This paper first appeared in Novi Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 64-73 and is reprinted in Opera Omnia: Series 1, Volume 2, pp.519-530. Its Eneström number is E270. Euler improves his results significantly in "On Three Square Numbers, of Which the Sum and the Sum of Products Two Apiece will be a Square" (E523).
Snavely, Mark R. and Woodruff, Philip
"The Solution of a Problem of Searching for Three Numbers, of Which the Sum, Product, and the Sum of Their Products Taken Two at a Time, Are Square Numbers,"
Euleriana: 2(1), pp.27-39.
Available at: https://scholarlycommons.pacific.edu/euleriana/vol2/iss1/5