Pagina:Werkecarlf02gausrich.djvu/151

$${\begin{array}{c|l}{\text{pro complexu}}&{\text{positivi esse debent numeri}}\\\hline C&\eta ,\xi -\eta ,p-\xi -\eta \\C^{\prime }&p-\xi ,\xi -\eta ,\xi +\eta -p\\C^{\prime \prime }&p-\eta ,\eta -\xi ,\xi +\eta -p\\C^{\prime \prime \prime }&\xi ,\eta -\xi ,p-\xi -\eta \\\end{array}}$$

Ceterum vel nobis non monentibus quisque facile intelliget, in repraesentatione figurata numerorum complexorum (vid. art. 39) numeros systematis ${\textstyle S}$ intra quadratum contineri, cuius latera iungant puncta numeros ${\textstyle 0}$, ${\textstyle a+bi}$, ${\textstyle (1+i)(a+bi)}$, ${\textstyle i(a+bi)}$ repraesentantia, et subdivisionem systematis ${\textstyle S}$ respondere partitioni quadrati per rectas diagonales. Sed hocce loco ratiocinationibus pure arithmeticis uti maluimus, illustrationem per intuitionem figuratam lectori perito brevitatis caussa linquentes.

70.

Si quatuor numeri complexi ${\textstyle r=x+yi}$, ${\textstyle r^{\prime }=x^{\prime }+y^{\prime }i}$, ${\textstyle r^{\prime \prime }=x^{\prime \prime }+y^{\prime \prime }i}$, ${\textstyle r^{\prime \prime \prime }=x^{\prime \prime \prime }+y^{\prime \prime \prime }i}$ ita inter se nexi sunt, ut habeatur ${\textstyle r^{\prime }=m+ir}$, ${\textstyle r^{\prime \prime }=m+ir^{\prime }}$ ${\textstyle =(1+i)m-r}$, ${\textstyle r^{\prime \prime \prime }=m+ir^{\prime \prime }=im-ir}$, atque primus ${\textstyle r}$ ad complexum ${\textstyle C}$ pertinere supponitur, reliqui ${\textstyle r^{\prime }}$, ${\textstyle r^{\prime \prime }}$, ${\textstyle r^{\prime \prime \prime }}$ resp. ad complexus ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$ pertinebunt. Statuendo enim ${\textstyle \xi =ax+by}$, ${\textstyle \eta =ay-bx}$, ${\textstyle \xi ^{\prime }=ax^{\prime }+by^{\prime }}$, ${\textstyle \eta ^{\prime }=ay^{\prime }-bx^{\prime }}$, ${\textstyle \xi ^{\prime \prime }=ax^{\prime \prime }+by^{\prime \prime }}$, ${\textstyle \eta ^{\prime \prime }=ay^{\prime \prime }-bx^{\prime \prime }}$, ${\textstyle \xi ^{\prime \prime \prime }=ax^{\prime \prime \prime }+by^{\prime \prime \prime }}$, ${\textstyle \eta ^{\prime \prime \prime }=ay^{\prime \prime \prime }-bx^{\prime \prime \prime }}$, invenitur

$${\begin{aligned}\eta =p-\xi ^{\prime }&=p-\eta ^{\prime \prime }=\xi ^{\prime \prime \prime }\\\xi -\eta =\xi ^{\prime }+\eta ^{\prime }-p&=\eta ^{\prime \prime }-\xi ^{\prime \prime }=p-\xi ^{\prime \prime \prime }-\eta ^{\prime \prime \prime }\\p-\xi -\eta =\xi ^{\prime }-\eta ^{\prime }&=\xi ^{\prime \prime }+\eta ^{\prime \prime }-p=\eta ^{\prime \prime \prime }-\xi ^{\prime \prime \prime }\end{aligned}}$$

unde adiumento criteriorum theorematis veritas sponte demanat. Et quum rursus fiat ${\textstyle r=m+ir^{\prime \prime \prime }}$, facile perspicietur, si ${\textstyle r}$ supponatur pertinere ad ${\textstyle C^{\prime }}$, numeros ${\textstyle r^{\prime }}$, ${\textstyle r^{\prime \prime }}$, ${\textstyle r^{\prime \prime \prime }}$ pertinere resp. ad ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$, ${\textstyle C}$; si ille ad ${\textstyle C^{\prime \prime }}$, hos ad ${\textstyle C^{\prime \prime \prime }}$, ${\textstyle C}$, ${\textstyle C^{\prime }}$; denique si ille ad ${\textstyle C^{\prime \prime \prime }}$, hos ad ${\textstyle C}$, ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$.

Simul hinc colligitur, in singulis complexibus ${\textstyle C}$, ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$ aeque multos numeros reperiri, puta ${\textstyle {\frac {1}{4}}(p-1)}$.

71.

Theorema. Si denotante ${\textstyle k}$ integrum per ${\textstyle m}$ non divisibilem singuli numeri complexus ${\textstyle C}$ per ${\textstyle k}$ multiplicantur, productorumque residuis simpliciter minimis secun-