One of the earliest significant mathematician-astronomers from the ancient era of Indian mathematics and astronomy was Aryabhata (ISO: Āryabhaṭa), sometimes known as Aryabhata I (476–550 CE). His writings include the Arya-Siddhanta and the Āryabhaṭīya, which states that he was 23 years old in 3600 Kali Yuga, or 499 CE.
He also meets the definition of a significant early physicist because of his clear reference to the relativity of motion.
An Autobiography
Name
The name Aryabhata is spelled correctly, despite the tendency to misspell it as “Aryabhatta” because of other names possessing the “bhatta” suffix. This is evident in all astronomical texts, including references to him made by Brahmagupta “in more than a hundred places by the name”. Moreover, “Aryabhatta” would not suit the meter in most cases.
Date and location of birth
It is not implied by Aryabhata’s statement in the Aryabhatiya that the work was written 3,600 years into the Kali Yuga when he was 23 years old. Given that the year indicated is 499 CE, it is likely that he was born in 476. Aryabhata is identified as a native of Pataliputra, or Kusumapura (modern-day Patna, Bihar).
Another theory
Aryabhata is identified by Bhāskara I as āśmakīya, which means “one belonging to the Aśmaka country.” Aūmaka people had established in central India between the Narmada and Godavari rivers during the time of the Buddha.
There have been claims that the aūmaka (Sanskrit for “stone”) where Aryabhata originated might be the location of Kodungallur today. Thiruvanchikkulam was the ancient Kerala state’s capital.
This is founded on the idea that Koṭuṅṅallūr was formerly known as Koṭum-Kal-l-ūr, or “city of hard stones”; historical documents, however, indicate that the city’s true name was Koṭum-kol-ūr, or “city of strict governance” Comparably, the fact that a number of Aryabhatiya commentary originate from Kerala has been used to imply that this was Aryabhata’s primary residence and center of activity; however, a large number of commentators originate from beyond Kerala, and the Aryasiddhanta was unknown in Kerala. K. Chandra Hari has used astronomical evidence to support the Kerala theory.
In the Aryabhatiya, Aryabhata refers to “Lanka” several times. However, “Lanka” is an abstraction that represents a place on the equator at the same longitude as his Ujjayini.
Learning
He most likely traveled to Kusumapura at some point to pursue more education and resided there for a while. Bhāskara I (CE 629), the Hindu and Buddhist traditions, and contemporary Patna are all in agreement that Kusumapura was Pāṭaliputra. The head of an institution (kulapa) in Kusumapura, Aryabhata, is mentioned in a poem. Since the University of Nalanda was located in Pataliputra during the period, there is conjecture that Aryabhata was also the head of the Nalanda University. It is also said that Aryabhata established an observatory in Taregana, Bihar’s Sun Temple.
Performs
Though Aryabhatiya is the only one that has survived, Aryabhata is the author of multiple treatises on mathematics and astronomy.
Numerous disciplines from astronomy, mathematics, physics, biology, medicine, and other domains were incorporated in the investigation. Major works of Aryabhata were predicated on earlier findings made by Mesopotamians and Greeks. The Indian mathematical literature made reference to the mathematics and astronomy compendium Aryabhatiya, which has persisted until the present day. Arithmetic, algebra, plane trigonometry, and spherical trigonometry are all covered in the Aryabhatiya’s mathematical section. In addition, it includes a table of sines, sums-of-power series, quadratic equations, and continuing fractions.
Through the works of Aryabhata’s contemporaries Varahamihira and later mathematicians and commentators such as Brahmagupta and Bhaskara I, we know of the Arya-siddhanta, a lost book on astronomical calculations. This book seems to be based on the earlier Surya Siddhanta, which employs midnight-day reckoning instead of dawn as in Aryabhatiya and was a Sanskrit compilation of Greek and Mesopotamian astronomical and mathematical systems. A number of astronomical instruments were also described in it, including the gnomon (shaku-yantra), the shadow instrument (chhAyA-yantra), two types of water clocks (bow-shaped and cylindrical) that may have been used to measure angles, as well as possibly semicircular and circular devices (dhanur-yantra / chakra-yantra) and cylindrical stick yasti-yantras.
Al ntf or Al-nanf is a third text that could have remained in the Arabic translation. Although it is said to be a translation by Aryabhata, it is unknown what this work’s Sanskrit title is. Abū Rayhān al-Bīrūnī, the Persian scholar and historian of India, mentions it, most likely from the ninth century.
Aryabhatiya
Only the Aryabhatiya has direct information about Aryabhata’s work. It was subsequent commentators who gave the term “Aryabhatiya”. It’s possible that Aryabhata did not give it a name. It is known as Ashmakatantra (or the Ashmaka treatise) by his student Bhaskara I. Because there are 108 verses in the book, it is also sometimes referred to as Arya-shatas-aShTa, which directly translates to “Aryabhata’s 108.”
It is written in the extremely condensed form that characterizes sutra literature, where each line serves as a memory aid for a multilayered structure. Thus, commentators are responsible for the meaning’s explanation. With 108 verses and 13 preface verses, the book is organized into four pādas or chapters:
- Large units of time—kalpa, manvantra, and yuga—presented in the 13-verse Gitikapada offer a cosmology distinct from those of older writings like Lagadha’s Vedanga Jyotisha (c. 1st century BCE). A table of sines (jya) is also provided; it is presented in a single stanza. 4.32 million years is the estimated period of planetary rotations during a mahayuga.
- Ganitapada is a collection of 33 poems that covers gnomon/shadows (shanku-chhAyA), arithmetic and geometric progressions, simple, quadratic, simultaneous, and indeterminate equations (kuṭṭaka), and mensuration (kṣetra vyāvahāra).
- The twenty-five-verse Kalakriyapada contains the following: a seven-day week with names for each day, calculations for the intercalary month (adhikamAsa), several time units, and a system for calculating the positions of the planets on a particular day.
- Golapada is a 50-verse poem that covers topics such as the geometric and trigonometric properties of the celestial sphere, the celestial equator, ecliptic features, the earth’s structure, the origin of day and night, and the rise of zodiacal signs on the horizon. Moreover, some renditions list a few epigraphs that were appended at the conclusion, praising the work’s merits, etc.
Many centuries-long breakthroughs in mathematics and astronomy were conveyed in poem form by the Aryabhatiya. His students Bhaskara I (Bhashya, c. 600 CE) and Nilakantha Somayaji (Aryabhatiya Bhasya, 1465 CE) both commented on the text’s remarkable shortness.
Another notable contribution of Aryabhatiya is his explanation of the relativity of motion. He put it this way: “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.”
Mathematics
Put zero and the value system in place.
He obviously used the place-value system, which was initially seen in the Bakhshali Manuscript from the third century. The French mathematician Georges Ifrah contends that although though Aryabhata did not employ a sign for zero, his place-value system implicitly knew what zero was, as it was a placeholder for powers of ten with null coefficients.
Aryabhata did not, however, employ the Brahmi numbers. He carried on the Sanskritic heritage from the Vedic era by representing numbers and quantities with alphabetic letters and creating mnemonic forms for things like the sine table.
An approximation for π
Aryabhata may have concluded that π is irrational after working on the estimate for pi (π). He writes in the second section of the Aryabhatiyam (gaṇitapāda 10).
caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
“Mix by eight, add four to 100 and then add 62,000. This rule allows one to approximate the circumference of a circle with a diameter of 20,000.”
This suggests that the circumference of a circle with a diameter of 20,000 will be 62832.
It is hypothesized that Aryabhata indicated that the value is incommensurable (or irrational) in addition to being an approximation when he used the word āsanna (approaching). If this is true, then Lambert’s demonstration of pi’s irrationality in 1761 was the first in Europe, making it a very advanced understanding.
This approximation was discussed in Al-Khwarizmi’s algebra book after Aryabhatiya was translated into Arabic (c. 820 CE).
Trigonometric Functions
Aryabhata describes the size of a triangle in Ganitapada 6 as
tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ
This means that “for a triangle, the result of a perpendicular with the half-side is the area.”
In his treatise known as ardha-jya—literally, “half-chord”—Aryabhata expounded upon the notion of sine. People started calling it jya for simplicity’s sake. His writings were translated into Arabic by Arabic authors, who called it jiba. Vowels are deleted and it is shortened to jb in Arabic texts. Subsequent authors replaced it with the term jaib, which means “fold (in a garment)” or “pocket”. (Jiba is a meaningless term in Arabic.) When Gherardo of Cremona translated these Arabic texts into Latin later in the 12th century, he substituted the Arabic term jaib with its Latin equivalent, sinus, which means “cove” or “bay” and is the source of the English word sine.
Uncertain Equations
Finding integer solutions to Diophantine equations with the form ax + by = c has long been of significant interest to Indian mathematicians. (This problem’s solution is commonly known as the Chinese remainder theorem; it was extensively investigated in ancient Chinese mathematics.) An instance from Bhāskara’s analysis of Aryabhatiya is as follows:
Determine the number that, when divided by 8, yields a residual of 5, when divided by 9, yields a remnant of 4, and when divided by 7, yields a remaining of 1.
Find N = 8x+5 = 9y+4 = 7z+1, in other words. As it happens, 85 is the least value for N. Diophantine equations like this one are generally known to be quite challenging. The Sulba Sutras, an old Vedic work, that may have been written as early as 800 BCE, had a thorough discussion of them. The technique developed by Bhaskara in 621 CE to solve such difficulties is known as the kuṭṭaka (कुट्टक) method, and it was first used by Aryabhata.
“Pulverizing” or “breaking into small pieces” is what Kuṭṭaka implies, and the process entails expressing the original components in smaller numbers using a recursive algorithm. In Indian mathematics, this approach has become the traditional method for solving first-order diophantine equations. Originally, algebra as a whole was known as kuṭṭaka-gaṇita, or simply kuṭṭaka.
Algebra
Aryabhata gave sophisticated answers in Aryabhatiya for the summation of square and cube series:
and
Astronomy
The astronomical system developed by Aryabhata, known as the audAyaka system, measured days starting on Uday, the dawn at Lanka, or the “equator”. While some of his later astronomical publications are lost, it is possible to piece together what he said about a second model (Ardha-rAtrikA, midnight) based on the debate in Brahmagupta’s Khandakhadyaka. He appears to attribute the apparent movements of the sky to the rotation of the Earth in several writings. It is possible that he thought the planet’s orbits were elliptical instead than circular.
Solar System Motions
Against the then-dominant belief that the sky rotated, Aryabhata rightly contended that the Earth revolves around its axis on a daily basis and that the apparent motion of the stars is a result of the Earth’s rotation. This is mentioned in the Aryabhatiya’s first chapter when the author states how many times the Earth rotates throughout a yuga. It is further explained in the gola chapter:
[Someone] on the equator sees the immovable stars traveling evenly westward, in the same manner as someone in a boat traveling ahead sees an immobile [thing] moving backward. The reason for rising and setting is because the cosmic wind continuously pushes the sphere of stars and planets [supposedly] straight west at the equator.
According to Aryabhata’s geocentric conception of the Solar System, the Moon and Sun are each supported by an epicycle. They circle around the Earth in turn. The planets’ movements are controlled by two epicycles in this concept, which is also included in the Paitāmahasiddhānta (c. 425 CE): a greater śīghra (rapid) and a smaller manda (slow). Based on their distance from Earth, the planets are considered to be in the following order: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.
The planets’ locations and periods were computed in relation to points that moved evenly. Venus and Mercury, on the other hand, orbit the Earth at the same average speed as the Sun. Mars, Jupiter, and Saturn orbit the Earth at different velocities, signifying the movement of each planet across the zodiac. The majority of astronomy historians believe that aspects of pre-Ptolemaic Greek astronomy may be found in this two-cycle paradigm. Some historians see the śīghrocca, the fundamental planetary time in reference to the Sun, as another component of Aryabhata’s concept that indicates an underlying heliocentric model.
Eclipses
Aryabhata provided a scientific explanation for solar and lunar eclipses. He claims that reflected sunlight gives the Moon and planets their brightness. He interprets eclipses in terms of shadows falling on Earth, as opposed to the prevalent cosmogony that attributes eclipses to Rahu and Ketu, the pseudo-planetary lunar nodes. Accordingly, the lunar eclipse happens when the Moon passes through the shadow of the Earth (verse gola.37). In verses GLa.38–48, he goes into great detail on the size and shape of the Earth’s shadow before giving the formula and the area that is obscured during an eclipse.
Aryabhata’s techniques established the basis, but other Indian astronomers refined the computations. The accuracy of his computational paradigm was so great that Guillaume Le Gentil, an 18th-century scientist, discovered, while visiting Pondicherry, India, that the Indian calculations for the duration of the August 30, 1765 lunar eclipse were off by 41 seconds, while his charts (made by Tobias Mayer, 1752) were correct by 68 seconds.
Intervals of sidereal time
Aryabhata determined the sidereal rotation, or the rotation of the earth with reference to the fixed stars, to be 23 hours, 56 minutes, and 4.1 seconds, which is equivalent to 23:56:4.091 in current English units of time. Similarly, there is an error of 3 minutes and 20 seconds over a year (365.25636 days) in his figure for the sidereal year’s duration, which is 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days).
The heliocentric paradigm
Aryabhata promoted an astronomical model in which the Earth rotates on its own axis, as was previously described. Additionally, his model included adjustments (the īgra anomaly) for the planets’ velocities in relation to the Sun’s mean speed. As a result, it has been refuted that Aryabhata’s calculations were predicated on an underlying heliocentric paradigm, according to which the planets orbit the Sun. Though there is little evidence to support this theory, it has also been proposed that some elements of Aryabhata’s system could have originated from an older, perhaps pre-Ptolemaic Greek, heliocentric model that Indian astronomers were not aware of.
The broad view is that Aryabhata’s system was not expressly heliocentric, and that a synodic anomaly (depending on the location of the Sun) does not indicate a physically heliocentric orbit (similar corrections being also evident in late Babylonian astronomical literature).
