Aryabhatiya by Aryabhata I

Aryabhata, potentially born in 476 in either Ashmaka or Kusumapura, India, achieved fame as a mathematician and astronomer. His sole surviving work, Aryabhatiya, is a comprehensive compilation of mathematics and astronomy, covering various subjects like arithmetic, algebra, plane trigonometry, and spherical trigonometry. The book also includes discussions on continued fractions, quadratic equations, sums-of-power series, and even features a table of sines.

Aryabhatiya systematically explores the composition of the Solar System, the characteristics of its planets, and provides explanations for solar and lunar eclipses. In its mathematical section, Aryabhatiya thoroughly addresses arithmetic, algebra, plane trigonometry, and spherical trigonometry.

The text is also referred to as Arya-status-ash, signifying Aryabhata's 108, denoting the presence of 108 verses in the book. Written in the form of a sutra, comprising aphorisms, it offers a concise and effective means of presenting scientific principles.

Written in Sanskrit, Aryabhatiya is divided into four sections, totaling 121 verses, following a mnemonic writing style commonly used in Indian works:

~Gitikapada (13 verses) - Explores cosmology, mentioning the planetary revolutions in a maha yuga, which spans up to 4.32 million years.
~Ganitapada (33 verses) - Focuses on Ganita, referring to calculations, with 33 verses dedicated to mathematics. This section explains various topics, including mensuration, simple and quadratic equations, as well as indeterminate and arithmetic and geometric equations.
~Kalakriyapada (25 verses) - Consists of 25 verses, providing insights into the measurement of days, weeks, and months using different units of time.
~Golapada (50 verses) - Contains 50 verses that delve into the causes of days and nights, the rising of zodiac signs, eclipses, the celestial equator, nodes, and the shape of the earth.
Aryabhatiya by Aryabhata I

Liu Hui, Chinese mathematician

Liu Hui, a Chinese mathematician who lived in the Kingdom of Wei. Kingdom of Wei was a north-central kingdom and one of the three separate kingdoms after the fall of the Han dynasty in the early 3rd century.

Liu Hui not only improves the algorithm of extraction, but also gives an explanation for extracting the square root.

Based on a method he developed, he obtained the most accurate approximation yet of the value of π (=3.1416). About 300 years later, another Chinese mathematician, Zu Chongzi, worked out π to 7 digits (3.1415926).

Liu Hui’s most influential work was his Island Mathematical Manual.  In this book, Liu Hui shows how to use pairs of similar right triangles to measure inaccessible distance.

He completed his commentary on The Nine Chapters on the Mathematical Art in 263. The Nine Chapters on the Mathematical Art was mathematical cannon of the 1st century BC that played a similar role in the East to Euclid’s Elements in the West.
Liu Hui, Chinese mathematician 

Aryabhata I (476–550 CE)

Little is known of the life of Aryabhata, who is called Aryabhata I in order to distinguish him from another mathematician of the same name who lived four centuries later.

Born in India, Aryabhata is considered to be one of the most brilliant original thinkers on mathematics and astronomy, making computations and explaining the nature of solar system some 1000 years before Nicolas Copernicus suggested the heliocentric system.

Aryabhata played a role in the development of the modern current number system and made contribution number theory which has become universal today.

Aryabhata wrote two works: the Aryabhatiya in 499 and another treatise, Arya-Siddhanta which has been lost. He wrote Aryabhatiya when he was 23 years old after completing his studies at the University of Nalanda.

His Aryabhatiya work is fairly advanced. It apparently was preceded by many generation of mathematical research. It has relevance even today.

The Arya-Siddhanta, a lost work on astronomical computations contained the description of several astronomical instruments, the gnomon, a shadow instrument possibly an angle measuring device, semi circle and circle shaped device etc.

Aryabhata was the first to deduce that the earth its round and that it rotate on its own axis causing day and night. He also declared that the moon is dark and shines only because of sunlight.

In measuring time, Aryabhata determined that the length of a year is 365 days, six hours, 12 minutes, 30 seconds, an extremely close calculation to the modern standard of 365 days, six hours.
Aryabhata I (476–550 CE)

Mahāvīra -Indian mathematician

The most celebrated mathematician of the ninth century AD is Mahaviracharya, well known as Mahāvīra (AD 817-875). Mahāvīra was a Jain by religion. He was born in Mysore, India and as a young man became familiar with the mathematics of southern India.

He was in the Royal Court of Karnataka Rastrakuta King, Amoghavarsha Nrpatunga who ruled from AD 815 to AD 874.

He wrote his work called ‘Ganita Sara Sangraha’ in AD 850. Ganita Sara Sangraha is te first book in the Indian mathematical tradition that confines its attention to pure mathematics.  This text contains nine chapters of about 1100 slokas dealing with arithmetic, algebra, geometry and mensuration.

The book is an ambitious attempt to summarize, improve upon and teach Indian mathematical knowledge as he understood it.

He provided methods for squaring numbers and determining the cube root of a number. He also gave a formula that approximately the perimeter and are of an ellipse.

Ganita Sara Sangraha served as a textbook in South India, where Vallabha and Varadaraja wrote commentaries on it and Pavaluri Mallana in translated it into Telugu.
Mahāvīra -Indian mathematician