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

In this paper Euler considers the problem of given a,b,c,d where a=v(x+y+z), b=x(v+y+z), c=y(v+x+z), d=z(v+x+y), determining v,x,y,z. He makes many changes of variable and ends up with four nicer equations, each in terms of a new variable u and the constants a,b,c,d. I do not understand exactly what Euler is doing, but I think the translation is “correct”. If anyone understands what Euler is doing, please email me so I can make the translation flow better.