How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?

Question

So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this:

Find four whole numbers, the sum of any two of which add up to a perfect square.

The numbers he found were: $$18530,~~~ 38114,~~~ 45986,~~~ 65570$$

I've searched everywhere on the net but I can't find how he solved this. Does anyone here know how he did it?

Thanks.

Answer 1

See page 129 and next of Yiu's book The Elementary Mathematical Works of Leonhard Euler (1707 – 1783) (Dunham's comment appears here).

Concerning the referenced Euler paper "Recherches sur le problème de trois nombres carrés tels que la somme de deux quelconques moins le troisième fasse un nombre carré" you may find the french, english and a synopsis here (from the excellent 'The works of Leonhard Euler online').