Sanskrit Literature in Science and Technology

Aryabhatta was the legendary mathematician of the Gupta Era. He wrote Aryabhattiya at the age of 23 years and later, Arya-Siddhanta. He worked on the approximation for pi to 3.1416. In trigonometry, he concluded for a triangle, the result of a perpendicular with the half-side is the area. He also worked on the motions of the solar system and calculated the length of the solar year to 365.8586805 days. Aryabhatta lived in Kusumpur in Pataliputra.

Contributions of Aryabhata (476 AD)

Aryabhata is the author Aryabhatiyam which sketches his mathematical, planetary, and cosmic theories. This book is divided into four chapters:

• The astronomical constants and the sine table
• Mathematics required for computations,
• Division of time and rules for computing the longitudes of planets using eccentrics and epicycles,
• The armillary sphere, rules relating to problems of trigonometry and the computation of eclipses.

Aryabhata took the earth to spin on its axis; this idea appears to have been his innovation. He also considered the heavenly motions to go through a cycle of 4.32 billion years; here he went with an older tradition, but he introduced a new scheme of subdivisions within this great cycle. According to the historian Hugh Thurston, Not only did Aryabhata believe that the earth rotates, but there are glimmerings in his system (and other similar systems) of a possible underlying theory in which the earth (and the planets) orbits the sun, rather than the sun orbiting the earth. The evidence is that the basic planetary periods are relative to the sun. That Aryabhata was aware of the relativity of motion is clear from this passage in his book “Just as a man in a boat sees the trees on the bank move in the opposite direction, so an observer on the equator sees the stationary stars as moving precisely toward the west.”

In his book named ‘Aryabhattium’, Aryabhatta has given lot of references of Suryasidhanta. He had developed instruments like chakra yantra (disk instrument), Gola yantra (type of armillary sphere) and shadow instruments.

Aryabhatta deduced that earth is a rotating sphere: the stars do not move, it is the earth that rotates. Its diameter is 1,050 yojanas. Its circumference is therefore 1050 x 13.6 x π = 44,860 km.

Aryabhatta also deduced that: “The moon eclipses the sun, and the great shadow of the earth eclipses the moon.”

Varahamihira lived in Ujjain and was one of the nine jewels (Navaratnas) of the court of Chandragupta II. He wrote Panchasiddhantaka, the five treatises on astronomy (NOT astrology). It summarises five earlier astronomical treatises, namely the Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhantas.

About Surya Siddhanta

In India, people had started the use of the astronomical instruments before 1000 BC. During this period one of the prominent books ‘Suryasidhanta’ was written for astronomical calculations. There are several works with the same name, BUT the Original writer of Surya Siddhanta is UNKNOWN.

The title ‘Suryasidhanta’ means sun theory and it highlights the calculations of positions of stars and planets. Some of the Indian mathematicians later have developed their own instruments and developed their own methods to facilitate the theory of ‘Suryasidhanta’. Introduction of zero in mathematics and the decimal method of calculation is one of such invaluable contribution. We should note that Varahamihira had contrasted Surya Siddhanta along with his 4 other treatises in the panchsiddhantika viz. Paitamaha Siddhantas, Paulisha , Romaka Siddhantas and Vasishta Siddhanta. Citation of the Surya Siddhanta is also found in the works of Aryabhata.

Panchasiddhanta

Varahamihira has done a valuable job of compilation of five astronomical theories, which were in use before Crist, and suryasidhanta is one of them. This compiled book is known as ‘Panchasidhanta’. He had developed some ring and string instruments.

Lalla was an Indian astronomer and mathematician who followed the tradition of Aryabhata I. Lalla’s most famous work was entitled Shishyadhividdhidatantra. He was well known because of twelve instruments which he brought into practice. One of the most discussed shloka of Lalla is as follows:

In the above Shloka, Lalla describes the 12 Instruments as follows:

Sphere, ring, dial, bow, time measuring water vessel, Gnomon, divider, scissor. Circular seat with central stick, semicircle with stick, combination of sticks, are the twelve instruments along with a stick.

The 12 instruments are as follows:

1. The Gola yantrais a type of armilliary sphere used to locate planetary positions.
2. Bhanganais a ring with angular graduations alonge its circumference, it is a type of protractor.
3. Chakrais a circular disk with angular graduations; it is also a type of protractor.
4. Dhanuis a semicircular disk with angular graduations and a stick pivoted at the center, it is a type of protractor with a plumb bob arrangement.
5. Ghati is a small vessel with a hole at the bottom. It was used to measure time.
6. Shankuis a type of gnomon, a long vertical cone used to identify East-West-North-South direction based on shadow of its tip.
7. A special geometrical construction known a ‘Matsya‘ was used for the above purpose. Altitude of sun and day time was also measured with this instrument based on the shadow.
8. Shakata consists of two ‘V’ shaped sticks, pivoted at the end.
9. Kartari means a seizer. This instrument is made up of two sticks both pivoted together. It was used like a caliper, and also to measure angle with the help of protractor.
10. Pithais a horizontal disk with a vertical stick at its center. It was used to measure local time based on its shadow, it was used to measure the height with the help of special geometrical contruction.
11. Shalakais combination of two sticks with a string.
12. Yasti is just a long stick having standard dimensions; it was used to measure height and distances. Special geometrical constructions were framed to facilitate the use of this stick. These proposed geometrical constructions were to construct the proportionate triangles with the help of which heights of terrestrial objects could be calculated.

Bhaskaracharya was one of the promonent Indian mathematician and astronomer, who wrote a book ‘Sidhantshiromani’. In his book he has documented valuable ancient liturature and given the references of many of the instruments used by the astronomers before him. Similarly he has documented the various methods for the use of these instruments.

Yasti means a stick. Yasti Yantra was developed by Bhaskaracharya and has also been refered to as Dhi Yantra. The same type of instrument has also been described by ancient sages and astronomers, but Bhaskaracharya has developed this Yantra as unique methods to calculate the height of terrestrial objects like trees and mountains. The usage and principles have been described in the ‘Shidhantashiromani’ of Bhaskaracharya.

The concept of this Yantra is to mount a stick on a pivot at a height d above the ground, and take sightings of the top and bottom of the object such as a tree using the stick.

The projected length of the stick on a horizontal line at the two sightings, L₁ and L₂, and the heights to which the stick is raised, h₁ and h₂, can be marked on an adjoining board. If the overall height of the object is H, and the horizontal line at the height at which the stick is mounted splits it into H₁ and H₂, the lengths form similar triangles, and we can write

h₁/L₁ = H₁/L and h₂/L₂ = H₂/L,

where L is the distance to the object. Eliminating L from the equations using L = H₂ (L₂/h₂), and since H₂=d, we get

H = H₁ + H₂ = (h₁/L₁) L + H₂ == ( (h₁/L₁) (L₂/h₂) + 1 ) d .

The Indian scholar Pingala (circa. 5th-2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables). This was very much similar to today’s Morse code, shown in the following picture. Kindly compare them:

Pingala used the above in his Chhandahshastra. The knowledge of binary numbers indicates his deep understanding of arithmetic. Binary repersentation has now become the basis of information storage in terms of sequences of 0s and 1s in modern-day computers.

Bhaskara (born 1114), who was from the Karnataka region, was an outstanding mathematician and astronomer. Amongst his mathematical contributions is the concept of differentials.  He was the author of Siddhanta Shiromani, a book in four parts:

• Lilavati on arithmetic
• Bijaganita on algebra
• Ganitadhyaya on astronomy
• Goladhyaya on astronomy

Bhaskara’s epicyclic eccentric theories of planetary motions are more developed than in the earlier siddhantas.