64.
Proficiscendo a regula ultima in art. praec. eruta invenimus esse
![{\displaystyle {\begin{array}{c|l}{\text{numeri}}&{\text{ characterem }}\equiv \\\hline -1+i&{\frac {1}{8}}(aa+2ab-bb-1)\\-1-i&{\frac {1}{8}}(-aa+2ab+bb+1)\\+1-i&{\frac {1}{8}}(aa+2ab+3bb-1)\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86b70776215b3ab74aa3b813b3a527b9067d11d7)
Hoc statim inde sequitur, quod character ipsius
![{\textstyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f327332497b21a059c479e7b2ce098baa1a7e)
est
![{\textstyle {\frac {1}{4}}(aa+bb-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c067f49cadd280b66f4a0538e233145edba534bf)
, character ipsius
![{\textstyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03d0251b380d21d87eb1cd07628008e254050c3)
autem
![{\textstyle {\frac {1}{2}}(aa+bb-1)\equiv {\frac {1}{2}}bb}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad61c653f348d5608f5bd0927405b688886de85)
, quum
![{\textstyle aa-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e02c236d33a0dc701340b4c654122435cb799368)
semper sit formae
![{\textstyle 8n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b2f9a2f3909383da379182411248cfbb891932)
. Manifesto hae quatuor regulae, etiamsi hactenus ab inductione mutuatae sint, ita inter se sunt nexae, ut quamprimum unius demonstratio absoluta fuerit, tres reliquae simul sint demonstratae. Vix opus est monere, etiam in his regulis tantummodo supponi
![{\textstyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8)
imparem,
![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2)
parem.
Si formulas ad modulos primarios restrictas adhibere non displicet, hac forma uti possumus. Est
![{\displaystyle {\begin{array}{c|l}{\text{numeri}}&{\text{ characterem }}\equiv \\\hline -1+i&{\frac {1}{4}}(-a-b+1-bb)\\-1-i&{\frac {1}{4}}(a-b-1+bb)\\+1-i&{\frac {1}{4}}(-a-b+1+bb)\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f69988343d126d31bb283e365a43da86f586b42)
Formulae simplicissimae prodeunt, si, ut initio inductionis nostrae feceramus, modulos primi et secundi generis distinguimus. Est scilicet character
![{\displaystyle {\begin{array}{c|c|c}{\text{numeri}}&{\text{pro modulis primi generis}}&{\text{pro modulis secundi generis}}\\\hline -1+i&{\frac {1}{4}}(-a-b+1)&{\frac {1}{4}}(-a-b-3)\\-1-i&{\frac {1}{4}}(a-b-1)&{\frac {1}{4}}(a-b+3)\\+1-i&{\frac {1}{4}}(-a-b+1)&{\frac {1}{4}}(-a-b+5)\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63818051da00ce10261180527947af9257d12a7b)
65.
Pro numero
, ad quem iam progredimur, eandem distinctionem inter modulos
eos, pro quibus
,
, atque eos, pro quibus
,
quoque adhibebimus. Tabula art. 62 docet, respectu illius numeri respondere