Science and Technology in the Gupta Period: Mathematics, Astronomy, Medicine and Metallurgy

Gupta Period Science: Patronage, Centres of Learning and Dates

Peace, Royal Patronage and the Centres of Learning

The prolonged peace and stability of the Gupta period strengthened the economy, and a prosperous state could support scholars, artists and scientists. Knowledge from earlier eras was consolidated and compiled into texts in these centuries, and some historians therefore label the period the "classical age" of India. Chandragupta II (c. 375‑415 CE) kept many learned men, poets and artists at his court.

Three places stand out as homes of this learning:

• Kusumapura: Near present-day Patna, a famous centre of learning, where Aryabhata lived and worked around 500 CE.
• Ujjayini (Ujjain): A city famous for its tradition of learning and a major centre of astronomy in central India, the home of Varahamihira and, in his later years, of Brahmagupta.
• Nalanda: The Buddhist monastery in Bihar founded by Kumaragupta I around 427 CE, whose library held texts on grammar, logic, astronomy and medicine as well as religious works; it flourished under Gupta patronage in the fifth and sixth centuries.

The Gupta kings were devout followers of Vishnu, yet the Buddhist monastery at Nalanda grew under their patronage.

Dates of the Gupta-Age Scientists and the Navaratna Legend

The scientists of this age lived across more than two centuries, and their dates rest on statements in their own works. Aryabhata records in the Aryabhatiya that he was 23 years old in the year 3600 of the Kali Yuga, which corresponds to 499 CE, so he was born in 476 CE. Varahamihira's Pancha-siddhantika refers to Shaka year 427, or 505 CE, most probably the year he composed or began the work, which gives his active life as c. 505‑587 CE; some scholars instead take 505 as his year of birth. Brahmagupta states that he was born in 598 CE and wrote his Brahmasphutasiddhanta in 628 CE, at the age of 30.

These dates also dispose of a famous legend. A later work, the Jyotirvidabharana, places Varahamihira among the navaratnas, the "nine gems" at the court of the legendary King Vikramaditya, and another version of the list puts Brahmagupta among the nine. The Jyotirvidabharana is a literary forgery dated variously from the twelfth to the eighteenth century. Even if the king of the legend is taken to be Chandragupta II, the Gupta king who bore the title Vikramaditya, he had died by about 415 CE, some 90 years before 505 CE and nearly two centuries before Brahmagupta was born.

Golden Age or Classical Age: The Historians' Debate

Some historians have called the Gupta period the golden age of India, while others dispute the description. Science shows why the label needs care. Much of what flowered under the Guptas had older roots: Ayurveda, the two great medical Samhitas and the decimal place-value system all began before them.

The finest work also continued well after the dynasty. Varahamihira likely lived in the kingdom of the Aulikaras, who ruled Avanti in the sixth century, and Brahmagupta and Bhaskara I worked in the seventh century, when Gupta rule had ended. The classical period of Indian mathematics runs from about 400 to 1200 CE, and it is this longer span that is often called the golden age of Indian mathematics. The Gupta centuries opened that long classical tradition rather than containing all of it.

Aryabhata and the Aryabhatiya: Gupta Period Mathematics

Aryabhata of Kusumapura and His Treatise

Aryabhata (also written Aryabhatta) was the first of the major mathematician-astronomers of the classical age of Indian mathematics and astronomy. He called himself a native of Kusumapura, or Pataliputra, near present-day Patna. Bhaskara I, his commentator, calls him ashmakiya, "one belonging to the Ashmaka country", after a people who had settled between the Narmada and the Godavari, and a separate theory places him in Kerala.

The Aryabhatiya is the only one of his works that survives. It is a short treatise of 108 verses with 13 introductory verses, written in the terse style of sutra literature, in which each line is an aid to memory; its meaning was unfolded by later commentators. The name Aryabhatiya itself was given by those commentators, and Bhaskara I called the book Ashmakatantra. A second work, the Arya-siddhanta, is lost and is known through the writings of Varahamihira, Brahmagupta and Bhaskara I.

Place-Value Numbers and Aryabhata's Value of Pi

The place-value system, in which a digit's worth depends on its position, was clearly at work in Aryabhata's calculations. He did not, however, use a symbol for zero, nor did he use the Brahmi numerals. Following an older Sanskrit practice, he wrote numbers with letters of the alphabet, which let him express a whole table of sines in a single verse. The French mathematician Georges Ifrah argued that knowledge of zero was implicit in this system, as a placeholder for a power of ten with nothing in it.

Aryabhata's rule for the circle, in Ganitapada 10, reads: "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." The steps give a circumference of 62,832 for a diameter of 20,000, and dividing the one by the other gives 3.1416, the value of pi.

This value is accurate to two parts in one million. Aryabhata used the word asanna, "approaching", for his result, and some scholars read it as a sign that he knew pi cannot be written exactly; its irrationality was proved in Europe only in 1761, by Lambert.

Sine Tables, Kuttaka and Aryabhata's Algebra

Aryabhata's word for the sine was ardha-jya, "half-chord", soon shortened to jya. Arabic translators wrote it as jiba; later writers read the unvowelled form as jaib, "fold" or "pocket", and in the twelfth century Gherardo of Cremona put that into Latin as sinus, from which the English word "sine" comes. His sine table of 24 numbers was the first sine table constructed in the history of mathematics; the older Greek tables of Hipparchus and Ptolemy were tables of chords, not half-chords. His sine and versine tables run at intervals of 3.75° from 0° to 90° and are accurate to four decimal places.

The kuttaka, meaning "pulverising", is Aryabhata's method for finding whole-number solutions of equations of the form ax + by = c. It works by breaking the given numbers down step by step into smaller ones. Aryabhata described it very briefly; Bhaskara I explained it with examples from astronomy and gave it the name kuttaka. The method became so central that Indian algebra itself was once called kuttaka-ganita. Aryabhata also gave, in the same work, formulas for the sums of the squares and of the cubes of the first n natural numbers.

Aryabhata's Astronomy: Rotation of the Earth, Eclipses and the Year

Rotation of the Earth and the Boat Analogy

Aryabhata held that the Earth is round and rotates on its axis every day, and that the daily movement of the stars across the sky is only apparent, caused by that rotation. Against the prevailing view that the sky itself turned, he used an image from everyday life in Verse 9 of the Golapada: "Just as a man in a boat moving forward sees stationary objects as moving backwards, so also the stars that are stationary are seen by people of Lanka as moving towards the west." His Lanka is not the island but an imagined point on the equator, at the same longitude as Ujjain.

The rotation of the Earth also explains the alternation of day and night. Aryabhata's value for one full rotation of the Earth relative to the stars, the sidereal day, works out in modern units to about 23 hours 56 minutes 4.1 seconds, against the modern value of 23 hours 56 minutes 4.091 seconds. He also gave a fair estimate of the size of the Earth. In the Aryabhatiya the day is reckoned from dawn at Lanka.

Scientific Explanation of Solar and Lunar Eclipses

Aryabhata explained eclipses by shadows. In the older belief, eclipses were caused by Rahu and Ketu, which astronomers identified with the lunar nodes. Aryabhata stated instead that the Moon and the planets shine by reflected sunlight, and that a lunar eclipse occurs when the Moon enters the Earth's shadow. A solar eclipse is the other case, in which the Moon's shadow falls on the Earth. He discussed at length the size and extent of the Earth's shadow and showed how to compute the eclipsed part of the Moon.

His methods stayed at the core of Indian eclipse computation. During a visit to Pondicherry in the eighteenth century, the scientist Guillaume Le Gentil found that Indian computations of the duration of the lunar eclipse of 30 August 1765 were short by 41 seconds, while the European tables he used, by Tobias Mayer, were long by 68 seconds.

Length of the Year and Aryabhata's Model of the Heavens

Aryabhata put the length of the year, measured against the stars, at 365 days, 6 hours, 12 minutes and 30 seconds. The modern value of this sidereal year is 365.25636 days, so his figure is too long by only about 3 minutes and 20 seconds.

His model of the heavens was still geocentric. The Sun and the Moon were carried on epicycles that revolved around the Earth, each planet's motion was governed by two epicycles, the smaller manda and the larger shighra, and the planets were ordered outward from the Earth as the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn. Some historians have suggested that his corrections for the planets rest on an underlying heliocentric idea, but the general view is that his system was not explicitly heliocentric.

Varahamihira of Ujjain: Pancha-siddhantika and Brihat Samhita

Varahamihira's Life and the Pancha-siddhantika

Varahamihira (active c. 505‑587 CE) was a mathematician, astronomer and astrologer who lived in Avanti, traditionally at Ujjain, in present-day Madhya Pradesh. He mastered all three divisions of jyotisha, namely mathematical astronomy (ganita), the natural sciences (samhita) and horoscopy (hora). He calls himself "of Avanti", and his own stanza in the Brihaj-jataka says that he was the son of Aditya-dasa and studied at Kapitthaka.

His Pancha-siddhantika, "treatise on the five siddhantas", summarises the five schools of astronomy current in India, whose own texts, at least in their original form, are lost; his summary is therefore the main record of them. The five are:

• Surya Siddhanta: Judged by Varahamihira the most accurate of the five.
• Romaka Siddhanta: The treatise of the Romans, or westerners.
• Paulisa Siddhanta: A system containing elements derived from Greek astronomy.
• Vasishtha Siddhanta: An older indigenous system named after the sage Vasishtha.
• Paitamaha Siddhanta: The oldest and least accurate of the five.

The Pancha-siddhantika uses the decimal place-value system, with addition and subtraction involving zero, and sine tables. Varahamihira's works use 35 Sanskritised Greek astronomical terms, and he praised the Greeks as "well trained in the sciences". His work shows Indian astronomers studying Greek and Indian systems side by side.

Brihat Samhita: Astronomy, Weather, Architecture and Farming

The Brihat Samhita is Varahamihira's encyclopaedic work. It covers astronomy and astrology, weather forecasting, architecture, town planning and farming; by its own penultimate verse it holds 100 chapters in fewer than 4,000 verses, composed in at least 63 different metres. It gave methods for predicting seasonal rainfall from cloud formation, wind patterns, the positions of the stars and the Moon, and other natural signs. In a chapter on perfumes, Varahamihira arranged the combinations of ingredients taken a few at a time in a table, a method for computing what are now called binomial coefficients, and the same work contains the oldest dateable magic square of the fourth order in the world.

Around 530 CE Varahamihira also gave an accurate rule for telling the time from the shadow of a vertical stick. His reputation outlived him. His works superseded nearly all earlier Indian texts in their field, the eleventh-century scholar al‑Biruni admired and quoted him, and in the fourteenth century the Brihat Samhita was translated into Persian for Sultan Firuz Shah Tughlaq.

Brahmagupta: Mathematics and Astronomy After the Gupta Empire

Brahmagupta's Rules for Zero and Negative Numbers

Brahmagupta was born in 598 CE at Bhillamala, modern Bhinmal in Rajasthan, the first capital of the Gurjara-Pratiharas, and later moved to Ujjain, where he is presumed to have died in 668 CE. He studied the five traditional siddhantas and the work of earlier astronomers, Aryabhata and Varahamihira among them. His Brahmasphutasiddhanta of 628 CE has 24 chapters and 1,008 verses; much of it is astronomy, but it also contains key chapters on algebra, geometry and trigonometry. In 665 CE, at the age of 67, he wrote the Khandakhadyaka, a practical manual of astronomy for students.

The Brahmasphutasiddhanta is the first book that gives rules of arithmetic for zero and for negative numbers, and the earliest known text to treat zero as a number in its own right rather than as a placeholder. Brahmagupta stated, for example, that a negative number added to zero is negative, a positive number added to zero is positive, and two zeros added give zero.

His treatment of division by zero differs from modern mathematics, for he held that zero divided by zero is zero and did not commit himself on a number divided by zero. He also gave rules for operations on fractions and pioneered new techniques for solving certain equations, which established him as one of the founders of modern algebra.

Gravity, Cyclic Quadrilaterals and Brahmagupta's Algebra

Brahmagupta was the first Indian scholar to describe gravity as an attractive force, for which he used the Sanskrit term gurutvakarshanam. In his words, all heavy things fall down to the Earth by a law of nature, "for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow".

In geometry his best-known result is a formula for the area of a cyclic quadrilateral, a four-sided figure whose corners lie on one circle, from the lengths of its four sides; Heron's formula for a triangle is a special case of it. In algebra he gave the first clear description of the solution of the quadratic equation, a method of generating solutions for equations of the type Nx² + 1 = y², now called Pell's equation, and rules for the sums of squares and cubes of the first n numbers. Bhaskara II later called him ganaka-chakra-chudamani, "the gem of the circle of mathematicians".

Zero and the Decimal Place-Value System: Evidence and Debate

From the Empty Place to Zero as a Number

The decimal place-value system was devised by Indian mathematicians between the first and fourth centuries CE, and zero grew out of it over several centuries rather than in a single discovery. The Sanskrit word for it, shunya ("void, empty"), was already used for zero by Pingala, a scholar of Sanskrit metre of the third or second century BCE. The evidence for zero inside the place-value system then builds up in stages:

1. 458 CE: The Lokavibhaga, a Jain text on cosmology that survives in a medieval Sanskrit translation of a Prakrit original and is internally dated to Shaka year 380, uses a decimal place-value system including a zero and calls it shunya.
2. c. 500 CE: Aryabhata works with place value in the Aryabhatiya, with an empty place but no written zero.
3. c. 505 CE: Varahamihira's Pancha-siddhantika adds and subtracts with zero.
4. 628 CE: Brahmagupta's Brahmasphutasiddhanta gives rules for zero as a number.
5. 629 CE: Bhaskara I, in his commentary on the Aryabhatiya, writes numbers in nine Brahmi numerals and a small circle, the shunya-bindu, for zero.

By the middle of the seventh century the system was known outside India, for the Syrian bishop Severus Sebokht wrote of the "nine signs" of the Indians for expressing numbers.

Bakhshali Manuscript and the Gwalior Inscription: Dating the Zero Symbol

The Bakhshali manuscript is a mathematical text on birch bark found in 1881 at the village of Bakhshali, near Mardan in present-day Pakistan; its 70 leaves are kept at the Bodleian Library of the University of Oxford. It uses a black dot for zero throughout.

In 2017 Oxford published radiocarbon dates that placed one folio as early as 224‑383 CE, after which the manuscript was regarded as the earliest known Indian use of a zero symbol; historians of Indian mathematics, among them Kim Plofker and Takao Hayashi, objected to the way the dates were announced. On 14 October 2024 Oxford revised its findings and dated the manuscript to 799‑1102 CE. Its script is an early form of Sharada, which was in use mainly from the eighth to the twelfth century.

In India, the Chaturbhuj Temple inscription at Gwalior, dated 876 CE, is the earliest known inscription to write zero as a small circle among special glyphs for the decimal digits. Outside India, an earlier stone inscription near Sambor in Cambodia writes the Shaka year 605, or 683 CE, in Khmer numerals, a set of glyphs for the Hindu-Arabic system.

Medicine in the Gupta Period: Ayurveda and Surgery

Charaka Samhita and Sushruta Samhita: Origins and Final Shape

Ayurveda was codified during the Gupta period, although this traditional system of medicine goes back several centuries BCE. Its two foundational texts, the Charaka Samhita and the Sushruta Samhita, are dated to about 100 CE in their early form and were compiled and given their final shape during the Gupta period. They deal with the cataloguing and diagnosis of diseases, their treatment, the importance of diet for good health, the preparation of medicines and surgical techniques advanced for their time, and they emphasise holistic healing and a connection between mind, body and nature.

Both texts grew in layers. The Charaka Samhita says its teaching came from Atreya and was first codified by Agnivesha; Charaka revised that work, and Dridhabala later edited it, writing a third of the book himself because that portion had been lost. In the Sushruta Samhita, the text that survives rests on a recension by Nagarjuna in the third or fourth century CE, on which Dallana's later commentary is based.

Surgery in the Sushruta Samhita: Instruments, Grafts and Rhinoplasty

The Sushruta Samhita is of great historical importance because it contains unique chapters on surgical training, instruments and procedures. It describes several kinds of skin graft, the sliding, rotation and pedicle grafts, and the reconstruction of a nose that has been cut off, using a flap of skin from the cheek. This is the first written record of cheek-flap rhinoplasty, a technique still used today, and the text gives more than 15 methods of repairing the nose. The text also counts the bones of the body, noting that teachers of Ayurveda spoke of 360 bones while books on surgery knew only 300, and explaining how its own count was checked.

Indian methods of nose repair later drew European attention. Gaspare Tagliacozzi of Bologna (1545‑1599) became familiar with Sushruta's techniques, and in the eighteenth century British physicians travelled to India to watch rhinoplasties performed by Indian methods. Reports of rhinoplasties performed by a Kumhar (potter) vaidya appeared in the Gentleman's Magazine by 1794.

A skin graft from the patient's own body is not the transplant of an internal organ. The first organ transplant in the modern sense was a thyroid transplant performed by Theodor Kocher in 1883, and the first successful kidney transplant followed in 1954.

Vagbhata, the Ashtanga Texts and the Bower Manuscript

Vagbhata is the name attached to the two later classics of Ayurveda, the Ashtangasangraha and the Ashtangahridayasamhita, although modern research holds that the two are unlikely to be the work of one author. Both draw heavily on Charaka and Sushruta and arrange Ayurveda in its eight-fold (ashtanga) division. The author of the Ashtangasangraha calls himself the son of Simhagupta and a pupil of Avalokita. Some scholars place Vagbhata around the sixth century CE, and Ayurvedic tradition counts Charaka, Sushruta and Vagbhata as its classical triad.

The Bower Manuscript shows how far this medicine travelled. It is a set of seven fragmentary Sanskrit treatises, three of them on Ayurvedic medicine, written in early Gupta script on birch bark and found buried in a Buddhist stupa near Kucha in north-western China. It is dated variously to the fifth or early sixth century and contains the oldest dated fragments of an Indian medical text, the Navanitaka. The British officer Hamilton Bower bought it in March 1890, and it is now in the Bodleian Library at Oxford.

Gupta Period Metallurgy: The Iron Pillar and the Sultanganj Buddha

Mehrauli Iron Pillar: Forge-Welded Iron That Resists Rust

The iron pillar in the Qutb complex at Mehrauli, Delhi, is 7.21 metres high, of which 1.12 metres is below ground, has a diameter of 41 centimetres and weighs more than six tonnes. Its Sanskrit inscription in Brahmi script, of the fourth century CE, records that the pillar was set up as a Vishnudhvaja, a standard of Vishnu, on a hill called Vishnupada in memory of a mighty king named Chandra, whom historians identify with Chandragupta II.

Where Vishnupada lay is debated; one view places it at the Udayagiri caves in Madhya Pradesh, where the pillar probably first stood. It was moved to its present place in Delhi in the eleventh century, probably by Anangpal Tomar. More than 1,600 years old, it still stands without rusting.

The pillar was made by forge-welding pieces of wrought iron. Its resistance to corrosion comes from a thin protective layer that forms on its surface from the special iron used and the oxygen of the air. The layer is made of crystalline iron hydrogen phosphate hydrate, which builds up evenly next to the metal. The iron is rich in phosphorus, about 0.25 per cent on average and never below 0.1 per cent, against less than 0.05 per cent in modern iron from blast furnaces.

The ancient smiths added no lime to their furnaces and used wood rich in phosphorus, whereas in a modern blast furnace most of the phosphorus is carried away by the slag.

Sultanganj Buddha and Copper Casting by the Lost-Wax Method

The Sultanganj Buddha is the largest substantially complete copper figure of the Buddha known from its time. It is 2.3 metres high and 1 metre wide, weighs over 500 kilograms, and was cast in pure, unrefined copper by the lost-wax (cire perdue) technique around a clay core mixed with rice husks, which has allowed radiocarbon dating. It is dated to between 500 and 700 CE and belongs to the transition from Gupta to Pala art.

Metal statues were probably about as common in the period as stone ones, but this is the only metal statue of any size to survive from Gupta art. It was found in 1861 at Sultanganj in Bhagalpur district, Bihar, during the building of the East Indian Railway, and is now in the Birmingham Museum and Art Gallery in England. Sultanganj lies in the old region of Anga, which was part of the Gupta empire. Its thin, clinging robe follows the style of the Sarnath Buddha images.

Transmission of Gupta-Age Science to the Arab World and Europe

Sindhind in Baghdad and the Hindu Numerals

Indian astronomy and arithmetic reached the Arab world in the eighth and ninth centuries. An embassy from Sindh to the court of the Caliph al‑Mansur (754‑775) included an astrologer named Kanaka who brought astronomical texts, Brahmagupta's among them. Al-Mansur had the Brahmasphutasiddhanta translated into Arabic, with Muhammad ibn Ibrahim al‑Fazari among the translators, and the resulting astronomical tables, the Zij al‑Sindhind, take their name from the Sanskrit word siddhanta. The Surya Siddhanta was translated in the same reign, and the Aryabhatiya followed around 820 CE.

In the ninth century Sanskrit texts of mathematics, astronomy and medicine were translated into Arabic, notably in Baghdad, and the Arabs adopted India's decimal numeral system with the zero. Al-Khwarizmi wrote On the Calculation with Hindu Numerals around 825 CE, and al‑Kindi wrote On the Use of the Hindu Numerals around 830 CE. Latin versions of al‑Khwarizmi's work, such as Algorithmi de numero indorum, brought the system into European mathematics from the twelfth century; Leonardo Fibonacci's Liber Abaci of the thirteenth century spread the "method of the Indians" further, and it entered common use from the fifteenth. Because they came through the Arabs, Europeans called these digits "Arabic numerals", and several dictionaries now add the term "Hindu-Arabic numerals".

Aryabhata's Legacy in Later Indian Science and Today

Indian astronomy built on Aryabhata for many centuries. Bhaskara I wrote his commentary on the Aryabhatiya in 629 CE, Brahmagupta referred to Aryabhata by name in more than a hundred places, Lalla in the eighth century corrected several of his assumptions, and Nilakantha Somayaji wrote another commentary in 1465. Calendar calculations devised by Aryabhata and his followers have stayed in continuous use for fixing the panchangam, the Hindu calendar.

Al-Biruni, who came to India with Mahmud of Ghazni around 1017, learned Sanskrit and compiled what he could learn of the astronomy of Aryabhata, Varahamihira and Brahmagupta. India's first satellite, Aryabhata, built by the Indian Space Research Organisation and launched by the Soviet Union on 19 April 1975, was named after the astronomer, and so are a lunar crater and the Aryabhatta Research Institute of Observational Sciences near Nainital.

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