elementary number theory - Reference for a Post about Gauss formula for primes written as sum of two squares?

In a post Efficiently finding two squares which sum to a prime

I read

"In 1825 Gauss gave the following construction for writing a prime congruent to $1 \pmod{4}$ as a sum of two squares: Let $p=4k+1$ be a prime number. Determine $x$ (this is uniquely possible...) so that

$$ x = \frac{(2k)!}{2(k!)^2} \pmod{p}, \quad |x| < \frac{p}{2}$$

Now determine $y$ so that

$$ y = x \cdot (2k)! \pmod{p}, \quad |y| < \frac{p}{2}$$

Gauss showed that $x^2+y^2=p$."

I checked the Stark's book and I did not see a direct reference to the Gauss original paper or book or other reference to explain the method how to come to this and find the formula. If you know the reference or you can explain its procedure, I will be grateful.