Short answers, pulled from the story.
Foundations of Economic Analysis was published by Harvard University Press in 1947. The book emerged two years after Samuelson submitted his doctoral dissertation to the David A. Wells Prize Committee of Harvard University in 1941.
The text serves as a canonical exposition of mathematical methods for economic theory and analysis. It demonstrates how operationally meaningful theorems can be described with a small number of analogous methods derived from equilibrium conditions and maximization principles.
J. Willard Gibbs provided the motto Mathematics is a language that appears on the front page of the first edition. This quote reflects Samuelson's fundamental principle of generalization by abstraction regarding analogies between central features of various theories.
Richard N. Cooper wrote in 1997 that the book drastically redirected advanced study toward greater use of mathematics after World War II. Roger E. Backhouse noted in 2015 that it played a major role defining how economic theory was undertaken for many years following the Second World War.
Chapter VIII provides a survey of welfare economics and develops what became known as the Bergson-Samuelson social welfare function. The section elucidates Pareto optimality and reveals the germ of truth in Adam Smith's doctrine of the invisible hand while outlining deficit financing effects.