# What is the meaning of "positive acceptus" in Gauss' Disquisitiones Arithmeticae §131?

Sī p est numerus prīmus fōrmae 4n+1, erit +p, sī vērō p fōrmae 4n+3, erit -p residuum vel nōn-residuum cuiusvīs numerī prīmī quī positīvē acceptus ipsīus p est residuum vel nōn-residuum.

(I have taken the liberty to add macrons.)

From what I know about this theorem, that is called quadratic reciprocity nowadays, the translation should roughly be:

If `p` is a prime number of the form `4n+1`, let `p*` denote `+p`; otherwise if `p` is of the form `4n+3`, let `p*` denote `-p`; then `p*` is a [quadratic] residue (resp. non-residue) of whichever prime number `q` that is a [quadratic] residue (resp. non-residue) of this very `p`.

(I have added `p*` as an intermediate definition to make the sentence flow better in English.)

The last sentence can also be rephrased as:

`p*` is a quadratic residue of `q` iff `q` is a quadratic residue of `p`.

However, I do not understand where the phrase `positīvē acceptus` fits in this translation.

• It seems to me that, after fiddling with the sign of p at the start of the sentence, Gauss is now saying that q is to be taken positive. (Caution: I claim no expertise in Latin, but I'm a mathematician, and this reading makes sense to me.) May 1, 2021 at 0:11
• @AndreasBlass As a bit of both mathematician and latinist, I agree. I tried to argue that in my answer too. I didn't manage to find whether Gauss in general required primes to be positive, but no other reading seems to make sense to me. May 1, 2021 at 14:54