So it’s true, the diagonal of a pentagon can be expressed in terms of square roots. In particular, it’s easy to see from the expression \[d = \frac{1}{2}\left(1 + \sqrt{5}\right)\]that \(d\) is an irrational number, approximately \(1,\!618\). Of course, we could also have obtained that information directly from \[d(d-1) = 1\]In fact, the two expressions are equivalent in every way and tell us exactly the same things about \(d\). There is not the slightest difference in mathematical content in the two.
I suppose the cynical view of the situation would be that we have expended a great deal of effort to go precisely nowhere. We began with a description of \(d\) as “the number that when multiplied by one less than itself equals 1”, and we ended with \(d\) as “half of one more than the number whose square is 5”. That’s progress ? If all the information about \(d\) is contained in the original equation, why did we bother solving it ?
On the other hand, why bother baking bread ? We could just eat the raw ingredients.
The point of doing algebra is not to solve equations; it’s to allow us to move back and forth between several equivalent representations, depending on the situation at hand and depending on our taste. In this sense, all algebraic manipulation is psychological. The numbers are making themselves known to us in various ways, and each different representation has its own feel to it and can give us ideas that might not occur to us otherwise.
Paul Lockhart, Measurement, Belknap Press, 2012
Gauss never pursued the matter. Ayoub speculates that Gauss “did not attach very much importance to solvability by radicals,” referring again to Gauss’s [1799] dissertation, in which he wrote :
what is called a solution to an equation is, in reality, nothing but the reduction of the equation to prime equations – the solution is not exhibited but symbolized – and if you express a root of the equation \[x^{n} = H\]by \[x = \sqrt[n]{H}\]you have not solve it nor done anything more than if you devise some symbol to denote a root of the equation \[x^{n} + Ax^{n-1} + \ \dots \ = 0\]and place the root equal to this symbol…
Ron Irving, Beyond the Quadratic Formula, MAA Press, 2013
Raymond G. Ayoub (1980), Paolo Ruffini’s contributions to the quintic, Archive for History of Exact Sciences 23