  Translated from the Latin original, "Observationes circa bina biquadrata
quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the
Enestroem index.
  This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect. 1,
Euler states his "quartic conjecture" that there do not exist any nontrivial
integer solutions to $A^4+B^4+C^4=D^4$. I do not know whether he stated this
conjecture previously.
  In sect. 3, Euler sets A=p+q, B=p-q, C=r+s,D=r-s. Taking r=p and s=q gives
the trivial solution of C=A and B=D, but this gives Euler the idea of making p
and r multiples of each other and q and s multiples of each other. If k=ab this
again gives the obvious solution, so in sect. 5: Perturb k to be $k=ab(1+z)$.
  Euler works out two solutions. One is A=2219449, B=-555617, C=1584749,
D=2061283.
  Hardy and Wright, fifth ed., p. 201 give a simpler parametric solution of
$A^4+B^4=C^4+D^4$. Thomas Heath in his Diophantus, second ed., pp. 377-380
gives a faithful explanation of Euler's solution.
