year was 365 days, 6 hours, 12 minutes, and 30 seconds (equivalent to 365.25858 days). This calculation has an error of only 3 minutes and 20 seconds compared to the modern value of 365.25636 days.

5.5. Astronomical Instruments

Although the Āryabhaṭīya does not detail the instruments Aryabhata used, his lost work, the Ārya-siddhānta, provided descriptions of several astronomical instruments. These include the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), angle-measuring devices (semicircular and circular, known as dhanur-yantra and chakra-yantra), a cylindrical stick (yasti-yantra), an umbrella-shaped device (chhatra-yantra), and at least two types of water clocks (bow-shaped and cylindrical).

6.1. Influence on Indian and Islamic Science

Aryabhata's mathematical and astronomical findings were widely disseminated and significantly shaped scientific thought in India. His work, particularly the Āryabhaṭīya, was translated into Arabic around 820 CE during the Islamic Golden Age, becoming highly influential. Some of his results were cited by the prominent mathematician Al-Khwarizmi. In the 10th century, Abū Rayhān al-Bīrūnī noted that Aryabhata's followers believed the Earth rotated on its axis. Aryabhata's astronomical calculation methods, along with his trigonometric tables, were extensively adopted in the Islamic world and used to compute numerous Arabic astronomical tables, known as zij. Notably, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo in the 12th century and remained the most accurate ephemeris used in Europe for centuries.

6.2. Trigonometric Terminology

Aryabhata's definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) were pivotal in the development of trigonometry. He was also the first to compile tables for sine and versine values at 3.75° intervals from 0° to 90°, with an accuracy of four decimal places. The modern terms "sine" and "cosine" are direct linguistic descendants of Aryabhata's original Sanskrit terms. As previously mentioned, jya was translated as jiba and kojya as kojiba in Arabic. Later, Gerard of Cremona, while translating an Arabic geometry text into Latin around 1150 CE, mistakenly assumed jiba was the Arabic word jaib (meaning "fold in a garment"), and replaced it with its Latin counterpart, sinus (meaning "cove" or "bay"), thus giving rise to the English word "sine."

6.3. Calendar Systems

The calendric calculations devised by Aryabhata and his followers have been continuously used in India for practical purposes, particularly in fixing the Panchangam (the Hindu calendar). In the Islamic world, his methods formed the basis of the Jalali calendar, introduced in 1073 CE by a group of astronomers including Omar Khayyam. Modified versions of the Jalali calendar are still the national calendars in Iran and Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, similar to Aryabhata's and earlier Siddhanta calendars. Although such calendars require an ephemeris for date calculation, they exhibit fewer seasonal errors compared to the Gregorian calendar.

6.4. Commemorations and Tributes

Aryabhata's profound contributions have been honored through various commemorations and tributes:

• India's first satellite, launched in 1975, was named "Aryabhata" in his honor. This satellite was also featured on the reverse of the Indian 2-rupee note.
• The lunar crater Aryabhata is named after him.
• The Aryabhatta Knowledge University (AKU) in Patna, Bihar, was established by the Government of Bihar to develop and manage educational infrastructure related to technical, medical, management, and allied professional education. It is governed by the Bihar State University Act 2008.
• The Aryabhatta Research Institute of Observational Sciences (ARIES), located near Nainital, India, is an institute dedicated to research in astronomy, astrophysics, and atmospheric sciences.
• The inter-school Aryabhata Maths Competition is also named after him.
• Bacillus aryabhata, a species of bacteria discovered in the stratosphere by ISRO scientists in 2009, bears his name.

7.1. Historical Assessment

Aryabhata is widely regarded as the first ācārya (scholar or master) in Indian astronomy, meaning he was an author who conducted his own research and presented original findings, rather than merely transmitting traditional knowledge. His scientific achievements were groundbreaking, particularly his systematic approach to mathematics and astronomy, which laid a strong foundation for future scholars.

7.2. Criticism and Controversy

Despite his significant contributions, Aryabhata's theories were not universally accepted by his contemporaries or later scholars. Brahmagupta, a prominent 7th-century Indian mathematician and astronomer, was a notable critic of Aryabhata. In his work Brāhmasphuṭasiddhānta, Brahmagupta harshly criticized Aryabhata, stating in chapter 1, verse 62: "Aryabhata's supporters do not openly confront like gazelles. They do not confront a lion even when they see one." This statement has been interpreted by scholars like Grigory Bongard-Levin as Brahmagupta's indirect defense of Aryabhata, suggesting that Aryabhata, as a scientist, maintained his stance and was therefore criticized by anti-science traditionalists who adhered strictly to Brahmanical traditions. It is almost certain that Aryabhata avoided direct confrontation to escape censure or persecution from orthodox Brahmins and their loyal followers.

Another area of debate concerns the potential heliocentric influences in Aryabhata's astronomical models. While some historians have suggested that his calculations, particularly the śīghra anomaly, might imply an underlying heliocentric model, this has been largely rebutted. The prevailing consensus among historians of astronomy is that Aryabhata's system was not explicitly heliocentric, and synodic anomalies do not necessarily indicate a physically heliocentric orbit.