An algebraic problem of finding four numbers given the products of each of the numbers with the sum of the other three

Abstract

This is a translation from the Latin of Euler's "Problema algebraicum de inveniendis quatuor numeris ex datis totidem productis uniuscuiusque horum numerorum in summas trium reliquorum", Opera Postuma 1 (1862), 282-287, reprinted in Leonhardi Euleri Opera omnia I.6. This paper is E808 in the E808 in the Eneström index. We are given the four products $a=v(x+y+z), b=x(v+y+z), c=y(v+x+z), d=z(v+x+y)$. We want to determine $v,x,y,z$ from this data. Euler wants to do this symmetrically: instead of successively eliminating variables and finding out, say, z in terms of a,b,c and then finding y, and then finding x, and then finding v, Euler introduces a new variable that has the same relation with each of the unknowns v,x,y,z and works with that. Euler finds that there will be up to eight choices of v,x,y,z for each choice of a,b,c,d.