Final answer:
Alexander Hamilton's quote on private interest and public prosperity closely aligns with the ideas of Adam Smith, who introduced the concept of the 'invisible hand' in economics.
Step-by-step explanation:
The quote by Alexander Hamilton regarding the guidance of private interest aligns most closely with the ideas of Adam Smith, a Scottish economist and philosopher known for his book 'The Wealth of Nations'. Smith's concept of the 'invisible hand' represents the notion that individuals pursuing their own interests often benefit society as a whole, even when that is not their intention. This theme is echoed in Hamilton's confidence that allowing private interests to operate freely will lead to the most profitable employment and by extension, the public prosperity.