Siddhānta Shiromani
Encyclopedia
Siddhānta Shiromani is the major work of Indian mathematician Bhāskara II.Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses.
and mensuration.
.
Bhaskara has given a very simple method to determine the circumference of the Earth. According to this method, first find out the distance between two places, which are on the same longitude. Then find the correct latitudes of those two places and difference between the latitudes. Knowing the distance between two latitudes, the distance that corresponds to 360 degrees can be easily found, which the circumference of is the Earth. For example, Satara and Kolhapur are two cities on almost the same longitude. The difference between their latitudes is one degree and the distance between them is 110 kilometers. Then the circumference of the Earth is 110 X 360 = 39600 kilometers. Once the circumference is fixed it is easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as 4967 ‘yojane’ (1 yojan = 8 km), which means 39736 kilometers. His value of the diameter of the Earth is 1581 yojane i.e. 12648 km. The modern values of the circumference and the diameter of the Earth are 40212 and 12800 kilometers respectively. The values given by Bhaskara are astonishingly close.
For astronomical calculations, Bhaskara selected a set of eight right angle triangles, similar to each other. The triangles are called ‘aksha kshetre’. One of the angles of all the triangles is the local latitude. If the complete information of one triangle is known, then the information of all the triangles is automatically known. Out of these eight triangles, complete information of one triangle can be obtained by an actual experiment. Then using all eight triangles virtually hundreds of ratios can be obtained. This method can be used to solve many problems in astronomy.
Ancient Indian Astronomers knew that there was a difference between the actual observed timing of a solar eclipse and timing of the eclipse calculated from mathematical formulae. This is because calculation of an eclipse is done with reference to the center of the Earth, while the eclipse is observed from the surface of the Earth. The angle made by the Sun or the Moon with respect to the Earth’s radius is known as parallax. Bhaskara knew the concept of parallax, which he has termed as ‘lamban’. He realized that parallax was maximum when the Sun or the Moon was on the horizon, while it was zero when they were at zenith. The maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact timing of a solar eclipse from the surface of the Earth can be determined.
In this chapter of Goladhyay, Bhaskar has discussed eight instruments, which were useful for observations. The names of these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Out of these eight instruments Bhaskara was fond of Phalak yantra, which he made with skill and efforts. He argued that ‘ this yantra will be extremely useful to astronomers to calculate accurate time and understand many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a precursor of the ‘astrolabe’ used during medieval times.
Dhee yantra
This instrument deserves to be mentioned specially. The word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of human being itself was an instrument. If an intelligent person gets a fine, straight and slender stick at his/her disposal he/she can find out many things just by using that stick. Here Bhaskara was talking about extracting astronomical information by using an ordinary stick. One can use the stick and its shadow to find the time, to fix geographical north, south, east, and west. One can find the latitude of a place by measuring the minimum length of the shadow on the equinoctial days or pointing the stick towards the North Pole. One can also use the stick to find the height and distance of a tree even if the tree is beyond a lake.
Lilavati
The name of the book comes from his daughter Līlāvatī.It is the first volume of Siddhānta Shiromani. The book contains thirteen chapters,278 verses, mainly arithmeticArithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
and mensuration.
Bijaganita
It is the second volume of Siddhānta Shiromani. It is divided into six parts, contains 213 verses and is devoted to algebraAlgebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
.
Ganitadhyaya and Goladhyaya
Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to astronomy. All put together there are about 1000 verses.(Ganitadhyaya has 451 and Goladhyaya has 501 verses). Almost all aspects of astronomy are considered in these two books. Some of the highlights are worth mentioning.- Earth’s circumference and diameter:
Bhaskara has given a very simple method to determine the circumference of the Earth. According to this method, first find out the distance between two places, which are on the same longitude. Then find the correct latitudes of those two places and difference between the latitudes. Knowing the distance between two latitudes, the distance that corresponds to 360 degrees can be easily found, which the circumference of is the Earth. For example, Satara and Kolhapur are two cities on almost the same longitude. The difference between their latitudes is one degree and the distance between them is 110 kilometers. Then the circumference of the Earth is 110 X 360 = 39600 kilometers. Once the circumference is fixed it is easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as 4967 ‘yojane’ (1 yojan = 8 km), which means 39736 kilometers. His value of the diameter of the Earth is 1581 yojane i.e. 12648 km. The modern values of the circumference and the diameter of the Earth are 40212 and 12800 kilometers respectively. The values given by Bhaskara are astonishingly close.
- Aksha kshetre:
For astronomical calculations, Bhaskara selected a set of eight right angle triangles, similar to each other. The triangles are called ‘aksha kshetre’. One of the angles of all the triangles is the local latitude. If the complete information of one triangle is known, then the information of all the triangles is automatically known. Out of these eight triangles, complete information of one triangle can be obtained by an actual experiment. Then using all eight triangles virtually hundreds of ratios can be obtained. This method can be used to solve many problems in astronomy.
- Geocentric parallax
Ancient Indian Astronomers knew that there was a difference between the actual observed timing of a solar eclipse and timing of the eclipse calculated from mathematical formulae. This is because calculation of an eclipse is done with reference to the center of the Earth, while the eclipse is observed from the surface of the Earth. The angle made by the Sun or the Moon with respect to the Earth’s radius is known as parallax. Bhaskara knew the concept of parallax, which he has termed as ‘lamban’. He realized that parallax was maximum when the Sun or the Moon was on the horizon, while it was zero when they were at zenith. The maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact timing of a solar eclipse from the surface of the Earth can be determined.
- Yantradhyay
In this chapter of Goladhyay, Bhaskar has discussed eight instruments, which were useful for observations. The names of these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Out of these eight instruments Bhaskara was fond of Phalak yantra, which he made with skill and efforts. He argued that ‘ this yantra will be extremely useful to astronomers to calculate accurate time and understand many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a precursor of the ‘astrolabe’ used during medieval times.
Dhee yantra
This instrument deserves to be mentioned specially. The word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of human being itself was an instrument. If an intelligent person gets a fine, straight and slender stick at his/her disposal he/she can find out many things just by using that stick. Here Bhaskara was talking about extracting astronomical information by using an ordinary stick. One can use the stick and its shadow to find the time, to fix geographical north, south, east, and west. One can find the latitude of a place by measuring the minimum length of the shadow on the equinoctial days or pointing the stick towards the North Pole. One can also use the stick to find the height and distance of a tree even if the tree is beyond a lake.