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:
pis a prime number of the form
+p; otherwise if
pis of the form
p*is a [quadratic] residue (resp. non-residue) of whichever prime number
qthat is a [quadratic] residue (resp. non-residue) of this very
(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
qis a quadratic residue of
However, I do not understand where the phrase
positīvē acceptus fits in this translation.