465. Find four terms of a geometric proportion given the sum of the extremes $2 S$, the sum of the means $2 S_{1}$, and the sum of their squares $4 C^{2}$.

The problem was proposed by the Paris Academy commission (Cauchy, Liouville, Sturm, and Arago) to the phenomenal calculator Henri Monde.
465. Let the proportion be $u: x=y: z$, then by the condition

$$
u+z=2 S ; x+y=2 S_{1} ; u^{2}+x^{2}+y^{2}+z^{2}=4 c^{2}
$$

Bourdon's Solution. Let $u z=p$, then combining with the equation $u+z=2 S$, we have $u$ and $z$ as the roots of a quadratic equation expressed in terms of $p$. Similarly, $x y=p$ and $x+y=2 S_{1}$ gives $x$ and $y$ in terms of $p$. Substituting into the equation

$$
u^{2}+x^{2}+y^{2}+z^{2}=4 c^{2}
$$

we find

$$
p=S^{2}+S_{1}^{2}-c^{2}
$$

then by substituting the found value of $p$, we obtain $x, y, z, u$.

Remark. Without the indicated artificial technique, the solution leads to very complex calculations.