In an unpublished fragment of Gauss entitled "on the theory of complex numbers", written by him around 1810, Gauss made early inroads into the theory of cyclotomic numbers, which are numbers that can be written as $f(\epsilon) = a_0+a_1\epsilon+a_2\epsilon^2+\dots+a_{n-1}\epsilon^{n-1}$ when $n$ is a prime number and $\epsilon^n = 1$ ($\epsilon$ is a non-trivial root of unity). Just to illustrate its importance, in chapter [I] he proves Fermat's last theorem for $n=3,5$, and in chapter [V] he states the following theorem on "cyclotomic units" later established by Ernst Kummer:
every unit of the cyclotomic field $\mathbb{Q}[\epsilon]$ is a product of numbers of the form $$\frac{\epsilon^{\alpha}-\epsilon^{\beta}}{\epsilon^{\gamma}-\epsilon^{\delta}}$$
According to Paul Bachmann, another theorem later established by Kummer (in 1847), which states that a cyclotomic integer $f(\rho)$ whose analytical modulus (analytical norm) is equal to 1 must be $\pm\rho^n$, also occurs here for the first time.
My question is concerned with a result Gauss gives in chapter [III], which posits a certain upper bound for a quantity Gauss calls "mensura" in the case $n=5$, in terms of a quantity he calls determinant $D$. The quantity he calls determinant is defined to be:
$D=f(\epsilon)f(\epsilon^2)\cdots f(\epsilon^{n-1})$
so it is almost the algebraic norm of the cyclotomic number. However, the deeper meaning of the notion of "mensura", which literally means "scale" , is not very clear to me. It is defined to be the sum of analytic norms of $f(\epsilon)$ and all of its complex conjugates except $f(\epsilon^0)$:
$$Mensura = \sum_{k=1}^{n-1}f(\epsilon^k)f(\epsilon^{-k})$$
Gauss states the following inequallity:
Highest mensura = $2(\frac{sin(2\pi/5)}{sin(\pi/5)}+\frac{sin(\pi/5)}{sin(2\pi/5)})\sqrt{D}\approx 4.472\sqrt{D}$
My question is how to prove this inequlity? and what is the meaning of the notion of "mensura"?