How did Great Britain Capture half of the world with A Small Army ever assemble?

 The population of Great Britain in 1750 was indeed 5.7 million, which is a relatively small number compared to the populations of other major powers at the time.

However, there are Many  factors that helped Britain to conquer and control with its small population.

Technology:

Britain's technological advantages were crucial in giving them an edge over their rivals during their imperial expansion .

Economic power: Britain was world's largest trader in 18th  and 19th centuries. This was  the more advantage to them to resource the fund their war. It is the empire that thrived on human trade known as slave trade

Diplomacy:

They are skilled and force to alliances with other power to achieve their goal.

Divide and rule:

Britain often used a policy of divide and rule to conquer and control its colonies. This involved divide different groups of people against each other and exploiting their land.

Indigenous allies: Britain often gained the support of indigenous peoples in their colonies. This helped them to fight their wars and maintain control over their territories.

These are just some of the factors that helped Britain to conquer half the world with a small population. 

It is important to note that Britain's imperial history was complex and often exploitative. 

However, it is undeniable that they were a major power in the 18th and 19th centuries and their empire had a profound impact on the world.