During the Hellenistic Age, Archimedes made advancements in "mathematics"--specifically in the realm of geometry, since he developed innovative ways to calculate volume, area, etc.
British Colonies in the Americas were established by royal charter. French colonies started as fur trading posts and then developed later to common colonies. Spanish colonies were crown-sponsored conquests to find riches such as gold and silver.
The British colonies had population mostly from colonists that were recruited among middle-class farmers, servants, and convicted criminals. French colonies population were initially traders, merchants and missionaries, later some soldiers were given land there. Spanish colonies population consisted of conquerors, soldiers and missionaries.
The economy from each colony was different, in the British colony the economy was diverse and included mostly farming, fishing and trading products like tobacco, rice, timber, and fish. The French colony economy like said before, was established over fur trade and some plantations on the Mississippi valley. The Spanish colonies economy was a trading economy that consisted of farming but it was heavily regulated by the Spanish crown.
The British colony started with a friendly relationship with the Native Americans, eventually, things changed when they needed to get new land. The French colony relationship with Native Americans was good, they respected them and forged alliances. Spanish colony relationship with Native Americans was bad when they were not killed by soldiers, they were subjugated or converted to Christianity.
