13. Legacy of Ancient Indian History: Important Observations

• Sulva Sutra also mentions a ritual which included ............ thus geometrically constructing a square having the same area as a given area. It has worked out the square root of 2 to 1.414215, up to last 5 decimals. Knowledge of Days of Year

"Squaring the circle" (and vice-versa),

Bhaskaracharya was one of the promonent Indian mathematicien and astronomer, who wrote a book .......... 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.

'Sidhantshiromani'.

Panchasiddhanta of Varahamihira

(505 AD)

The twelve Yantras of Lalla

(700 AD)

Madhava

(c. 1340-1425)

The Indian scholar Pingala........... used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables).

(circa. 5th-2nd century BC)

The Panini's grammar was composed in 6th century B.C or earlier. This grammar has provided .......that describe the Sanskrit of his day completely. It has been shown that grammar of Panini represents a universal grammatical and computing system. From this perspective it anticipates the logical framework of modern computers

.4,000 rules

Seidenberg's conclusion of India being the source of the geometric and mathematical knowledge of the ancient world has been included now in chronology of the texts. Please note that Sulva sutras belong to a bigger text Shrauta Sutras. The four major Sulva Sutras, which are mathematically the most significant, are those composed by ..............Apastamba and Katyayana. Out of them the oldest belongs to Baudhayana and dates back to 600BC. They discuss the cases of the Pythagorean Theorem and Pythagorean triples.

.Baudhayana, Manava

It has been hypothesized that symbol of Zero was devised by unknown Indians who were inspired with the Fish sign of ........... that refered to 10.

.Brahmi script

Pingala used the above in his ......... 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.

.Chhandahshastra

Bhaskara was the author of Siddhanta Shiromani, a book in four parts: .......................

1. Lilavati on arithmetic 2. Bijaganita on algebra 3. Ganitadhyaya on astronomy 4. Goladhyaya on astronomy

SaptaDhatu or 7 constituent elements viz.

1. Rasa dhatu (Chyle, Lymph, Plasma) 2. Rakta dhatu (Haemoglobin fraction in blood) 3. Mamsa dhatu (Muscle tissue) 4. Medas dhatu (Fat or Adipose tissue) 5. Asthi dhatu (Bone-tissue including cartilage) 6. Majja dhatu (Bone Marrow) 7. Sukra dhatu (Semen, Sperm, Ovum)

Lagadha

1st millennium BCe

The gahtiyantra theory interprets that the vessel sinks in

24 Minutes

Plato in Greek medicine has also described the same ........ and it appears to be derived from the earlier tridosha theory of Ayurveda.

3 issues

There have been found symbols that prove that the Aryans knew about the year. In a ritual, pebbles were placed around the altars for the earth, the atmosphere, and the sky. The number of these pebbles was 21, 78, and 261, respectively. These numbers add up to the 360 days of the year. There were other features related to the design of the altars which suggested that the ritualists were aware that the length of the year was between

365 and 366 days.

Madhava (c. 1340-1425) developed a procedure to determine the positions of the moon every 3.........

6 minutes

However, it was not till ........., when practical calculations were carried out using zero.

6-8th century AD

Kindly go thru the following Shloka The meaning of the above shloka is that a Ghatika or (bowl) is a hemispherical vessel made of copper with a small aperature at the bottom so that it sinks into the water........

60 times in a day and night.

The Panini's grammar was composed in ..........

6th century B.C or earlier.

His school, which was a rival to that of Aryabhata, has been very infuential in western and northern India. Brahmagupta's work was translated into Arabic in .......... at Baghdad and it became famous in the Arabic world as Sindhind.

771 or 773

Valmiki has also written ............ in which he deals with mathematics in general, algebra, trigonometry, chemicals and heat

Akshara Lakshana

Citation of the Surya Siddhanta is also found in the works of

Aryabhata.

The ........... says: The diagonal of a rectangle produces both areas produced seperately by its two sides. In the Baudhayan's Sulva Sutras we should note that the Right angles were made by ropes marked to give the triads 3, 4, 5 and 5, 12, 13 (32 + 42 = 52, 52 + 122 = 132)

Baudhayan Sulbasutra 1.48

The Lilavati of

Bhaskara

The Siddhant Shiromani & Yasti Yantra of ............

Bhaskaracharya (1072 AD)

The above Shloka is of Bharamagupta. Brahmagupta of Bhilamala in Rajasthan, who was born in 598, wrote his masterpiece,,,,,,,,,,,,,,,,, in 628.

Brahmasphuta Siddhanta,

The Vedic People took Earth as .............Thus Earth was taken on mid of the planetary system.The system they identified was as follows:

Centre of Universe.

Sixty Vipala makes one pala. Sixty pala makes one ghati. 150 Pala makes one Hora. 24 Hora (Hours) makes one ...........

Dina (a day).

The ........... is a type of armilliary sphere used to locate planetary positions.

Gola yantra

Bhaskara (born 1114), who was from the........... was an outstanding mathematician and astronomer.

Karnataka region,

............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.

Kartari

One of Brahmagupta's chief contributions is the solution of a certain second order indeterminate equation which is of great significance in number theory. Another of his books, the...........remained a popular handbook for astronomical computations for centuries

Khandakhadyaka,

• A special geometrical construction known a '..............was used for the above purpose. Altitude of sun and day time was also measured with this instrument based on the shadow.

Matsya'

Knowledge of Binary Numbers- ...........

Pingala

Ayurveda adopts the physics of the five elements viz.

Prithvi(earth), Jala(water), Agni (fire), Vayu(air) and Akasha (ether)

Ayurveda divides the human body in

SaptaDhatu or 7 constituent elements viz.

Most probably, the first true alphabetic script was .............that was practiced on the banks of Saraswati River. \

Saraswati Script

Some scholars have shown on the basis of evidence in ......... Indian geometry predates Greek geometry by centuries. It has been argued that Geometry and Mathematics had a ritualistic beginning in India centuries before Greeks or Babylon. In these rituals, Earth was represented by Circular altar and heavens were represented in Squar altar. There wer eagle shaped altars also. The examples are as follows:

Shatapatha Brahmana that

Lalla was an Indian astronomer and mathematician who followed the tradition of Aryabhata I. Lalla's most famous work was entitled ...........He was well known because of twelve instruments which he brought into practice.

Shishyadhividdhidatantra

Bhaskara was the author of ............. a book in four parts:

Siddhanta Shiromani,

One of the most discussed shloka of Lalla is 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 instuments along with a stick

The Ritual consisted of coverting the Circle into a square of identical area. As per a paper by Seidenberg: Babylonia [1700 BC] got the theorem of Pythagoras from India or that both Old-Babylonia and India got it from a third source. The source quoted was ............. deal with complex fire altars of various shapes constructed with bricks of specific shapes and area: the total area of the altar must always be carefully respected. This proves that despite of no existance of algebra, there was an awareness of precise purely geometrical calculations.

Sulvasutras. Sulvasutras

Unbalance in any of the Vata: Air Pitta : Bile and Kapha: Phlegm causes illness and this is called .

Tridosha

Madhava provided methods to estimate the motions of the planets. He gave power series expansions for ...............correct to eleven decimal places

Trigonometric functions, and for pi

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. Please note that there are several works with the same name, BUT the Original writer of Surya Siddhanta is

UNKNOWN.

We refer to the following shloka from Surya Siddhanta The above shloka means Sixty vikala is one kala, sixty units (kala) is one bhaga, it is said this way that, thirty units (bhaga) makes one rashi, twelve rashi makes one bhagana. The Vikala, Kala, Bhaga,Bhagna are the .......... used in Surya Sidhnata. This knowledge about the division of the angles in a circle is one of the earliest knowledge. Today, Vikala, Kala and Bhaga are coterminous with the Second, Minute and Degree respectively. Please note that Rashi was also used as a unit of angle.

Units of Angle

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 theoryof 'Suryasidhanta'. Introduction of zero in mathematics and the decimal method of calculation is one of such invaluable contribution. We should note that ............ had contrasted Surya Siddhanta along with his 4 other treatises in the panchsiddhantika viz. Paitamaha Siddhantas, Paulisha , Romaka Siddhantas and Vasishta Siddhanta.

Varahamihira

Ayurveda talks about the balance of three elemental energies called

Vata: Air • Pitta : Bile and • Kapha: Phlegm

Yasti means a stick. ........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.

Yasti Yantra

Valmiki, the author of Ramayana has written................. in which there are some passages on scientific point of view relating to the description of the nature of space, time, matter, and consciousness.

Yoga Vashishtha

...............of Valmiki

Yogavashishtha

The ........ was used in India for astronomical purposes until recent times.

clepsydra (Ghatī -yantra)

Bhaskara (born 1114), who was from the Karnataka region, was an outstanding mathematician and astronomer. Amongst his mathematical contributions is the concept of ............

diffeentials.

• Shanku is a type of............a long vertical cone used to identify East-West-North-South direction based on shadow of its tip.

gnomon,

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 andbottom 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 .................................. 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 .

h1/L1 = H1/L and h2/L2 = H2/L,

Panchasiddhanta of Varahamihira He 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.

Bhaskara's epicycliceccentric theories of planetary motions are more developed than in the earlier ............

siddhantas.

Valmiki, the author of Ramayana has written Yoga Vashishtha in one of the passage says: The world is like a potter's wheel: the wheel looks as if it stands still, though it revolves at a..............

terriffic speed.

• Shakata consists of

two 'V' shaped sticks, pivoted at the end.

Length: Yahodara, Angula, Hatha, Danda, Krosha and YojanasWe refer to this shloka from Lilavati Meaning of above shloka is as follows: Eight rice grains (yahodara) makes one fingure width (angula), twentyfore fingures make one hand (hasta), four hands make one stick (danda), 2000 danda make one krosha (unit of length). Four krosha make one ............., ten hasta make one vamsha (unit of length), one nirwatan (unit of area) is equal to 400 square vamsha.

yojana (unit of length)

.........is a ring with angular graduations alonge its circumference, it is a type of protractor.

• Bhangana

............... is a circular disk with angular graduations; it is also a type of protractor.

• Chakra

........ is a semicircular disk with angular graduations and a stick pivoted at the center, it is a type of protractor with a plumb bob arrangement.

• Dhanu

........... is a small vessel with a hole at the bottom. It was used to measure time.

• Ghati

..........is 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.

• Pitha

............ is combination of two sticks with a string.

• Shalaka

.........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.

• Yasti