Discuss main stages and accomplishments in the development of astronomy and mathematics in ancient India?10 M

Note : Question looks Tough but arrange periodical arrangement by  observation in ancient India may easier 

observe on astronomy, mathematics, Vedanga Jyotisha, Shulbasutra, Siddhanta, Aryabhata, Bhaskara, Brahmagupta, Bhaskaracharya.

Introduction is your choice I just Mention for commonly

Introduction:

As in all ancient cultures, astronomy was born in India before mathematics, beginning with observations of the periodicity of the moon’s phases, a few identifiable planets, the northward or southward journey of the sunrise on the eastern horizon through the year, or to trace imaginary lines between the stars. 

1. Harappan Beginnings:

Astronomy: In India, those beginnings are not adequately documented. But  Ring-stones found at Mohenjo Daro, the largest city of the Indus civilization (2600-1900 BCE), have been interpreted as calendrical devices keeping track of the sunrise at different times of the year. The alignment of main streets along the cardinal directions has been attributed to the sighting of the star cluster Pleiades (Krittikā), which rose due east at the time. These interpretations are sound but remain conjectural.

Mathematical :for trading purposes the Harappans developed a standardized system of weights  in which, initially, each weight was double the preceding one, then, 10, 100 or 1,000 times the value of a smaller weight.

2. Science from the Vedic to the Early Historical Era:

Astronomy :

• ,the Rig-Veda, the oldest of the four Vedas, spoke of a year of 360 days divided into twelve equal. parts. 
•  the Yajur-Veda considered a lunar year of 354 days and a solar year of 365 days, and divided the year into six ritus or seasons of two months each. 

Mathematics:

• The first Indian texts dealing explicitly with mathematics are the Shulbasūtras, dated between the 8th and 6th centuries BCE. They were written in Sanskrit in the highly concise sūtra style.
• The Shulbasūtras also introduced a system of linear units based on dimensions of the human body; they were later slightly modified and became the traditional units used across India .

3.The Siddhāntic or Classical Era:

Astronomy:s, astronomy’s interface with the society was mostly through calendars and pañchāngas (almanacs), and the prediction of eclipses, which had great religious and social significance. Indeed, an astronomer’s fame was guaranteed if he could accurately predict the occurrence, nature and duration of eclipses, and many inscriptions record a king’s reward to such an astronomer

Mathematics:

• The Siddhāntic era opened in the 5th century CE, when texts called siddhāntas were composed — a Sanskrit word meaning ‘principle’ or ‘conclusion’, but which applies here to a collection of conclusions or a treatise. Āryabhata I (born 476 CE), working near what is today Patna, ushered in this era with his Āryabhatīya, which dealt concisely but systematically with developments in mathematics and astronomy. Brilliant scholars followed, such as Varāhamihira, Bhāskara I, Brahmagupta, Lalla, Srīdhara, Mahāvīra, Āryabhata II and many more. This was the time of great advances in algebra, geometry and the first steps in calculus, and in astronomy effective algorithms to track the paths of celestial bodies and predict eclipses. 

4 Cross Cultural Developments :

Astrnomy :

• Islamic astronomy took place, which, among other benefits, brought instruments such as the astrolabe to India. The famous and massive yantramantra or Jantar Mantar observatories built in the early 18th century by the Maharaja of Jaipur, Sawai Jai Singh (1688-1743 CE), represent a convergence between Indian, Arabic and European astronomy. Indian astronomy interacted not only with Islamic (or Zīj) and European astronomies, but also with Chinese astronomy, in complex interplays that invariably enriched both players.

Mathematics:

• Bhāskarācārya states that presenting proofs (upapattis) is part of the teaching tradition, while Jyeshthadeva devotes considerable space to them in his Yukti Bhāshā. 

Conclusion :In India, those beginnings are not adequately documented,Indian science limited the further growth of Indian mathematics is open to debate.