On Double Integral Formulas: An English Translation of E391

In this paper, Euler investigates the theory of double intergrals to uses them to find areas, volumes and surface areas. He considers transformations of variables in double integrals. After showing that simple multiplication of the transformed differentials is not appropriate, he derives the formula for a change of variables in double integrals. Euler then uses such changes of variables to investigate areas and volumes, and then turns his attention to the so-called ``Florentine problem.'' This problem requires one to remove four equal windows from a hemispherical surface so that the remaining area is equal to the area of a square. He exhibits two solutions to this problem and shows how more solutions could be generated. In conclusion, Euler notes how double integrals might be used to investigate isoperimetric problems, particularly in finding the minimum surface area needed to enclose a given volume.