Aryabhata I

Statue of Aryabhata

Aryabhata (Āryabhaṭa) or Aryabhata I (476–550 CE) roughly 1,474 to 1,548 years ago, was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.

For his explicit mention of the relativity of motion, he also qualifies as a major early physicist. 

Learning about Aryabhata is usually kind of confusing to note why is he prompt to be an important figure, imagine living around 1000 years ago, you didn't have a journal or articles to know the world about, some scholars set into research works just like how modern-day now scientists collaborate works. One example is Aryabhata, it is very common whenever to think that the past is always about spiritual and religious information we often hear or share around but the past is not only about spirituality. 

Precisions Tools 

Ancient mathematicians and scientists relied on highly precise mechanical instruments and keen observational skills, which often resulted in remarkably accurate measurements. These primary instruments served as the standard for calibrating other tools, ensuring consistency.

Modern instruments, on the other hand, depend on advanced technologies like ADC converters, which can introduce variability due to electrical noise and require regular calibration. While we benefit from the speed and sophistication of electronic tools, there's something to be said about the meticulous craftsmanship and practical ingenuity of ancient methods.

Gnomon: A simple yet effective instrument used to measure the angle of the sun and determine the time of day. It's essentially a vertical stick or pillar that casts a shadow, and the length and position of the shadow can be used to calculate various astronomical parameters.

1. Shadow Instrument: Similar to the gnomon, this instrument was used to measure the length of shadows cast by objects to determine the position of celestial bodies.
2. Aryabhatiya: While not a physical tool, this is Aryabhata's most famous work, which contains a wealth of mathematical and astronomical knowledge. It includes methods for calculating the positions of planets, the length of the solar year, and the value of π.
3. Arya-siddhanta: Another significant work by Aryabhata, which includes detailed astronomical calculations and tables. It was influential in the development of Islamic astronomy.

Knowing The π (pi) Value With Rope

Pic Egyptians rope. Vector illustration of making a right-angled triangle using a rope.

Pic ref to Egytian Rope

Ancient mathematicians used a simple yet effective method involving ropes to estimate the value of π (pi). Here's how they did it:

1. Inscribed and Circumscribed Polygons: They would draw a circle and then inscribe and circumscribe regular polygons (like hexagons, octagons, etc.) within and around the circle.
2. Measuring Perimeters: Using a rope, they would measure the perimeters (circumferences) of these polygons

Calculating Ratios: By comparing the perimeters of the inscribed and circumscribed polygons to the diameter of the circle, they could approximate the value of π. 

For example, the ancient Greek mathematician Archimedes used this method to approximate π to be between 3 1/7 and 3 10/712. This method, known as the method of exhaustion, was a precursor to integral calculus.

If we notice that the Egyptians during the pyramids knew the value of π, also Baudhayana, ancient Chinese, etc .. they were similarly using techniques on mathematical discovery. At the point during the collaboration works 1000-2000 years ago or older in past they exchange knowledge. 


Aryabhatas mathematics and astronomy made several significant contributions:

1. Approximation of π: Aryabhata calculated the value of π to be approx 3.1416, which was a remarkable achievement for his time.
2. Concept of Zero and Place Value System: He introduced the concept of zero and the place-value system, which revolutionized mathematics.
3. Indeterminate Equations: Aryabhata was one of the first to solve indeterminate (or Diophantine) equations.
4. Trigonometry: He made significant advancements in trigonometry, including the introduction of sine and cosine functions.
5. Astronomy: Aryabhata proposed that the Earth rotates on its axis and explained the motion of the planets and stars. He also accurately predicted solar and lunar eclipses.

* We often see bramins wearing scared ropes on their shoulder, does that primarily used as used for calculating? Well, I do not get a scientific answer for these curious thought-provoking observations. History says it was used only for spiritual traditional practice.  

Time and place of birth

Aryabhata mentions in the Aryabhatiya that he was 23 years old 3,600 years into the Kali Yuga, but this is not to mean that the text was composed at that time. This mentioned year corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra (present-day Patna, Bihar).

Concept of Kali Yuga

we often hear the yugas and cycle in Indian ancient discoveries and there are reasons why: Kali Yuga is rooted in ancient Hindu scriptures, particularly the Mahabharata, Manusmriti, and various Puranas

It is believed that Kali Yuga, the age of darkness and moral decline, began around 3102 BCE following the departure of Lord Krishna. This transition marks the end of Dvapara Yuga and the start of Kali Yuga.

The idea of Kali Yuga was not "discovered" by a single individual but rather developed and elaborated upon by ancient sages and scholars over time. It's a collective understanding that has been passed down through generations. * Kali Yuga is a debatable timeline, we will delve deep into it in another blog post. 

Hypothesis 

Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India.

It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present-day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr 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 Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala. K. Chandra Hari has argued for the Kerala hypothesis based on astronomical evidence. 

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.

Education

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist traditions, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the University of Nalanda was in Pataliputra at the time, 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. 

Mathematics

Place value system and zero

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 argues 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.

Approximation of π

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:

caturadhikaṃ śatamaṣṭaguṇaṃ 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 for a circle whose diameter is 20000, the circumference will be 62832

i.e, ${\displaystyle \pi }$ = $\frac{{\displaystyle 62832}}{20000}$ = ${\displaystyle 3.1416}$, which is accurate to two parts in one million.

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 (π) was proved in Europe only in 1761 by Lambert.

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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, which literally 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"; thence comes the English word sine.

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem.) 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, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka (कुट्टक) method. Kuṭṭaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially, the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.

Algebra

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

${\displaystyle 1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}}$

and

${\displaystyle 1^3+2^3+\cdots +n^3=(1+2+\cdots +n{)}^2}$ (see squared triangular number)

Astronomy

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 are elliptical rather than circular.

Motions of the Solar System

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, 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. 425 CE), 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 were 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.

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by Rahu and Ketu (identified as the pseudo-planetary lunar nodes), 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 (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) 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.

Sidereal periods

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).

Heliocentrism

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.

India's first satellite named after Aryabhata

In conclusion, Aryabhata stands as a monumental figure in the history of science and mathematics. Living in a time with limited means of knowledge dissemination and dominant non-scientific worldviews, his groundbreaking contributions in astronomy – including the revolutionary concept of Earth's rotation, accurate explanations of eclipses, and remarkably precise calculations of celestial periods – were far ahead of his time. Simultaneously, his advancements in mathematics, such as a highly accurate approximation of pi, the implicit understanding of zero and the place-value system, and methods for solving indeterminate equations, laid crucial foundations for future mathematical developments.

Aryabhata's work, exemplified by the Aryabhatiya, represents a significant shift towards systematic observation and mathematical reasoning in understanding the universe. His insights not only influenced subsequent generations of Indian scholars but also, through translations, contributed to the progress of astronomy and mathematics in other parts of the world. His legacy serves as a powerful reminder of the capacity for human intellect and rigorous inquiry to unlock the mysteries of nature, even with the constraints of the distant past. He remains a testament to the enduring power of scientific curiosity and a source of inspiration for the ongoing quest to understand our cosmos.