Final answer:
The Mughal Empire was the ruling dynasty in India when Europeans arrived in the 1600's and 1700's, but eventually the British gained control through the British East India Company.
Step-by-step explanation:
The name of India's ruling dynasty when Europeans began to arrive in the region in the 1600's and 1700's was the Mughal Empire. However, by the beginning of the 18th century, the Mughal Empire had collapsed and multiple weak Indian states were formed. This allowed the British East India Company to establish control and eventually rule most of the southern part of India.