Inferences on the forms of roots of equations and of their orders
For an equation of degree n, Euler wants to define a "resolvent equation" of degree n-1 whose roots are related to the roots of the original equation. Thus, by solving the resolvent one can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic, and biquadratic equations. In section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper he tries to figure out in what cases resolvents will work.
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 6, pp. 216-231.
Opera Omnia Citation
Series 1, Volume 6, pp.1-19.