Summatio quarandum serierum singularium

by Carolus Fridericus Gauss

Carolus Fridericus Gauss

Summatio Quarandum Serierum Singularium

1811

editio: incognita
fons: incognitus

1.

Inter veritates insigniores, ad quas theoria divisionis circuli aditum aperuit, locum haud ultimum sibi vindicat summatio in Disquiss. Arithmet. art. 356 proposita, non modo propter elegantiam suam peculiarem, miramque foecunditatem, quam fusius exponendi occasionem posthac dabit alia disquisitio, sed ideo quoque, quod eius demonstratio rigorosa atque completa difficultatibus haud vulgaribus premitur. Quae sane eo minus exspectari debuissent, quum non tam in ipsum theorema cadant, quam potius in aliquam theorematis limitationem, qua neglecta demonstratio statim in promtu est, facillimeque e theoria in opere isto explicata derivatur. Theorema illic exhibitum est in forma sequente. Supponendo ${\textstyle n}$ esse numerum primum, denotandoque indefinite omnia residua quadratica ipsius ${\textstyle n}$ inter limites 1 et ${\textstyle n-1}$ incl. sita per ${\textstyle a}$, omniaque non-residua inter eosdem limites iacentia per ${\textstyle b}$, denique per ${\textstyle \omega }$ arcum ${\textstyle {\frac {360^{\circ }}{n}}}$, et per ${\textstyle k}$ integrum determinatum quemcunque per ${\textstyle n}$ non divisibilem, erit

I. pro valore ipsius ${\textstyle n}$, qui est formae ${\textstyle 4m+1}$, $${\begin{aligned}\Sigma \cos ak\omega &=-{\frac {1}{2}}\pm {\frac {1}{2}}\surd n\\\Sigma \cos bk\omega &=-{\frac {1}{2}}\mp {\frac {1}{2}}\surd n,{\text{ adeoque }}\\\Sigma \cos ak\omega -\Sigma \cos bk\omega &=\pm \surd n\\\Sigma \sin ak\omega &=0\\\Sigma \sin bk\omega &=0\end{aligned}}$$

II. pro valore ipsius ${\textstyle n}$, qui est formae ${\textstyle 4m+3}$, $${\begin{aligned}\Sigma \cos ak\omega &=-{\frac {1}{2}}\\\Sigma \cos bk\omega &=-{\frac {1}{2}}\\\Sigma \sin ak\omega &=\pm {\frac {1}{2}}\surd n\\\Sigma \sin bk\omega &=\mp {\frac {1}{2}}\surd n\\\Sigma \sin ak\omega -\Sigma \sin bk\omega &=\pm \surd n\end{aligned}}$$

Hae summationes l.c. omni rigore demonstratae sunt, neque alia difficultas hic remanet nisi in determinatione signi quantitati radicali praefigendi. Nullo quidem negotio ostendi potest, hoc signum eatenus a numero ${\textstyle k}$ pendere, quod semper pro cunctis valoribus ipsius ${\textstyle k}$, qui sint residua quadratica ipsius ${\textstyle n}$, signum idem valere debeat, et contra signum huic oppositum pro omnibus valoribus ipsius ${\textstyle k}$, qui sint non-residua quadratica ipsius ${\textstyle n}$. Hinc totum negotium in valore ${\textstyle k=1}$ versabitur, patetque, quam primum signum pro hoc valore valens innotuerit, pro omnibus quoque reliquis valoribus ipsius ${\textstyle k}$ signa statim in promtu fore. Verum enim vero in hac ipsa quaestione, quae primo aspectu inter faciliores referenda videtur, in difficultates improvisas incidimus, methodusque, qua ducente sine impedimentis hucusque progressi eramus, auxilium ulterius prorsus denegat.

2.

Haud abs re erit, antequam ulterius progrediamur, quaedam exempla summationis nostrae per calculum numericum evolvisse: huic vero quasdam observationes generales praemittere conveniet.
I. Si in casu eo, ubi ${\textstyle n}$ est numerus primus formae ${\textstyle 4m+1}$, omnia residua quadratica ipsius ${\textstyle n}$ inter 1 et ${\textstyle {\frac {1}{2}}(n-1)}$ incl. iacentia indefinite per ${\textstyle a^{\prime }}$ exhibentur, omniaque non-residua inter eosdem limites per ${\textstyle b^{\prime }}$, constat, omnes ${\textstyle n-a^{\prime }}$ inter ipsos ${\textstyle a}$, omnesque ${\textstyle n-b^{\prime }}$ inter ${\textstyle b}$ comprehensos fore: quamobrem quum omnes ${\textstyle a^{\prime }}$, ${\textstyle b^{\prime }}$, ${\textstyle n-a^{\prime }}$, ${\textstyle n-b^{\prime }}$ manifesto totum complexum numerorum ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots n-1}$ expleant, omnes ${\textstyle a^{\prime }}$ cum omnibus ${\textstyle n-a^{\prime }}$ iuncti omnes ${\textstyle a}$ complectentur, et perinde omnes ${\textstyle b^{\prime }}$ cum omnibus ${\textstyle n-b^{\prime }}$ iuncti omnes ${\textstyle b}$ comprehendent. Hinc erit $${\begin{aligned}&\Sigma \cos ak\omega =\Sigma \cos a^{\prime }k\omega +\Sigma \cos(n-a^{\prime })k\omega \\&\Sigma \cos bk\omega =\Sigma \cos b^{\prime }k\omega +\Sigma \cos(n-b^{\prime })k\omega \\&\Sigma \sin ak\omega =\Sigma \sin a^{\prime }k\omega +\Sigma \sin(n-a^{\prime })k\omega \\&\Sigma \sin bk\omega =\Sigma \sin b^{\prime }k\omega +\Sigma \sin(n-b^{\prime })k\omega \end{aligned}}$$ Iam quum habeatur ${\textstyle \cos(n-a^{\prime })k\omega =\cos a^{\prime }k\omega }$, ${\textstyle \cos(n-b^{\prime })k\omega =\cos b^{\prime }k\omega }$, ${\textstyle \sin(n-a^{\prime })k\omega =-\sin a^{\prime }k\omega }$, ${\textstyle \sin(n-b^{\prime })k\omega =-\sin b^{\prime }k\omega }$, patet sponte fieri $${\begin{aligned}&\Sigma \sin ak\omega =\Sigma \sin a^{\prime }k\omega -\Sigma \sin a^{\prime }k\omega =0\\&\Sigma \sin bk\omega =\Sigma \sin b^{\prime }k\omega -\Sigma \sin b^{\prime }k\omega =0\end{aligned}}$$ Summatio cosinuum vero hanc formam assumit $${\begin{aligned}&\Sigma \cos ak\omega =2\Sigma \cos a^{\prime }k\omega \\&\Sigma \cos bk\omega =2\Sigma \cos b^{\prime }k\omega \end{aligned}}$$ unde fieri debebit $${\begin{aligned}1+4\Sigma \cos a^{\prime }k\omega &=\pm \surd n\\1+4\Sigma \cos b^{\prime }k\omega &=\mp \surd n\\2\Sigma \cos a^{\prime }k\omega -2\Sigma \cos b^{\prime }k\omega &=\pm \surd n\end{aligned}}$$

II. In casu eo, ubi ${\textstyle n}$ est formae ${\textstyle 4m+3}$, complementum cuiusvis residui ${\textstyle a}$ ad ${\textstyle n}$ erit non-residuum, complementumque cuiusvis ${\textstyle b}$ erit residuum; quocirca omnes ${\textstyle n-a}$ convenient cum omnibus ${\textstyle b}$, omnesque ${\textstyle n-b}$ cum omnibus ${\textstyle a}$. Hinc colligitur $$\Sigma \cos ak\omega =\Sigma \cos(n-b)k\omega =\Sigma \cos bk\omega$$ quare quum omnes ${\textstyle a}$ et ${\textstyle b}$ iuncti omnes numeros ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots n-1}$ expleant, adeoque fiat ${\textstyle \Sigma \cos ak\omega +\Sigma \cos bk\omega =\cos k\omega +\cos 2k\omega +\cos 3k\omega +{\text{etc.}}+\cos(n-1)k\omega =-1}$, summationes $${\begin{aligned}&\Sigma \cos ak\omega =-{\frac {1}{2}}\\&\Sigma \cos bk\omega =-{\frac {1}{2}}\end{aligned}}$$ sponte sunt obviae. Perinde erit $$\Sigma \sin ak\omega =\Sigma \sin(n-b)k\omega =-\Sigma \sin bk\omega$$ unde patet, quomodo summationum $${\begin{aligned}&2\Sigma \sin ak\omega =\pm \surd n\\&2\Sigma \sin bk\omega =\mp \surd n\end{aligned}}$$ altera ab altera pendeat.

3.

Ecce iam computum numericum pro aliquot exemplis:
I. Pro ${\textstyle n=5}$ adest valor unus ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle a^{\prime }=1}$, valorque unus ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle b^{\prime }=2}$; est autem $$\cos \omega =+0{,}3090169944\qquad \qquad \cos 2\omega =-0{,}8090169944$$ adeoque ${\textstyle 1+4\cos \omega =+\surd 5}$, ${\textstyle 1+4\cos 2\omega =-\surd 5}$.
II. Pro ${\textstyle n=13}$ adsunt tres valores ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle 1}$, ${\textstyle 3}$, ${\textstyle 4}$, totidemque valores ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle 2}$, ${\textstyle 5}$, ${\textstyle 6}$, unde computamus $${\begin{array}{r}\cos \omega {\phantom {1}}=+0{,}8854560257\\\cos 3\omega =+0{,}1205366803\\\cos 4\omega =-0{,}3546048870\\\hline {\text{Summa}}=+0{,}6513878190\end{array}}\qquad \qquad {\begin{array}{r}\cos 2\omega =+0{,}5680647467\\\cos 5\omega =-0{,}7485107482\\\cos 6\omega =-0{,}9709418174\\\hline {\text{Summa}}=-1{,}1513878189\end{array}}$$

Hinc ${\textstyle 1+4\sum \cos a^{\prime }\omega =+\surd 13}$, ${\textstyle 1+4\Sigma \cos b^{\prime }\omega =-\surd 13}$.
III. Pro ${\textstyle n=17}$ habemus quatuor valores ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 4}$, ${\textstyle 8}$, totidemque valores ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle 3}$, ${\textstyle 5}$, ${\textstyle 6}$, ${\textstyle 7}$. Hinc computantur cosinus $${\begin{array}{r}\cos \omega {\phantom {1}}=+0{,}9324722294\\\cos 2\omega =+0{,}7390089172\\\cos 4\omega =+0{,}0922683595\\\cos 8\omega =-0{,}9829730997\\\hline {\text{Summa}}=+0{,}7807764064\end{array}}\qquad \qquad {\begin{array}{r}\cos 3\omega =+0{,}4457383558\\\cos 5\omega =-0{,}2736629901\\\cos 6\omega =-0{,}6026346364\\\cos 7\omega =-0{,}8502171357\\\hline {\text{Summa}}=-1{,}2807764065\end{array}}$$

Hinc ${\textstyle 1+4\Sigma \cos a^{\prime }\omega =+\surd 17,1+4\Sigma \cos b^{\prime }\omega =-\surd 17}$.

IV. Pro ${\textstyle n=3}$ adest valor unicus ipsius ${\textstyle a}$, puta ${\textstyle a=1}$, cui respondet $$\sin \omega =+0{,}8660254038\quad \quad {\phantom {\sin \omega =+0{,}8660254038}}$$

Hinc ${\textstyle 2\sin \omega =+\surd 3}$.
V. Pro ${\textstyle n=7}$ adsunt valores tres ipsius ${\textstyle a}$, puta ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 4}$: hinc habentur sinus $${\begin{array}{rl}\sin \omega {\phantom {1}}=+0{,}7818314825{\phantom {,}}&\\\sin 2\omega =+0{,}9749279122{\phantom {,}}&\\\sin 4\omega =-0{,}4338837391{\phantom {,}}&\\{\begin{array}{l}\hline {\text{Summa}}=+1{,}3228756556,\end{array}}&{\text{adeoque }}2\Sigma \sin a\omega =+\surd 7.\end{array}}$$

VI. Pro ${\textstyle n=11}$ valores ipsius ${\textstyle a}$ sunt ${\textstyle 1}$, ${\textstyle 3}$, ${\textstyle 4}$, ${\textstyle 5}$, ${\textstyle 9}$, quibus respondent sinus $${\begin{array}{rl}\sin \omega {\phantom {1}}=+0{,}5406408175{\phantom {,}}&\\\sin 3\omega =+0{,}9898214419{\phantom {,}}&\\\sin 4\omega =+0{,}7557495744{\phantom {,}}&\\\sin 5\omega =+0{,}2817325568{\phantom {,}}&\\\sin 9\omega =-0{,}9096319954{\phantom {,}}&\\{\begin{array}{l}\hline {\text{Summa}}=+1{,}6583123952,\end{array}}&{\text{et proin }}2\Sigma \sin a\omega =+\surd 11.\end{array}}$$

VII. Pro ${\textstyle n=19}$ valores ipsius ${\textstyle a}$ sunt ${\textstyle 1}$, ${\textstyle 4}$, ${\textstyle 5}$, ${\textstyle 6}$, ${\textstyle 7}$, ${\textstyle 9}$, ${\textstyle 11}$, ${\textstyle 16}$, ${\textstyle 17}$, quibus respondent sinus $${\begin{array}{rl}\sin \omega {\phantom {11}}=+0{,}3246994692{\phantom {,}}&\\\sin 4\omega {\phantom {1}}=+0{,}9694002659{\phantom {,}}&\\\sin 5\omega {\phantom {1}}=+0{,}9965844930{\phantom {,}}&\\\sin 6\omega {\phantom {1}}=+0{,}9157733267{\phantom {,}}&\\\sin 7\omega {\phantom {1}}=+0{,}7357239107{\phantom {,}}&\\\sin 9\omega {\phantom {1}}=+0{,}1645945903{\phantom {,}}&\\\sin 11\omega =-0{,}4759473930{\phantom {,}}&\\\sin 16\omega =-0{,}8371664783{\phantom {,}}&\\\sin 17\omega =-0{,}6142127127{\phantom {,}}&\\{\begin{array}{l}\hline {\text{Summa}}=+2{,}1794494718,\end{array}}&{\text{adeoque }}2\Sigma \sin a\omega =+{\sqrt {19}}.\end{array}}$$

4.

In omnibus hisce exemplis quantitas radicalis signum positivum obtinet, idemque facile pro valoribus maioribus ${\textstyle n=23}$, ${\textstyle n=29}$ etc. confirmatur, unde fartis iam probabilitas oritur, hoc generaliter perinde se habere. Sed demonstratio huius phaenomeni e principiis l.c. expositis peti nequit, plenissimoque iure altioris indaginis aestimanda est. Propositum itaque huius commentationis eo tendit, ut demonstrationem rigorosam huius elegantissimi theorematis, per plures annos olim variis modis incassum tentatam, tandemque per considerationes singulares satisque subtiles feliciter perfectam in medium proferamus, simulque theorema ipsum salva seu potius aucta elegantia sua ad longe maiorem generalitatem evehamus. Coronidis denique loco nexum mirabilem arctissimum inter hanc summationem aliudque theorema arithmeticum gravissimum docebimus. Speramus, hasce disquisitiones non modo per se geometris gratas fore, sed methodos quoque, per quas haec omnia efficere licuit, quaeque in aliis quoque occasionibus utiles esse poterunt, ipsorum attentione dignas visum iri.

5.

Petita est demonstratio nostra e consideratione generis singularis progressionum, quarum termini pendent ab expressionibus talibus $${\frac {(1-x^{m})(1-x^{m-1})(1-x^{m-2})\ldots (1-x^{m-\mu +1})}{(1-x)(1-xx)(1-x^{3})\ldots (1-x^{\mu })}}$$ Brevitatis caussa talem fractionem per ${\textstyle (m,\mu )}$ denotabimus, et primo quasdam observationes generales circa huiusmodi functiones praemittemus.

I. Quoties ${\textstyle m}$ est integer positivus minor quam ${\textstyle \mu }$, functio ${\textstyle (m,\mu )}$ manifesto evanescit, numeratore factorem ${\textstyle 1-x^{0}}$ implicante. Pro ${\textstyle m=\mu }$, factores in numeratore identici erunt ordine inverso cum factoribus in denominatore, unde erit ${\textstyle (\mu ,\mu )=1}$: denique pro casu eo, ubi ${\textstyle m}$ est integer positivus maior quam ${\textstyle \mu }$, habentur formulae $${\begin{aligned}&(\mu +1,\mu )={\frac {1-x^{\mu +1}}{1-x}}=(\mu +1,1)\\&(\mu +2,\mu )={\frac {(1-x^{\mu +2})(1-x^{\mu +1})}{(1-x)(1-xx)}}=(\mu +2,2)\\&(\mu +3,\mu )={\frac {(1-x^{\mu +3})(1-x^{\mu +2})(1-x^{\mu +1})}{(1-x)(1-xx)(1-x^{3})}}=(\mu +3,3){\text{ etc.}}\end{aligned}}$$ sive generaliter $$(m,\mu )=(m,m-\mu )$$

II. Porro facile confirmatur, haberi generaliter $$(m,\mu +1)=(m-1,\mu +1)+x^{m-\mu -1}(m-1,\mu )$$ quamobrem, quum perinde sit $${\begin{aligned}&(m-1,\mu +1)=(m-2,\mu +1)+x^{m-\mu -2}(m-2,\mu )\\&(m-2,\mu +1)=(m-3,\mu +1)+x^{m-\mu -3}(m-3,\mu )\\&(m-3,\mu +1)=(m-4,\mu +1)+x^{m-\mu -4}(m-4,\mu ){\text{ etc.,}}\end{aligned}}$$ quae series continuari poterit usque ad $${\begin{aligned}(\mu +2,\mu +1)&=(\mu +1,\mu +1)+x(\mu +1,\mu )\\&=(\mu ,\mu )+x(\mu +1,\mu )\end{aligned}}$$ siquidem ${\textstyle m}$ est integer positivus maior quam ${\textstyle \mu +1}$, erit $$(m,\mu +1)=(\mu ,\mu )+x(\mu +1,\mu )+xx(\mu +2,\mu )+x^{3}(\mu +3,\mu )+{\text{etc.}}+x^{m-\mu -1}(m-1,\mu )$$

Hinc patet, si pro aliquo valore determinato ipsius ${\textstyle \mu }$ quaevis functio ${\textstyle (m,\mu )}$ integra sit, existente ${\textstyle m}$ integro positivo, etiam quamvis functionem ${\textstyle (m,\mu +1)}$ integram evadere debere. Quare quum suppositio illa pro ${\textstyle \mu =1}$ locum habeat, eadem etiam pro ${\textstyle \mu =2}$ valebit, atque hinc etiam pro ${\textstyle \mu =3}$ etc., i.e. generaliter pro valore quocunque integro positivo ipsius ${\textstyle m}$ erit ${\textstyle (m,\mu )}$ functio integra, sive productum $$(1-x^{m})(1-x^{m-1})(1-x^{m-2})\ldots (1-x^{m-\mu +1})$$ divisibile per $$(1-x)(1-x^{2})(1-x^{3})\ldots (1-x^{\mu })$$

6.

Duas iam progressiones considerabimus, quae ambae ad scopum nostrum ducere possunt. Progressio prima haec est $$1-{\frac {1-x^{m}}{1-x}}+{\frac {(1-x^{m})(1-x^{m-1})}{(1-x)(1-xx)}}-{\frac {(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}}+{\text{etc.}}$$ sive $$1-(m,1)+(m,2)-(m,3)+(m,4)-{\text{etc.}}$$ quam brevitatis caussa per ${\textstyle f(x,m)}$ denotabimus. Primo statim obvium est, quoties ${\textstyle m}$ sit numerus integer positivus, hanc seriem post terminum suum ${\textstyle {m+1}^{\text{tum }}}$ (qui fit ${\textstyle =\pm 1}$) abrumpi, adeoque in hoc casu summam fieri debere functionem finitam integram ipsius ${\textstyle x}$. Porro per art. 5. II. patet, generaliter pro valore quocunque ipsius ${\textstyle m}$ haberi $${\begin{aligned}1&=1\\-(m,1)&=-(m-1,1)-x^{m-1}\\+(m,2)&=+(m-1,2)+x^{m-2}(m-1,1)\\-(m,3)&=-(m-1,3)-x^{m-3}(m-1,2){\text{ etc.}}\end{aligned}}$$ adeoque $${\begin{aligned}f(x,m)=1-x^{m-1}&-(1-x^{m-2})(m-1,1)+(1-x^{m-3})(m-1,2)\\&-(1-x^{m-4})(m-1.3)+{\text{etc.}}\end{aligned}}$$ Sed manifesto fit $${\begin{aligned}&(1-x^{m-2})(m-1,1)=(1-x^{m-1})(m-2,1)\\&(1-x^{m-3})(m-1,2)=(1-x^{m-1})(m-2,2)\\&(1-x^{m-4})(m-1,3)=(1-x^{m-1})(m-2,3){\text{ etc.}}\end{aligned}}$$ unde deducimus aequationem $${\begin{aligned}f(x,m)=(1-x^{m-1})f(x,m-2)&\dots \dots \dots [1]\end{aligned}}$$

7.

Quum pro ${\textstyle m=0}$ fiat ${\textstyle f(x,m)=1}$, per formulam modo inventam erit $${\begin{aligned}&f(x,2)=1-x\\&f(x,4)=(1-x)(1-x^{3})\\&f(x,6)=(1-x)(1-x^{3})(1-x^{5})\\&f(x,8)=(1-x)(1-x^{3})(1-x^{5})(1-x^{7}){\text{ etc.}}\end{aligned}}$$ sive generaliter pro valore quocunque pari ipsius ${\textstyle m}$ $$f(x,m)=(1-x)(1-x^{3})(1-x^{5})\ldots (1-x^{m-1})\dots \dots \dots [2]$$

Contra quum pro ${\textstyle m=1}$ fiat ${\textstyle f(x,m)=0}$, erit etiam $${\begin{aligned}&f(x,3)=0\\&f(x,5)=0\\&f(x,7)=0{\text{ etc.}}\end{aligned}}$$ sive generaliter pro valore quocunque impari ipsius ${\textstyle m}$ $$f(x,m)=0$$

Ceterum summatio posterior iam inde derivari potuisset, quod in progressione $$1-(m,1)+(m,2)-(m,3)+{\text{etc.}}+(m,m-1)-(m,m)$$ terminus ultimus primum destruit, penultimus secundum etc.

8.

Ad scopum quidem nostrum sufficit casus is, ubi ${\textstyle m}$ est integer positivus impar: sed propter rei singularitatem etiam de casibus iis, ubi ${\textstyle m}$ vel fractus vel negativus est, pauca adiecisse haud poenitebit. Manifesto tunc series nostra haud amplius abrumpetur, sed in infinitum excurret, facileque insuper perspicitur, divergentem eam fieri, quoties ipsi ${\textstyle x}$ valor minor quam 1 tribuatur, quapropter ipsius summatio ad valores ipsius ${\textstyle x}$ qui sint maiores quam 1 restringi debebit.

Per formulam [1] art. 6. habemus $${\begin{aligned}&f(x,-2)={\frac {1}{1-{\frac {1}{x}}}}\\&f(x,-4)={\frac {1}{1-{\frac {1}{x}}}}\cdot {\frac {1}{1-{\frac {1}{x^{3}}}}}\\&f(x,-6)={\frac {1}{1-{\frac {1}{x}}}}\cdot {\frac {1}{1-{\frac {1}{x^{3}}}}}\cdot {\frac {1}{1-{\frac {1}{x^{5}}}}}{\text{ etc.}}\end{aligned}}$$ ita ut valor functionis ${\textstyle f(x,m)}$ etiam pro valore negativo integro pari ipsius ${\textstyle m}$ in terminis finitis assignabilis sit. Pro reliquis vero valoribus ipsius ${\textstyle m}$ functionem ${\textstyle f(x,m)}$ in productum infinitum sequenti modo convertemus.

Crescente ${\textstyle m}$ in valorem negativum infinitum, functio ${\textstyle f(x,m)}$ transit in $$1+{\frac {1}{x-1}}+{\frac {1}{x-1}}\cdot {\frac {1}{xx-1}}+{\frac {1}{x-1}}\cdot {\frac {1}{xx-1}}\cdot {\frac {1}{x^{3}-1}}+{\text{etc.}}$$ Haec itaque series aequalis est producto infinito $${\frac {1}{1-{\frac {1}{x}}}}\cdot {\frac {1}{1-{\frac {1}{x^{3}}}}}\cdot {\frac {1}{1-{\frac {1}{x^{5}}}}}\cdot {\frac {1}{1-{\frac {1}{x^{7}}}}}{\text{etc. in infin. }}$$ Porro quum generaliter sit $$f(x,m)=f(x,m-2\lambda ).(1-x^{m-1})(1-x^{m-3})(1-x^{m-5})\ldots (1-x^{m-2\lambda +1})$$ erit $${\begin{aligned}f(x,m)&=f(x,-\infty )\cdot (1-x^{m-1})(1-x^{m-3})(1-x^{m-5}){\text{ etc. in infin. }}\\&={\frac {1-x^{m-1}}{1-x^{-1}}}\cdot {\frac {1-x^{m-3}}{1-x^{-3}}}\cdot {\frac {1-x^{m-5}}{1-x^{-5}}}\cdot {\frac {1-x^{m-7}}{1-x^{-7}}}{\text{ etc. in infin. }}\end{aligned}}$$ quos factores tandem continuo magis ad unitatem convergere palam est.

Attentionem peculiarem meretur casus ${\textstyle m=-1}$, ubi fit $$f(x,-1)=1+x^{-1}+x^{-3}+x^{-6}+x^{-10}+{\text{etc.}}$$ Haec itaque series aequatur producto infinito $${\frac {1-x^{-2}}{1-x^{-1}}}\cdot {\frac {1-x^{-2}}{1-x^{-3}}}\cdot {\frac {1-x^{-6}}{1-x^{-5}}}{\text{ etc.}}$$ sive scribendo ${\textstyle x}$ pro ${\textstyle x^{-1}}$, erit $$1+x+x^{3}+x^{6}+{\text{etc.}}={\frac {1-xx}{1-x}}\cdot {\frac {1-x^{4}}{1-x^{3}}}\cdot {\frac {1-x^{6}}{1-x^{5}}}\cdot {\frac {1-x^{8}}{1-x^{7}}}{\text{ etc.}}$$ Haec aequalitas inter duas expressiones abstrusiores, ad quas alia occasione reveniemus, valde sane est memorabilis.

9.

Secundo loco considerabimus progressionem hancce $$1+x^{\frac {1}{2}}{\frac {1-x^{m}}{1-x}}+x{\frac {(1-x^{m})}{(1-x)}}{\frac {(1-x^{m-1})}{(1-xx)}}+x^{\frac {3}{2}}{\frac {(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}}+{\text{etc.}}$$ sive $$1+x^{\frac {1}{2}}(m,1)+x(m,2)+x^{\frac {3}{2}}(m,3)+xx(m,4)+{\text{etc.}}$$ quam per ${\textstyle F(x,m)}$ denotabimus. Restringemus hanc disquisitionem ad casum eum, ubi ${\textstyle m}$ est integer positivus, ita ut haec quoque series semper abrumpatur cum termino ${\textstyle m+1^{\text{to}}}$, qui est ${\textstyle =x^{{\frac {1}{2}}m}(m,m)}$. Quum sit $$(m,m)=1,\quad (m,m-1)=(m,1),\quad (m,m-2)=(m,2){\text{ etc.}}$$ progressio ita quoque exhiberi poterit: $$F(x,m)=x^{{\frac {1}{2}}m}+x^{{\frac {1}{2}}(m-1)}(m,1)+x^{{\frac {1}{2}}(m-2)}(m,2)+x^{{\frac {1}{2}}(m-3)}(m,3)+{\text{etc.}}$$ Hinc fit $${\begin{aligned}(1+x^{{\frac {1}{2}}m+{\frac {1}{2}}})F(x,m)=1&+x^{\frac {1}{2}}(m,1)+x(m,2)+x^{\frac {3}{2}}(m,3)+{\text{etc.}}\\&+x^{\frac {1}{2}}.x^{m}+x.x^{m-1}(m,1)+x^{\frac {3}{2}}.x^{m-2}(m,2)+{\text{etc.}}\end{aligned}}$$ Quare quum habeatur (art. 5. II) $${\begin{aligned}(m,1)+x^{m}&=(m+1,1)\\(m,2)+x^{m-1}(m,1)&=(m+1,2)\\(m,3)+x^{m-2}(m,2)&=(m+1,3){\text{ etc.,}}\end{aligned}}$$ provenit $$(1+x^{{\frac {1}{2}}m+{\frac {1}{2}}})F(x,m)=F(x,m+1)\dots \dots \dots [3]$$ Sed fit ${\textstyle F(x,0)=1}$: quamobrem erit $${\begin{aligned}&F(x,1)=1+x^{\frac {1}{2}}\\&F(x,2)=(1+x^{\frac {1}{2}})(1+x)\\&F(x,3)=(1+x^{\frac {1}{2}})(1+x)(1+x^{3}){\text{ etc.,}}\end{aligned}}$$ sive generaliter $$F(x,m)=(1+x^{\frac {1}{2}})(1+x)(1+x^{\frac {3}{2}})\ldots (1+x^{{\frac {1}{2}}m})\dots \dots \dots [4]$$

10.

Praemissis hisce disquisitionibus praeliminaribus iam propius ad propositum nostrum accedamus. Quum pro valore primo ipsius ${\textstyle n}$ quadrata ${\textstyle 1}$, ${\textstyle 4}$, ${\textstyle 9\ldots ({\frac {1}{2}}(n-1))^{2}}$ omnia inter se incongrua sint secundum modulum ${\textstyle n}$, patet, illorum residua minima secundum hunc modulum cum numeris ${\textstyle a}$ identica esse debere, adeoque $${\begin{aligned}&\Sigma \cos ak\omega =\cos k\omega +\cos 4k\omega +\cos 9k\omega +{\text{etc.}}+\cos({\frac {1}{2}}(n-1))^{2}k\omega \\&\Sigma \sin ak\omega =\sin k\omega +\sin 4k\omega +\sin 9k\omega +{\text{etc.}}+\sin({\frac {1}{2}}(n-1))^{2}k\omega \end{aligned}}$$ Perinde quum eadem quadrata ${\textstyle 1,4,9\ldots ({\frac {1}{2}}(n-1))^{2}}$ ordine inverso congrua sint his ${\textstyle ({\frac {1}{2}}(n+1))^{2}}$,${\textstyle ({\frac {1}{2}}(n+3))^{2}}$,${\textstyle ({\frac {1}{2}}(n+5))^{2}\ldots (n-1)^{2}}$, etiam erit $${\begin{aligned}&\Sigma \cos ak\omega =\cos({\frac {1}{2}}(n+1))^{2}k\omega +\cos({\frac {1}{2}}(n+3))^{2}k\omega +{\text{etc.}}+\cos(n-1)^{2}k\omega \\&\Sigma \sin ak\omega =\sin({\frac {1}{2}}(n+1))^{2}k\omega +\sin({\frac {1}{2}}(n+3))^{2}k\omega +{\text{etc.}}+\sin(n-1)^{2}k\omega \end{aligned}}$$ Statuendo itaque $${\begin{alignedat}{2}&T=&1+\cos k\omega +\cos 4k\omega +\cos 9k\omega +{\text{etc.}}+\cos(n-1)^{2}k\omega \\&U=&\sin k\omega +\sin 4k\omega +\sin 9k\omega +{\text{etc.}}+\sin(n-1)^{2}k\omega \end{alignedat}}$$ erit $${\begin{aligned}1+2\Sigma \cos ak\omega &=T\\2\Sigma \sin ak\omega &=U\end{aligned}}$$ Hinc patet, summationes, quales in art. 1. propositae sunt, pendere a summatione serierum ${\textstyle T}$ et ${\textstyle U}$, quocirca, missis illis, disquisitionem nostram his adaptabimus, eaque generalitate absolvemus, ut non modo valores primos ipsius ${\textstyle n}$, sed quoscunque compositos complectatur. Numerum ${\textstyle k}$ autem supponemus ad ${\textstyle n}$ primum esse: nullo enim negotio casus is, ubi ${\textstyle k}$ et ${\textstyle n}$ divisorem communem haberent, ad hunc reduci poterit.

11.

Designemus quantitatem imaginariam ${\textstyle \surd {-1}}$ per ${\textstyle i}$, statuamusque $$\cos k\omega +i\sin k\omega =r$$ unde erit ${\textstyle r^{n}=1}$, sive ${\textstyle r}$ radix aequationis ${\textstyle r^{n}-1=0}$. Facile perspicietur, omnes numeros ${\textstyle k}$, ${\textstyle 2k}$, ${\textstyle 3k\ldots (n-1)k}$ per ${\textstyle n}$ non divisibiles atque inter se secundum modulum ${\textstyle n}$ incongruos esse: hinc potestates ipsius ${\textstyle r}$ $$1,r,rr,r^{3}\ldots r^{n-1}$$ omnes erunt inaequales, singulae vero quoque aequationi ${\textstyle x^{n}-1=0}$ satisfacient. Hanc ob caussam hae potestates omnes radices aequationis ${\textstyle x^{n}-1=0}$ repraesentabunt.

Hae conclusiones non valerent, si ${\textstyle k}$ divisorem communem haberet cum ${\textstyle n}$. Si enim ${\textstyle \nu }$ esset talis divisor communis, foret ${\textstyle k.{\frac {n}{\nu }}}$ per ${\textstyle n}$ divisibilis, adeoque potestas inferior quam ${\textstyle r^{n}}$, puta ${\textstyle r^{\frac {n}{\nu }}}$, unitati aequalis. In hoc itaque casu potestates ipsius ${\textstyle r}$ ad summum ${\textstyle {\frac {n}{\nu }}}$ radices aequationis ${\textstyle x^{n}-1=0}$ exhibebunt, et quidem revera tot radices diversas sistent, si ${\textstyle \nu }$ est divisor communis maximus numerorum ${\textstyle k}$, ${\textstyle n}$. In casu nostro, ubi ${\textstyle k}$ et ${\textstyle n}$ supponuntur inter se primi, ${\textstyle r}$ commode dici potest radix propria aequationis ${\textstyle x^{n}-1=0}$: contra in casu altero, ubi ${\textstyle k}$ et ${\textstyle n}$ haberent divisorem communem (maximum) ${\textstyle \nu }$, ${\textstyle r}$ vocaretur radix impropria illius aequationis, manifesto autem tunc eadem ${\textstyle r}$ foret radix propria aequationis ${\textstyle x^{\frac {n}{\nu }}-1=0}$. Radix impropria simplicissima est unitas, in eoque casu, ubi ${\textstyle n}$ est numerus primus, impropriae aliae omnino non dabuntur.

12.

Quodsi iam statuimus $$W=1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$ patet fieri ${\textstyle W=T+iU}$, adeoque ${\textstyle T}$ esse partem realem ipsius ${\textstyle W}$, atque ${\textstyle U}$ prodire ex parte imaginaria ipsius ${\textstyle W}$ factore ${\textstyle i}$ suppresso. Totum itaque negotium reducitur ad inventionem summae ${\textstyle W}$: ad hunc finem vel series in art. 6 considerata, vel ea quam in art. 9 summare docuimus, adhiberi potest, prior tamen minus idonea est in casu eo, ubi ${\textstyle n}$ est numerus par. Nihilominus lectoribus gratum fore speramus, si casum eum, ubi ${\textstyle n}$ impar est, secundum methodum duplicem tractemus.

Supponamus itaque primo, ${\textstyle n}$ esse numerum imparem, ${\textstyle r}$ designare radicem propriam aequationis ${\textstyle x^{n}-1=0}$ quamcunque, et in functione ${\textstyle f(x,m)}$ statui ${\textstyle x=r}$, atque ${\textstyle m=n-1}$. Hinc patet fieri $${\begin{alignedat}{3}&{\frac {1-x^{m}}{1-x}}&={\frac {1-r^{-1}}{1-r}}&=-r^{-1}\\&{\frac {1-x^{m-1}}{1-xx}}&={\frac {1-r^{-2}}{1-rr}}&=-r^{-2}\\&{\frac {1-x^{m-2}}{1-x^{3}}}&={\frac {1-r^{-3}}{1-r^{3}}}&=-r^{-3}{\text{ etc.}}\end{alignedat}}$$ usque ad $${\begin{alignedat}{3}&{\frac {1-x}{1-x^{m}}}&={\frac {1-r^{-m}}{1-r^{m}}}&=-r^{-m}{\phantom {\text{etc.}}}\end{alignedat}}$$ (Haud superfluum erit monere, has aequationes eatenus tantum valere, quatenus ${\textstyle r}$ supponitur radix propria: si enim esset ${\textstyle r}$ radix impropria, in quibusdam illarum fractionum numerator et denominator simul evanescerent, adeoque fractiones indeterminatae fierent).

Hinc deducimus aequationem sequentem $${\begin{aligned}f(r,n-1)&=1+r^{-1}+r^{-3}+r^{-6}+{\text{etc.}}+r^{-{\frac {1}{2}}(n-1)n}\\&=(1-r)(1-r^{3})(1-r^{5})\ldots (1-r^{n-2})\end{aligned}}$$

Eadem aequatio etiamnum valebit, si pro ${\textstyle r}$ substituitur ${\textstyle r^{\lambda }}$, designante ${\textstyle \lambda }$ integrum quemcunque ad ${\textstyle n}$ primum: tunc enim etiam ${\textstyle r^{\lambda }}$ erit radix propria aequationis ${\textstyle x^{n}-1=0}$. Scribamus itaque pro ${\textstyle r}$, ${\textstyle r^{n-2}}$ sive quod idem est ${\textstyle r^{-2}}$, eritque $$1+r^{2}+r^{6}+r^{12}+{\text{etc.}}+r^{(n-1)n}=(1-r^{-2})(1-r^{-6})(1-r^{-10})\ldots (1-r^{-2(n-2)})$$ Multiplicemus utramque partem huius aequationis per $$r\cdot r^{3}\cdot r^{5}\ldots \cdot r^{(n-2)}=r^{{\frac {1}{4}}(n-1)^{2}}$$ prodibitque, propter $${\begin{alignedat}{4}r^{2+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-3)^{2}},\quad &r^{(n-1)n+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+1)^{2}}\\r^{6+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-5)^{2}},\quad &r^{(n-2)(n-1)+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+3)^{2}}\\r^{12+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-7)^{2}},\quad &r^{(n-3)(n-2)+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+5)^{2}}{\text{ etc.}}\end{alignedat}}$$ aequatio sequens $${\begin{aligned}r^{{\frac {1}{4}}(n-1)^{2}}&+r^{{\frac {1}{4}}(n-3)^{2}}+r^{{\frac {1}{4}}(n-5)^{2}}+{\text{etc.}}+r+1\\+r^{{\frac {1}{4}}(n+1)^{2}}&+r^{{\frac {1}{4}}(n+3)^{2}}+r^{{\frac {1}{4}}(n+5)^{2}}+{\text{etc.}}+r^{{\frac {1}{4}}(2n-2)^{2}}\\&=(r-r^{-1})(r^{3}-r^{-3})(r^{5}-r^{-5})\ldots (r^{n-2}-r^{-n+2})\end{aligned}}$$ aut, partibus membri primi aliter dispositis, $$1+r+r^{4}+{\text{etc.}}+r^{(n-1)^{2}}=(r-r^{-1})(r^{3}-r^{-3})\ldots (r^{n-2}-r^{-n+2})\dots \dots \dots [5]$$

13.

Factores membri secundi aequationis [5] ita quoque exhiberi possunt $${\begin{aligned}r-r^{-1}&=-(r^{n-1}-r^{-n+1})\\r^{3}-r^{-3}&=-(r^{n-3}-r^{-n+3})\\r^{5}-r^{-5}&=-(r^{n-5}-r^{-n+5}){\text{ etc.}}\end{aligned}}$$ usque ad $$r^{n-2}-r^{-n+2}=-(r^{2}-r^{-2}){\phantom {\text{ etc.}}}$$ quo pacto aequatio ista hanc formam assumit: $$W=(-1)^{{\frac {1}{2}}(n-1)}(r^{2}-r^{-2})(r^{4}-r^{-4})(r^{6}-r^{-6})\ldots (r^{n-1}-r^{-n+1})$$ Multiplicando hanc aequationem per [5] in forma primitiva, prodit $$W^{2}=(-1)^{{\frac {1}{2}}(n-1)}(r-r^{-1})(r^{2}-r^{-2})(r^{3}-r^{-3})\ldots (r^{n-1}-r^{-n+1})$$ ubi ${\textstyle (-1)^{{\frac {1}{2}}(n-1)}}$ est vel ${\textstyle =+1}$ vel ${\textstyle =-1}$, prout ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, vel formae ${\textstyle 4\mu +3}$. Hinc $$W^{2}=\pm r^{{\frac {1}{2}}n(n-1)}(1-r^{-2})(1-r^{-4})(1-r^{-6})\ldots (1-r^{-2(n-1)})$$ Sed nullo negotio perspicitur, ${\textstyle r^{-2}}$, ${\textstyle r^{-4}}$, ${\textstyle r^{-6}\ldots r^{-2n+2}}$ exhibere omnes radices aequationis ${\textstyle x^{n}-1=0}$, radice ${\textstyle x=1}$ excepta, unde locum habere debebit aequatio identica indefinita $$(x-r^{-2})(x-r^{-4})(x-r^{-6})\ldots (x-r^{-2n+2})=x^{n-1}+x^{n-2}+x^{n-3}+{\text{etc.}}+x+1$$ Quamobrem statuendo ${\textstyle x=1}$, fiet $$(1-r^{-2})(1-r^{-4})(1-r^{-6})\ldots (1-r^{-2n+2})=n$$ et quum manifesto sit ${\textstyle r^{{\frac {1}{2}}n(n-1)}=1}$, aequatio nostra transit in hanc $$W^{2}=\pm n\dots \dots \dots [6]$$ In casu itaque eo, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, fiet $$W=\pm \surd n,{\text{ et proin }}T=\pm \surd n,\quad U=0$$ Contra in casu altero, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, fiet $$W=\pm i\surd n,{\text{ adeoque }}T=0,\quad U=\pm \surd n$$

14.

Methodus art. praec. valorem tantummodo absolutum aggregatorum ${\textstyle T}$, ${\textstyle U}$ assignat, ambiguumque linquit, utrum statuere oporteat ${\textstyle T}$ in casu priori atque ${\textstyle U}$ in casu posteriori ${\textstyle =+\surd n}$, an ${\textstyle =-\surd n}$. Hoc autem, saltem pro casu eo ubi ${\textstyle k=1}$, ex aequatione [5] sequenti modo decidere licebit. Quum sit, pro ${\textstyle k=1}$, $${\begin{aligned}r-r^{-1}&=2i\sin \omega \\r^{3}-r^{-3}&=2i\sin 3\omega \\r^{5}-r^{-5}&=2i\sin 5\omega {\text{ etc., }}\end{aligned}}$$ aequatio ista transmutatur in $$W=(2i)^{{\frac {1}{2}}(n-1)}\sin \omega \sin 3\omega \sin 5\omega \ldots \sin(n-2)\omega$$ Iam in casu eo, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, in serie numerorum imparium $$1,3,5,7\ldots {\frac {1}{2}}(n-3),{\frac {1}{2}}(n+1)\ldots (n-2)$$ reperiuntur ${\textstyle {\frac {1}{4}}(n-1)}$, qui sunt minores quam ${\textstyle {\frac {1}{2}}n}$, hisque manifesto respondent sinus positivi; contra reliqui ${\textstyle {\frac {1}{4}}(n-1)}$ erunt maiores quam ${\textstyle {\frac {1}{2}}n}$, hisque sinus negativi respondebunt: quapropter productum omnium sinuum statuendum est aequale producto e quantitate positiva in multiplicatorem ${\textstyle (-1)^{{\frac {1}{4}}(n-1)}}$, adeoque ${\textstyle W}$ aequalis erit producto e quantitate reali positiva in ${\textstyle i^{n-1}}$ sive in 1, quoniam ${\textstyle i^{4}=1}$, atque ${\textstyle n-1}$ per 4 divisibilis: i.e. quantitas ${\textstyle W}$ erit realis positiva, unde necessario esse debebit $$W=+\surd n,\quad T=+\surd n$$

In casu altero, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$ in serie numerorum imparium $$1,3,5,7\ldots {\frac {1}{2}}(n-1),{\frac {1}{2}}(n+3)\ldots (n-2)$$ priores ${\textstyle {\frac {1}{4}}(n+1)}$ erunt minores quam ${\textstyle {\frac {1}{2}}n}$, reliqui ${\textstyle {\frac {1}{4}}(n-3)}$ autem maiores. Hinc inter sinus arcuum ${\textstyle \omega ,3\omega ,5\omega \ldots (n-2)\omega }$ negativi erunt ${\textstyle {\frac {1}{4}}(n-3)}$, adeoque ${\textstyle W}$ erit productum ex ${\textstyle i^{{\frac {1}{2}}(n-1)}}$ in quantitatem realem positivam in ${\textstyle (-1)^{{\frac {1}{4}}(n-3)}}$; factor tertius est ${\textstyle =i^{{\frac {1}{2}}(n-3)}}$, qui cum primo iunctus producit ${\textstyle i^{n-2}=i}$, quoniam ${\textstyle i^{n-3}=1}$. Quamobrem necessario erit $$W=+i\surd n,{\text{ atque }}U=+\surd n$$

15.

Iam ostendemus, quo pacto eaedem conclusiones e progressione in art. 9 considerata deduci possint. Scribamus in aequ. [4] pro ${\textstyle x^{\frac {1}{2}}}$, ${\textstyle -y^{-1}}$, eritque $$1-y^{-1}{\frac {1-y^{-2m}}{1-y^{-2}}}+y^{-2}{\frac {(1-y^{-2m})(1-y^{-2m+2})}{(1-y^{-2})(1-y^{-4})}}-y^{-3}{\frac {(1-y^{-2m})(1-y^{-2m+2})(1-y^{-2m+2})}{(1-y^{-2})(1-y^{-4})(1-y^{-6})}}+{\text{etc.}}$$ usque ad terminum ${\textstyle m+1^{\text{tum }}}$ $$=(1-y^{-1})(1+y^{-2})(1-y^{-3})(1+y^{-4})\ldots (1\pm y^{-m})\dots \dots \dots [7]$$ Quodsi hic pro ${\textstyle y}$ accipitur radix propria aequationis ${\textstyle y^{n}-1=0}$, puta ${\textstyle r}$, atque simul statuitur ${\textstyle m=n-1}$, erit $${\begin{aligned}&{\frac {1-y^{-2m}}{1-y^{-2}}}={\frac {1-r^{2}}{1-r^{-2}}}=-r^{2}\\&{\frac {1-y^{-2m+2}}{1-y^{-4}}}={\frac {1-r^{4}}{1-r^{-4}}}=-r^{4}\\&{\frac {1-y^{-2m+4}}{1-y^{-6}}}={\frac {1-r^{6}}{1-r^{-6}}}=-r^{6}{\text{ etc.}}\end{aligned}}$$ usque ad $${\frac {1-y^{-2}}{1-y^{-2m}}}={\frac {1-r^{2n-2}}{1-r^{-2n+2}}}=-r^{2n-2}$$ ubi notandum, nullum denominatorum ${\textstyle 1-r^{-2}}$, ${\textstyle 1-r^{-4}}$ etc. fieri ${\textstyle =0}$. Hinc aequatio [7] hancce formam assumit $$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}=(1-r^{-1})(1+r^{-2})(1-r^{-3})\ldots (1+r^{-n+1})$$ Multiplicando in membro secundo huius aequationis terminum primum per ultimum, secundum per penultimum etc., habemus $${\begin{aligned}&(1-r^{-1})(1+r^{-n+1})=r-r^{-1}\\&(1+r^{-2})(1-r^{-n+2})=r^{n-2}-r^{-n+2}\\&(1-r^{-3})(1+r^{-n+3})=r^{3}-r^{-3}\\&(1+r^{-4})(1-r^{-n+4})=r^{n-4}-r^{-n+4}{\text{ etc.}}\end{aligned}}$$ Ex his productis partialibus facile perspicietur conflari productum $$(r-r^{-1})(r^{3}-r^{-3})(r^{5}-r^{-5})\ldots (r^{n-4}-r^{-n+4})(r^{n-2}-r^{-n+2})$$ quod itaque erit $$=1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}=W$$ Haec aequatio identica est cum aequ. [5] in art. 12 e progressione prima derivata, ratiociniaque dein reliqua eodem modo adstruentur, ut in artt. 13 et 14.

16.

Transimus ad casum alterum, ubi ${\textstyle n}$ est numerus par. Sit primo ${\textstyle n}$ formae ${\textstyle 4\mu +2}$ sive impariter par, patetque, numeros ${\textstyle {\frac {1}{4}}nn}$, ${\textstyle ({\frac {1}{2}}n+1)^{2}-1}$, ${\textstyle ({\frac {1}{2}}n+2)^{2}-4}$ etc. sive generaliter ${\textstyle ({\frac {1}{2}}n+\lambda )^{2}-\lambda \lambda }$ per ${\textstyle {\frac {1}{2}}n}$ divisos producere quotientes impares, adeoque secundum modulum ${\textstyle n}$ congruos fieri ipsi ${\textstyle {\frac {1}{2}}n}$. Hinc colligitur, si ${\textstyle r}$ sit radix propria aequationis ${\textstyle x^{n}-1=0}$, adeoque ${\textstyle r^{{\frac {1}{2}}n}=-1}$, fieri $${\begin{aligned}r^{({\frac {1}{2}}n)^{2}}&=-1\\r^{({\frac {1}{2}}n+1)^{2}}&=-r\\r^{({\frac {1}{2}}n+2)^{2}}&=-r^{4}\\r^{({\frac {1}{2}}n+3)^{2}}&=-r^{9}{\text{ etc.}}\end{aligned}}$$ Hinc in progressione $$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$ terminus ${\textstyle r^{({\frac {1}{2}}n)^{2}}}$ destruet primum, sequens secundum etc., adeoque erit $$W=0,\quad T=0,\quad U=0$$

17.

Superest casus, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu }$ sive pariter par. Hic generaliter ${\textstyle ({\frac {1}{2}}n+\lambda )^{2}-\lambda \lambda }$ divisibilis erit per ${\textstyle n}$, adeoque $$r^{({\frac {1}{2}}n+\lambda )^{2}}=r^{\lambda \lambda }$$ Hinc in serie $$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$ terminus ${\textstyle r^{({\frac {1}{2}}n)^{2}}}$ aequalis erit primo, sequens secundo etc., ita ut fiat $$W=2(1+r+r^{4}+r^{9}+{\text{etc.}}+r^{({\frac {1}{2}}n-1)^{2}})$$

Iam supponamus, in aequ. [7] art. 15 statui ${\textstyle m={\frac {1}{2}}n-1}$, et pro ${\textstyle y}$ accipi radicem propriam aequationis ${\textstyle y^{n}-1=0}$, puta ${\textstyle r}$. Tunc perinde ut in art. 15 aequatio sequentem formam obtinet: $$1+r+r^{4}+{\text{etc.}}+r^{({\frac {1}{2}}n-1)^{2}}=(1-r^{-1})(1+r^{-2})(1-r^{-3})\ldots (1-r^{-{\frac {1}{2}}n+1})$$ sive $$W=2(1-r^{-1})(1+r^{-2})(1-r^{-3})(1+r^{-4})\ldots (1-r^{-{\frac {1}{2}}n+1})\dots \dots \dots [8]$$

Porro quum sit ${\textstyle r^{{\frac {1}{2}}n}=-1}$, adeoque $${\begin{aligned}&1+r^{-2}=-r^{{\frac {1}{2}}n-2}(1-r^{-{\frac {1}{2}}n+2})\\&1+r^{-4}=-r^{{\frac {1}{2}}n-4}(1-r^{-{\frac {1}{2}}n+4})\\&1+r^{-6}=-r^{{\frac {1}{2}}n-6}(1-r^{-{\frac {1}{2}}n+6}){\text{ etc.}}\end{aligned}}$$ productumque e factoribus ${\textstyle -r^{{\frac {1}{2}}n-2}}$,${\textstyle -r^{{\frac {1}{2}}n-4}}$,${\textstyle -r^{{\frac {1}{2}}n-6}}$ etc. usque ad ${\textstyle -r^{2}}$ fiat ${\textstyle =(-1)^{{\frac {1}{4}}n-1}r^{{\frac {1}{16}}nn-{\frac {1}{4}}n}}$, aequatio praecedens ita quoque exhiberi potest $$W=2(-1)^{{\frac {1}{4}}n-1}r^{{\frac {1}{16}}nn-{\frac {1}{4}}n}(1-r^{-1})(1-r^{-2})(1-r^{-3})(1-r^{-4})\ldots (1-r^{-{\frac {1}{2}}n+1})$$

Quum habeatur $${\begin{aligned}&1-r^{-1}=-r^{-1}(1-r^{-n+1})\\&1-r^{-2}=-r^{-2}(1-r^{-n+2})\\&1-r^{-3}=-r^{-3}(1-r^{-n+3}){\text{ etc.}}\end{aligned}}$$ erit $${\begin{aligned}(1-r^{-1})&(1-r^{-2})(1-r^{-3})\ldots (1-r^{-{\frac {1}{2}}n+1})\\=&(-1)^{{\frac {1}{2}}n-1}r^{-{\frac {1}{8}}nn+{\frac {1}{4}}n}(1-r^{-{\frac {1}{2}}n-1})(1-r^{-{\frac {1}{2}}n-2})(1-r^{-{\frac {1}{2}}n-3})\ldots (1-r^{-n+1})\end{aligned}}$$ adeoque $$W=2(-1)^{{\frac {3}{4}}n-2}r^{-{\frac {1}{16}}nn}(1-r^{-{\frac {1}{2}}n-1})(1-r^{-{\frac {1}{2}}n-2})(1-r^{-{\frac {1}{2}}n-3})\ldots (1-r^{-n+1})$$ Multiplicando hunc valorem ipsius ${\textstyle W}$ per prius inventum, adiungendoque utrimque factorem ${\textstyle 1-r^{-{\frac {1}{2}}n}}$, prodit $$(1-r^{-{\frac {1}{2}}n})W^{2}=4(-1)^{n-3}r^{-{\frac {1}{4}}n}(1-r^{-1})(1-r^{-2})(1-r^{-3})\ldots (1-r^{-n+1})$$ Sed fit $${\begin{alignedat}{2}1-r^{-{\frac {1}{2}}n}&=2\\(-1)^{n-3}&=-1\\r^{-{\frac {1}{4}}n}&=-r^{{\frac {1}{4}}n}\\(1-r^{-1})(1-r^{-2})(1-r^{-3})&\ldots (1-r^{-n+1})=n\end{alignedat}}$$ Unde tandem concluditur $$W^{2}=2r^{{\frac {1}{4}}n}n\dots \dots \dots [9]$$ Iam facile perspicietur, ${\textstyle r^{{\frac {1}{4}}n}}$ esse vel ${\textstyle =+i}$ vel ${\textstyle =-i}$, prout scilicet ${\textstyle k}$ vel formae ${\textstyle 4\mu +1}$ sit, vel formae ${\textstyle 4\mu +3}$. Et quum sit $$2i=(1+i)^{2},\quad -2i=(1-i)^{2}$$ erit in casu eo, ubi ${\textstyle k}$ est formae ${\textstyle 4\mu +1}$, $$W=\pm (1+i)\surd n,{\text{ adeoque }}\quad T=U=\pm \surd n$$ in casu altero autem, ubi ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$, $$W=\pm (1-i)\surd n,{\text{ adeoque }}T=-U=\pm \surd n$$

18.

Methodus art. praec. valores absolutos functionum ${\textstyle T}$, ${\textstyle U}$ suppeditavit, conditionesque assignavit, sub quibus signa aequalia vel opposita illis tribuenda sint: sed signa ipsa hinc nondum determinantur. Hoc pro eo casu, ubi statuitur ${\textstyle k=1}$, sequenti modo supplebimus.

Statuamus ${\textstyle \rho =\cos {\frac {1}{2}}\omega +i\sin {\frac {1}{2}}\omega }$, ita ut fiat ${\textstyle r=\rho \rho }$, patetque, propter ${\textstyle \rho ^{n}=-1}$ aequationem [8] ita exhiberi posse $$W=2(1+\rho ^{n-2})(1+\rho ^{-4})(1+\rho ^{n-6})(1+\rho ^{-8})\ldots (1+\rho ^{-n+4})(1+\rho ^{2})$$ sive factoribus alio ordine dispositis $$W=2(1+\rho ^{2})(1+\rho ^{-4})(1+\rho ^{6})(1+\rho ^{-8})\ldots (1+\rho ^{-n+4})(1+\rho ^{n-2})$$ Iam fit $${\begin{aligned}&1+\rho ^{2}=2\rho \cos {\frac {1}{2}}\omega \\&1+\rho ^{-4}=2\rho ^{-2}\cos \omega \\&1+\rho ^{+6}=2\rho ^{3}\cos {\frac {3}{2}}\omega \\&1+\rho ^{-8}=2\rho ^{-4}\cos 2\omega {\text{ etc.}}\end{aligned}}$$ usque ad $${\begin{aligned}&1+\rho ^{-n+4}=2\rho ^{-{\frac {1}{2}}n+2}\cos({\frac {1}{4}}n-1)\omega \\&1+\rho ^{n-2}=2\rho ^{{\frac {1}{2}}n-1}\cos({\frac {1}{4}}n-{\frac {1}{2}})\omega \end{aligned}}$$ Quamobrem habetur $$W=2^{{\frac {1}{2}}n}\rho ^{{\frac {1}{4}}n}\cos {\frac {1}{2}}\omega \cos \omega \cos {\frac {3}{2}}\omega \ldots \cos({\frac {1}{4}}n-{\frac {1}{2}})\omega$$ Cosinus in hoc productum ingredientes manifesto omnes positivi sunt, factor ${\textstyle \rho ^{{\frac {1}{4}}n}}$ autem fit ${\textstyle =\cos 45^{\circ }+i\sin 45^{\circ }=(1+i)\surd {\frac {1}{2}}}$. Hinc colligimus, ${\textstyle W}$ esse productum ex ${\textstyle 1+i}$ in quantitatem realem positivam, unde necessario esse debebit $$W=(1+i)\cdot \surd n,\quad T=+\surd n,\quad U=+\surd n$$

19.

Operae pretium erit, omnes summationes hactenus evolutas, hic in unum conspectum colligere. Generaliter scilicet est

${\textstyle T=}$	${\textstyle U=}$	prout ${\textstyle n}$ est formae
${\textstyle \pm \surd n}$	${\textstyle \pm \surd n}$	${\textstyle 4\mu }$
${\textstyle \pm \surd n}$	${\textstyle 0}$	${\textstyle 4\mu +1}$
${\textstyle 0}$	${\textstyle 0}$	${\textstyle 4\mu +2}$
${\textstyle 0}$	${\textstyle \pm \surd n}$	${\textstyle 4\mu +3}$

et in casu eo, ubi ${\textstyle k}$ supponitur ${\textstyle =1}$, quantitati radicali signum positivum tribui debet. Omni itaque iam rigore ea, quae pro valoribus primis ipsius ${\textstyle n}$ in art. 3 per inductionem animadverteramus, demonstrata sunt, nihilque superest, nisi ut signa pro valoribus quibuscunque ipsius ${\textstyle k}$ in omnibus casibus determinare doceamus. Sed antequam hoc negotium in omni generalitate aggredi liceat, primo casus eos, ubi ${\textstyle n}$ est numerus primus vel numeri primi potestas, propius considerare oportebit.

20.

Sit primo ${\textstyle n}$ numerus primus impar, patetque per ea, quae in art. 10 exposuimus, esse ${\textstyle W=1+2\Sigma r^{a}=1+2\Sigma R^{ak}}$, si statuatur ${\textstyle R=\cos \omega +i\sin \omega }$, denotante ${\textstyle a}$ ut illic indefinite omnia residua quadratica ipsius ${\textstyle n}$ inter 1 et ${\textstyle n-1}$ contenta. Quodsi quoque per ${\textstyle b}$ indefinite omnia non-residua quadratica inter eosdem limites exprimimus, nullo negotio perspicitur, omnes numeros ${\textstyle ak}$ congruos fieri secundum modulum ${\textstyle n}$ vel omnibus ${\textstyle a}$ vel omnibus ${\textstyle b}$ (nullo ordinis respectu habito), prout ${\textstyle k}$ vel residuum sit vel non-residuum. Quamobrem in casu priori erit $$W=1+2\sum R^{a}=1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}}$$ adeoque ${\textstyle W=+\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=+i\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$.

Contra in casu altero, ubi ${\textstyle k}$ est non-residuum ipsius ${\textstyle n}$, erit $$W=1+2\Sigma R^{b}$$ Hinc quum manifesto omnes ${\textstyle a}$, ${\textstyle b}$ complexum integrum numerorum ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots }$ expleant, adeoque sit $$\Sigma R^{a}+\Sigma R^{b}=R+R^{2}+R^{3}+{\text{etc.}}+R^{n-1}=-1$$ fiet $$W=-1-2\Sigma R^{a}=-(1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}})$$ adeoque ${\textstyle W=-\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=-i\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$.

Hinc itaque colligitur
primo, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle k}$ residuum quadraticum ipsius ${\textstyle n}$, $$T=+\surd n,\quad U=0$$ secundo, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle k}$ non-residuum ipsius ${\textstyle n}$, $$T=-\surd n,\quad U=0$$ tertio, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle k}$ residuum ipsius ${\textstyle n}$, $$T=0,\quad U=+\surd n$$ quarto, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle k}$ non-residuum ipsius ${\textstyle n}$, $$T=0,\quad U=-\surd n$$

21.

Sit secundo ${\textstyle n}$ quadratum altiorve potestas numeri primi imparis ${\textstyle p}$, statuaturque ${\textstyle n=p^{2\chi }q}$, ita ut sit ${\textstyle q}$ vel ${\textstyle =1}$ vel ${\textstyle =p}$. Hic ante omnia observare convenit, si ${\textstyle \lambda }$ sit integer quicunque per ${\textstyle p^{\chi }}$ non divisibilis, fieri $${\begin{aligned}&r^{\lambda \lambda }+r^{(\lambda +p^{\chi }q)^{2}}+r^{(\lambda +2p^{\chi }q)^{2}}+r^{(\lambda +3p^{\chi }q)^{2}}+{\text{etc.}}+r^{(\lambda +n-p^{\chi }q)^{2}}\\&\quad =r^{\lambda \lambda }\left\{1+r^{2\lambda p^{\chi }q}+r^{4\lambda p^{\chi }q}+r^{6\lambda p^{\chi }q}+{\text{etc.}}+r^{2\lambda (n-p^{\chi }q)}\right\}={\frac {r^{\lambda \lambda }(1-r^{2\lambda n})}{1-r^{2\lambda p^{\chi }q}}}=0\end{aligned}}$$ Hinc facile perspicietur, fieri $$W=1+r^{p^{2\chi }}+r^{4p^{2\chi }}+r^{9p^{2\chi }}+{\text{etc.}}+r^{(n-p^{\chi })^{2}}$$ Termini enim reliqui progressionis $$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$ distribui poterunt in ${\textstyle (p^{\chi }-1)q}$ progressiones partiales, quae singulae sint ${\textstyle p^{\chi }}$ terminorum, et per transformationem modo traditam summas evanescentes conficiant.

Hinc colligitur, in casu eo, ubi fit ${\textstyle q=1}$, sive ubi ${\textstyle n}$ est potestas numeri primi cum exponente pari, fieri $$W=p^{\chi }=+\surd n,{\text{ adeoque }}T=+\surd n,U=0$$ Contra in casu eo, ubi ${\textstyle q=p}$, sive ubi ${\textstyle n}$ est potestas numeri primi cum exponente impari, statuemus ${\textstyle r^{p^{2\chi }}=\rho }$, unde ${\textstyle \rho }$ erit radix propria aequationis ${\textstyle x^{p}-1=0}$, et quidem ${\textstyle \rho =\cos {\frac {k}{p}}360^{\circ }+i\sin {\frac {k}{p}}360^{\circ }}$, ac dein $$W=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(p^{\chi +1}-1)^{2}}=p^{\chi }(1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(p-1)^{2}})$$

Sed summa șeriei ${\textstyle 1+p+p^{4}+p^{9}+{\text{etc.}}+p^{(p-1)^{2}}}$ per art. praec. determinatur, unde sponte concluditur, fieri $${\begin{aligned}&W=\pm {\sqrt {n}}=T,{\text{ si fuerit }}p{\text{ formae }}4\mu +1\\&W=\pm i{\sqrt {n}}=iU,{\text{ si fuerit }}p{\text{ formae }}4\mu +3\end{aligned}}$$ signo positivo vel negativo valente, prout ${\textstyle k}$ fuerit residuum vel non-residuum ipsius ${\textstyle p}$.

22.

Facile quoque ex iis, quae in artt. 20. et 21 exposita sunt, derivatur propositio sequens, quae infra usum notabilem nobis praestabit. Statuatur $$W^{\prime }=1+r^{h}+r^{4h}+r^{9h}+{\text{etc.}}+r^{h(n-1)^{2}}$$ denotante ${\textstyle h}$ integrum quemcunque per ${\textstyle p}$ non divisibilem, eritque in casu eo, ubi ${\textstyle n=p}$, vel ubi ${\textstyle n}$ est potestas ipsius ${\textstyle p}$ cum exponente impari, $${\begin{aligned}&W^{\prime }=W,{\text{ si fuerit }}h{\text{ residuum quadraticum ipsius }}p\\&W^{\prime }=-W,{\text{ si fuerit }}h{\text{ non-residuum quadraticum ipsius }}p\end{aligned}}$$ Patet enim, ${\textstyle W^{\prime }}$ oriri ex ${\textstyle W}$, si pro ${\textstyle k}$ substituatur ${\textstyle kh}$; in casu priori autem ${\textstyle k}$ et ${\textstyle kh}$ similes erunt, in posteriori dissimiles, quatenus sunt residua vel non-residua ipsius ${\textstyle p}$.

In casu eo autem, ubi ${\textstyle n}$ est potestas ipsius ${\textstyle p}$ cum exponente pari, manifesto fit ${\textstyle W^{\prime }=+\surd n}$, adeoque semper ${\textstyle W^{\prime }=W}$.

23.

In artt. 20, 21, 22 consideravimus numeros primos impares, taliumque potestates: superest itaque casus, ubi ${\textstyle n}$ est potestas binarii.

Pro ${\textstyle n=2}$ manifesto fit ${\textstyle W=1+r=0}$.

Pro ${\textstyle n=4}$ prodit ${\textstyle W=1+r+r^{4}+r^{9}=2+2r}$: hinc ${\textstyle W=2+2i}$, quoties ${\textstyle k}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=2-2i}$, quoties ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$.

Pro ${\textstyle n=8}$ habemus ${\textstyle W=1+r+r^{4}+r^{9}+r^{16}+r^{25}+r^{36}+r^{49}=2+4r+2r^{4}}$ ${\textstyle =4r}$. Hinc erit $${\begin{aligned}&W=(1+i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +1\\&W=(-1+i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +3\\&W=(-1-i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +5\\&W=(1-i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +7\end{aligned}}$$

Si ${\textstyle n}$ est altior potestas binarii, statuamus ${\textstyle n=2^{2\chi }q}$, ita ut ${\textstyle q}$ sit vel ${\textstyle =1}$ vel ${\textstyle =2}$, atque ${\textstyle \chi }$ maior quam ${\textstyle 1}$. Hic ante omnia observari debet, si ${\textstyle \lambda }$ sit integer quicunque per ${\textstyle 2^{\chi -1}}$ non divisibilis, fieri $${\begin{aligned}&r^{\lambda \lambda }+r^{(\lambda +2^{\chi }q)^{2}}+r^{(\lambda +2.2^{\chi }q)^{2}}+r^{(\lambda +3.2^{\chi }q)^{2}}+{\text{etc.}}+r^{(\lambda +n-2^{\chi }q)^{2}}\\&\quad =r^{\lambda \lambda }\left\{1+r^{2^{\chi +1}\lambda q}+r^{2.2^{\chi +1}\lambda q}+r^{3.2^{\chi +1}\lambda q}+{\text{etc.}}+r^{(2n-2^{\chi +1}q)\lambda }\right\}={\frac {r^{\lambda \lambda }(1-r^{2\lambda n})}{1-r^{2^{\chi +1}\lambda q}}}=0\end{aligned}}$$ Hinc facile perspicietur, fieri $$W=1+r^{2^{2\chi -2}}+r^{4.2^{2\chi -2}}+r^{9.2^{2\chi -2}}+{\text{etc.}}+r^{(n-2^{\chi -1})^{2}}$$ Statuamus ${\textstyle r^{2^{2\chi -2}}=\rho }$, eritque ${\textstyle \rho }$ radix aequationis ${\textstyle x^{4q}-1=0}$, et quidem ${\textstyle \rho =\cos {\frac {k}{4q}}360^{\circ }+i\sin {\frac {k}{4q}}360^{\circ };}$ dein fiet $${\begin{aligned}W&=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(2^{\chi +1}q-1)^{2}}\\&=2^{\chi -1}(1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(4q-1)^{2}})\end{aligned}}$$ Sed summa seriei ${\textstyle 1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(4q-1)^{2}}}$ per ea, quae de casibus ${\textstyle n=4}$, ${\textstyle n=8}$ explicavimus, determinatur, unde colligimus
in casu eo, ubi ${\textstyle q=1}$, sive ubi ${\textstyle n}$ est potestas numeri 4, fieri $${\begin{aligned}&W=(1+i)2^{\chi }=(1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}4\mu +1\\&W=(1-i)2^{\chi }=(1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}4\mu +3\end{aligned}}$$ quae sunt ipsissimae formulae pro ${\textstyle n=4}$ traditae;
in casu eo autem, ubi ${\textstyle q=2}$, sive ubi ${\textstyle n}$ est potestas binarii cum exponente impari maiori quam 3, fieri $${\begin{alignedat}{2}W&=(1+i)2^{\chi }\surd 2&&=(1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +1\\W&=(-1+i)2^{\chi }\surd 2&&=(-1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +3\\W&=(-1-i)2^{\chi }\surd 2&&=(-1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +5\\W&=(1-i)2^{\chi }\surd 2&&=(1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +7\end{alignedat}}$$ quae quoque prorsus conveniunt cum iis, quae pro ${\textstyle n=8}$ tradidimus.

24.

Etiam hic operae pretium erit, rationem summae progressionis $$W^{\prime }=1+r^{h}+r^{4h}+r^{9h}+{\text{etc.}}+r^{h(n-1)^{2}}$$ ad ${\textstyle W}$ determinare, ubi ${\textstyle h}$ integrum quemcunque imparem denotat. Quum ${\textstyle W^{\prime }}$ oriatur ex ${\textstyle W}$, mutando ${\textstyle k}$ in ${\textstyle kh}$, valor ipsius ${\textstyle W^{\prime }}$ perinde a forma numeri ${\textstyle kh}$ pendebit, ut ${\textstyle W}$ a forma ipsius ${\textstyle k}$. Statuamus ${\textstyle {\frac {W^{\prime }}{W}}=l}$, patetque fieri

I. in casu eo, ubi ${\textstyle n=4}$, vel altior potestas binarii cum exponente pari, fieri
$${\begin{array}{l}l=1,{\text{ si fuerit }}h{\text{ formae }}4\mu +1\\l=-i,{\text{ si fuerit }}h{\text{ formae }}4\mu +3,{\text{ atque }}k{\text{ formae }}4\mu +1\\l=+i,{\text{ si fuerit }}h{\text{ formae }}4\mu +3,{\text{ atque }}k{\text{ eiusdem formae}}\\\end{array}}$$ II. in casu eo, ubi ${\textstyle n=8}$, vel altior potestas binarii cum exponente impari, fieri $${\begin{array}{l}l=1,{\text{ si fuerit }}h{\text{ formae }}8\mu +1,\\l=-1,{\text{ si fuerit }}h{\text{ formae }}8\mu +5,\\l=+i,{\text{ si fuerit vel }}h{\text{ formae }}8\mu +3{\text{, atque }}k{\text{ formae }}4\mu +1,\\{\phantom {l=+i}}\quad {\text{ vel }}h{\text{ formae }}8\mu +7,{\text{ atque }}k{\text{ formae }}4\mu +3,\\l=-i,{\text{ si fuerit vel }}h{\text{ formae }}8\mu +3,{\text{ atque }}k{\text{ formae }}4\mu +3,\\{\phantom {l=-i}}\quad {\text{ vel }}h{\text{ formae }}8\mu +7,{\text{ atque }}k{\text{ formae }}4\mu +1.\end{array}}$$

Per praecc. determinatio summae ${\textstyle W}$ pro iis casibus, ubi ${\textstyle n}$ est numerus primus vel numeri primi potestas, complete perfecta est: superest itaque, ut eos quoque casus absolvamus, ubi ${\textstyle n}$ e pluribus numeris primis compositus est, huc viam nobis sternet theorema sequens.

25.

Theorema. Sit n productum e duobus integris positivis inter se primis ${\textstyle a}$, ${\textstyle b}$, statuaturque $${\begin{aligned}&P=1+r^{aa}+r^{4aa}+r^{9aa}+{\text{etc.}}+r^{(b-1)^{2}aa}\\&Q=1+r^{bb}+r^{4bb}+r^{9bb}+{\text{etc.}}+r^{(a-1)^{2}bb}\end{aligned}}$$ Tum dico fore ${\textstyle W=PQ}$.

Demonstr. Designet ${\textstyle \alpha }$ indefinite numeros ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots a-1}$, ${\textstyle \beta }$ indefinite numeros ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots b-1}$, ${\textstyle \nu }$ indefinite numeros ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots n-1}$. Tunc patet esse $$P=\Sigma r^{aa\beta \beta },\quad Q=\Sigma r^{bb\alpha \alpha },\quad W=\Sigma r^{\nu \nu }$$ Hinc erit ${\textstyle PQ=\Sigma r^{aa\beta \beta +bb\alpha \alpha }}$, substituendo pro ${\textstyle \alpha }$ et ${\textstyle \beta }$ omnes valores, omnibus modis inter se combinatos; hinc porro propter ${\textstyle 2ab\alpha \beta =2\alpha \beta n}$, erit ${\textstyle PQ=\Sigma r^{(a\beta +b\alpha )^{2}}}$. Sed nullo negotio perspicitur, singulos valores ipsius ${\textstyle a\beta +b\alpha }$ inter se diversos esse, atque alicui valori ipsius ${\textstyle \nu }$ aequales. Hinc erit ${\textstyle PQ=\Sigma r^{\nu \nu }=W}$.
Ceterum notandum est, ${\textstyle r^{aa}}$ esse radicem propriam aequationis ${\textstyle x^{b}-1=0}$, atque ${\textstyle r^{bb}}$ radicem propriam aequationis ${\textstyle x^{a}-1=0}$.

26.

Sit porro ${\textstyle n}$ productum e tribus numeris inter se primis ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$, patetque, si statuatur ${\textstyle bc=b^{\prime }}$, etiam ${\textstyle a}$ et ${\textstyle b^{\prime }}$ inter se primos fore; adeoque ${\textstyle W}$ productum e duobus factoribus $${\begin{aligned}&1+r^{aa}+r^{4aa}+r^{9aa}+{\text{etc.}}+r^{(b^{\prime }-1)^{2}aa}\\&1+r^{b^{\prime }b^{\prime }}+r^{4b^{\prime }b^{\prime }}+r^{9b^{\prime }b^{\prime }}+{\text{etc.}}+r^{(a-1)^{2}b^{\prime }b^{\prime }}\end{aligned}}$$ Sed quum ${\textstyle r^{aa}}$ sit radix propria aequationis ${\textstyle x^{bc}-1=0}$, erit ipse factor prior productum ex $${\begin{aligned}&1+\rho ^{bb}+\rho ^{4bb}+\rho ^{9bb}+{\text{etc.}}+\rho ^{(c-1)^{2}bb}\\&1+\rho ^{cc}+\rho ^{4cc}+\rho ^{9cc}+{\text{etc.}}+\rho ^{(b-1)^{2}cc}\end{aligned}}$$ si statuitur ${\textstyle r^{aa}=\rho }$. Hinc patet, ${\textstyle W}$ esse productum e factoribus tribus $${\begin{aligned}&1+r^{bbcc}+r^{4bbcc}+r^{9bbcc}+{\text{etc.}}+r^{(a-1)^{2}bbcc}\\&1+r^{aacc}+r^{4aacc}+r^{9aacc}+{\text{etc.}}+r^{(b-1)^{2}aacc}\\&1+r^{aabb}+r^{4aabb}+r^{9aabb}+{\text{etc.}}+r^{(c-1)^{2}aabb}\end{aligned}}$$ ubi ${\textstyle r^{bbcc}}$, ${\textstyle r^{aacc}}$, ${\textstyle r^{aabb}}$ erunt resp. radices propriae aequationum ${\textstyle x^{a}-1=0}$, ${\textstyle x^{b}-1=0}$, ${\textstyle x^{c}-1=0}$.

27.

Hinc facile concluditur generaliter, si ${\textstyle n}$ sit productum e factoribus quotcunque inter se primis ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc., ${\textstyle W}$ fieri productum e totidem factoribus, qui sint $${\begin{aligned}&1+r^{\frac {nn}{aa}}+r^{\frac {4nn}{aa}}+r^{\frac {9nn}{aa}}+{\text{etc.}}+r^{\frac {(a-1)^{2}nn}{aa}}\\&1+r^{\frac {nn}{bb}}+r^{\frac {4nn}{bb}}+r^{\frac {9nn}{bb}}+{\text{etc.}}+r^{\frac {(b-1)^{2}nn}{bb}}\\&1+r^{\frac {nn}{cc}}+r^{\frac {4nn}{cc}}+r^{\frac {9nn}{cc}}+{\text{etc.}}+r^{\frac {(c-1)^{2}nn}{cc}}{\text{ etc.}}\end{aligned}}$$ ubi ${\textstyle r^{\frac {nn}{ab}},r^{\frac {nn}{bb}},r^{\frac {nn}{cc}}}$ etc. erunt radices propriae aequationum ${\textstyle x^{a}-1=0,x^{b}-1=0}$, ${\textstyle x^{c}-1=0}$ etc.

28.

Ex his principiis transitus ad determinationem completam ipsius ${\textstyle W}$ pro valore quocunque ipsius ${\textstyle n}$ sponte iam obvius est. Decomponatur scilicet ${\textstyle n}$ in factores ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. tales, qui sint vel numeri primi inaequales, vel potestates numerorum primorum inaequalium, statuatur ${\textstyle r^{\frac {nn}{aa}}=A}$, ${\textstyle r^{\frac {nn}{bb}}=B}$, ${\textstyle r^{\frac {nn}{cc}}=C}$ etc., eruntque ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$ etc. radices propriae aequationum ${\textstyle x^{a}-1=0}$, ${\textstyle x^{b}-1=0}$, ${\textstyle x^{c}-1=0}$ etc., atque ${\textstyle W}$ productum e factoribus $${\begin{aligned}&1+A+A^{4}+A^{9}+{\text{etc.}}+A^{(u-1)^{2}}\\&1+B+B^{4}+B^{9}+{\text{etc.}}+B^{(b-1)^{2}}\\&1+C+C^{4}+C^{9}+{\text{etc.}}+C^{(c-1)^{2}}{\text{ etc.}}\end{aligned}}$$ Sed hi singuli factores per ea, quae in artt 20, 21, 23 docuimus, determinari poterunt, unde etiam valor producti innotescet. Regulas pro determinandis illis factoribus hic in unum obtutum collegisse haud inutile erit. Quum radix ${\textstyle A}$ fiat ${\textstyle ={\frac {kn}{a}}\cdot {\frac {360^{0}}{a}}}$, aggregatum ${\textstyle 1+A+A^{4}+A^{9}+{\text{etc.}}+A^{(a-1)^{2}}}$, quod per ${\textstyle L}$ denotabimus, perinde per numerum ${\textstyle {\frac {kn}{a}}}$ determinabitur, ut in disquisitione nostra generali ${\textstyle {W}}$ per ${\textstyle k}$. Duodecim iam casus sunt distinguendi.

I. Si ${\textstyle a}$ est numerus primus formae ${\textstyle 4\mu +1}$, puta ${\textstyle =p}$, vel potestas talis numeri primi cum exponente impari, simulque ${\textstyle {\frac {kn}{a}}}$ residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle L=+\surd a}$.

II. Si manentibus reliquis ${\textstyle {\frac {kn}{a}}}$ est non-residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle L=-\surd a}$.

III. Si ${\textstyle a}$ est numerus primus formae ${\textstyle 4\mu +3}$, puta ${\textstyle =p}$, vel potestas talis numeri primi cum exponente impari, simulque ${\textstyle {\frac {kn}{a}}}$ residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle L=+i\surd a}$.

IV. Si, manentibus reliquis ut in III, ${\textstyle {\frac {kn}{a}}}$ est non-residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle L=-i\surd a}$.

V. Si ${\textstyle a}$ est quadratum, altiorve potestas numeri primi (imparis) cum exponente pari, erit ${\textstyle L=+\surd a}$.

VI. Si ${\textstyle a=2}$, erit ${\textstyle L=0}$.

VII. Si ${\textstyle a=4}$, altiorve potestas binarii cum exponente pari, simulque ${\textstyle {\frac {kn}{a}}}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle L=(1+i)\surd a}$.

VIII. Si, manentibus reliquis ut in VII, ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 4\mu +3}$, erit ${\textstyle L=(1-i)\surd a}$.

IX. Si ${\textstyle a=8}$, altiorve potestas binarii cum exponente impari, simulque ${\textstyle {\frac {kn}{a}}}$ formae ${\textstyle 8\mu +1}$, erit ${\textstyle L=(1+i)\surd a}$.

X. Si, manentibus reliquis ut in IX, ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +3}$, erit ${\textstyle L=(-1+i)\surd a}$.

XI. Si manentibus reliquis ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +5}$, erit ${\textstyle L=(-1-i)\surd a}$.

XII. Si manentibus reliquis ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +7}$, erit ${\textstyle L=(1-i)\surd a}$.

29.

Sit exempli caussa ${\textstyle n=2520=8.9.5.7}$, atque ${\textstyle k=13}$. Hic erit

pro ${\textstyle a=8}$, per casum XII, ${\textstyle L=(1-i)\surd 8}$
pro factore 9, per casum V, summa respondens erit ${\textstyle =\surd 9}$
pro factore 5, per casum II, summa respondens erit ${\textstyle =-\surd 5}$

pro factore 7, per casum III, summa respondens erit ${\textstyle =+i\surd 7}$

Hinc fit ${\textstyle W=(1-i)\cdot (-i)\cdot \surd 2520=(-1-i)\surd 2520}$.

Sit pro eodem valore ipsius ${\textstyle n}$, ${\textstyle k=1}$: tunc respondebit

factori 8 summa ${\textstyle (-1+i)\surd 8}$
factori 9 summa ${\textstyle \surd 9}$
factori 5 summa ${\textstyle \surd 5}$

factori 7 summa ${\textstyle -i\surd 7}$

Hinc conflatur productum ${\textstyle W=(1+i)\surd 2520}$.

30.

Methodus alia, summam ${\textstyle W}$ generaliter determinandi, petitur ex iis, quae in artt. 22, 24 exposita sunt. Statuamus ${\textstyle \cos \omega +i\sin \omega =\rho }$, atque $$\rho ^{\frac {nn}{aa}}=\alpha ,\quad \rho ^{\frac {nn}{bb}}=\beta ,\quad \rho ^{\frac {nn}{cc}}=\gamma {\text{ etc.}}$$ ita ut habeatur ${\textstyle r=\rho ^{k}}$, ${\textstyle A=\alpha ^{k}}$, ${\textstyle B=\beta ^{k}}$, ${\textstyle C=\gamma ^{k}}$ etc. Tunc erit $$1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(n-1)^{2}}$$ productum e factoribus $${\begin{aligned}&1+\alpha +\alpha ^{4}+\alpha ^{9}+{\text{etc.}}+\alpha ^{(a-1)^{2}}\\&1+\beta +\beta ^{4}+\beta ^{9}+{\text{etc.}}+\beta ^{(b-1)^{2}}\\&1+\gamma +\gamma ^{4}+\gamma ^{9}+{\text{etc.}}+\gamma ^{(c-1)^{2}}{\text{ etc.}}\end{aligned}}$$ adeoque ${\textstyle W}$ productum e factoribus $${\begin{aligned}&w=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(n-1)^{2}}\\&{\mathfrak {A}}={\frac {1+A+A^{4}+A^{9}+{\text{etc.}}+A^{(a-1)^{2}}}{1+\alpha +\alpha ^{4}+\alpha ^{9}+{\text{etc.}}+a^{(a-1)^{2}}}}\\&{\mathfrak {B}}={\frac {1+B+B^{4}+B^{9}+{\text{etc.}}+B^{(b-1)^{2}}}{1+\beta +\beta ^{4}+\beta ^{9}+{\text{etc.}}+\beta ^{(b-1)^{2}}}}\\&{\mathfrak {C}}={\frac {1+C+C^{4}+C^{9}+{\text{etc.}}+C^{(c-1)^{2}}}{1+\gamma +\gamma ^{4}+\gamma ^{9}+{\text{etc.}}+\gamma ^{(c-1)^{2}}}}{\text{ etc.}}\end{aligned}}$$ Iam factor primus ${\textstyle w}$ determinatus est per disquisitiones supra traditas (art. 19); factores reliqui vero ${\textstyle {\mathfrak {A}}}$, ${\textstyle {\mathfrak {B}}}$, ${\textstyle {\mathfrak {C}}}$ etc. prodeunt per formulas artt. 22, 24, quas ut omnia iuncta habeantur, hic denuo colligimus^[1]. Duodecim casus hic sunt distinguendi, scilicet
I. Si ${\textstyle a}$ est numerus primus (impar) ${\textstyle =p}$, vel talis numeri potestas cum exponente impari, atque ${\textstyle k}$ residuum quadraticum ipsius ${\textstyle p}$, erit factor respondens ${\textstyle {\mathfrak {A}}=+1}$.

II. Si manentibus reliquis ${\textstyle k}$ est non-residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle {\mathfrak {A}}=-1}$.

III. Si ${\textstyle a}$ est quadratum numeri primi imparis, altiorve eius potestas cum exponente pari, erit ${\textstyle {\mathfrak {A}}=+1}$.

IV. Si ${\textstyle a}$ est ${\textstyle =4}$, aut altior binarii potestas cum exponente pari, simulque ${\textstyle k}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+1}$.

V. Si, manentibus reliquis ut in IV, ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=-i}$.

VI. Si, manentibus reliquis ut in IV, ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +3}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

VII. Si ${\textstyle a}$ est ${\textstyle =8}$, aut altior binarii potestas cum exponente impari, atque ${\textstyle k}$ formae ${\textstyle 8\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+1}$.

VIII. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +5}$, erit ${\textstyle {\mathfrak {A}}=-1}$.

IX. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

XI. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +7}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$ erit ${\textstyle {\mathfrak {A}}=-i}$.

XII. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +7}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +3}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

Casum eum, ubi ${\textstyle a=2}$, praeterimus; hic quidem ${\textstyle {\mathfrak {A}}}$ foret ${\textstyle ={\frac {0}{0}}}$ sive indeterminatus, sed tunc semper ${\textstyle W=0}$.

Factores reliqui ${\textstyle {\mathfrak {B}}}$, ${\textstyle {\mathfrak {C}}}$ etc. perinde pendent a ${\textstyle b}$, ${\textstyle c}$ etc., ut ${\textstyle {\mathfrak {A}}}$ ab ${\textstyle a}$, quatenus in illorum determinationem ingrediuntur.

31.

Secundum hanc methodum alteram exemplum primum art. 29 ita se habet:

Factor ${\textstyle w}$ fit ${\textstyle =(1+i)\surd 2520}$
Pro ${\textstyle a=8}$ factor respondens ${\textstyle {\mathfrak {A}}}$ fit, per casum VIII, ${\textstyle =-1}$
Factori ipsius ${\textstyle n}$ secundo ${\textstyle 9}$ respondet factor ${\textstyle +1}$ (per casum III.)
Factori ${\textstyle 5}$ respondet factor ${\textstyle -1}$ (per casum II.)

Factori ${\textstyle 7}$ respondet factor ${\textstyle -1}$ (per casum II.)

Hinc conflatur productum ${\textstyle W=(-1-i)\surd 2520}$, ut in art. 29.

32.

Quum valor ipsius ${\textstyle W}$ per methodos duas determinari possit, quarum altera relationibus numerorum ${\textstyle {\frac {nk}{a}}}$, ${\textstyle {\frac {nk}{b}}}$, ${\textstyle {\frac {nk}{c}}}$ etc. ad numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. innititur, altera vero a relationibus ipsius ${\textstyle k}$ ad numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. pendet, inter omnes has relationes nexus quidam conditionalis intercedere debet, ita ut quaevis e reliquis determinabilis esse debeat. Supponamus, omnes numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. esse numeros primos impares, atque ${\textstyle k}$ accipi ${\textstyle =1}$; distribuanturque factores ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. in duas classes, quarum altera contineat eos, qui sunt formae ${\textstyle 4\mu +1}$, et qui denotentur per ${\textstyle p}$, ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$ etc., altera vero constet ex iis, qui sunt formae ${\textstyle 4\mu +3}$, et qui exprimantur per ${\textstyle q}$, ${\textstyle q^{\prime }}$, ${\textstyle q^{\prime \prime }}$ etc.: multitudinem posteriorum designabimus per ${\textstyle m}$. His ita factis, observamus primo, ${\textstyle n}$ fieri formae ${\textstyle 4\mu +1}$, si ${\textstyle m}$ fuerit par (quorsum etiam referri debet casus is, ubi factores classis alterius omnino desunt, sive ubi ${\textstyle m=0}$), contra ${\textstyle n}$ fieri formae ${\textstyle 4\mu +3}$, si ${\textstyle m}$ fuerit impar. Iam determinatio ipsius ${\textstyle W}$ per methodum primam ita perficitur. Pendeant numeri ${\textstyle P}$, ${\textstyle P^{\prime }}$, ${\textstyle P^{\prime \prime }}$ etc., ${\textstyle Q}$, ${\textstyle Q^{\prime }}$, ${\textstyle Q^{\prime \prime }}$ etc. ita a relationibus numerorum ${\textstyle {\frac {n}{p}}}$, ${\textstyle {\frac {n}{p^{\prime }}}}$, ${\textstyle {\frac {n}{p^{\prime \prime }}}}$ etc., ${\textstyle {\frac {n}{q}}}$, ${\textstyle {\frac {n}{q^{\prime }}}}$, ${\textstyle {\frac {n}{q^{\prime \prime }}}}$ etc. ad numeros ${\textstyle p}$, ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$ etc., ${\textstyle q}$, ${\textstyle q^{\prime }}$, ${\textstyle q^{\prime \prime }}$ etc. resp., ut statuatur $${\begin{aligned}&P=+1,{\text{ si }}{\frac {n}{p}}{\text{ est residuum quadraticum ipsius }}p\\&P=-1,{\text{ si }}{\frac {n}{p}}{\text{ est non-residuum quadraticum ipsius }}p\end{aligned}}$$ et perinde de reliquis. Tunc erit ${\textstyle W}$ productum e factoribus ${\textstyle P\surd p}$, ${\textstyle P^{\prime }\surd p^{\prime }}$, ${\textstyle P^{\prime \prime }\surd p^{\prime \prime }}$ etc., ${\textstyle iQ\surd q}$, ${\textstyle iQ^{\prime }\surd q^{\prime }}$, ${\textstyle iQ^{\prime \prime }\surd q^{\prime \prime }}$ etc., adeoque $$W=PP^{\prime }P^{\prime \prime }\ldots QQ^{\prime }Q^{\prime \prime }\ldots i^{m}\surd n$$ Per methodum secundam, aut potius statim per praecepta art. 19, erit

${\textstyle W=+\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, vel quod eodem redit, si ${\textstyle m}$ est par

${\textstyle W=+i{\sqrt {n}}}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, vel si ${\textstyle m}$ est impar

Utrumque casum simul complecti licet per formulam sequentem: $$W=i^{mm}\surd n$$ Hinc itaque colligitur $$PP^{\prime }P^{\prime \prime }\ldots QQ^{\prime }Q^{\prime \prime }\ldots =i^{mm-m}$$ Sed ${\textstyle i^{mm-m}}$ fit ${\textstyle =1}$, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu }$ vel ${\textstyle 4\mu +1}$, atque ${\textstyle =-1}$, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu +2}$ vel ${\textstyle 4\mu +3}$, unde deducimus sequens elegantissimum

Theorema. Denotantibus ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. numeros primos impares positivos inaequales, quorum productum statuitur ${\textstyle =n}$, et inter quos ${\textstyle m}$ sint formae ${\textstyle 4\mu +3}$, reliqui formae ${\textstyle 4\mu +1}$: multitudo eorum ex his numeris ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc., quorum non-residua resp. sunt ${\textstyle {\frac {n}{a}}}$, ${\textstyle {\frac {n}{b}}}$, ${\textstyle {\frac {n}{c}}}$ etc., par erit, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu }$ vel ${\textstyle 4\mu +1}$, impar vero, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu +2}$ vel ${\textstyle 4\mu +3}$.

Ita e.g. statuendo ${\textstyle a=3}$, ${\textstyle b=5}$, ${\textstyle c=7}$, ${\textstyle d=11}$, habemus tres numeros formae ${\textstyle 4\mu +3}$, puta ${\textstyle 3}$, ${\textstyle 7}$ et ${\textstyle 11}$; est autem ${\textstyle 5.7.11R3}$; ${\textstyle 3.7.11R5}$; ${\textstyle 3.5.11R7}$; ${\textstyle 3.5.7N11}$, sive unicus ${\textstyle {\frac {n}{d}}}$ est non-residuum ipsius ${\textstyle d}$.

33.

Celeberrimum theorema fundamentale circa residua quadratica nihil aliud est, nisi casus specialis theorematis modo evoluti. Limitando scilicet multitudinem numerorum ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. ad duos, patet, si unus tantum ex ipsis, vel neuter, sit formae ${\textstyle 4\mu +3}$, fieri debere vel simul ${\textstyle aRb}$, ${\textstyle bRa}$, vel simul ${\textstyle aNb}$, ${\textstyle bNa}$; contra si uterque est formae ${\textstyle 4\mu +3}$, unus ex ipsis alterius non-residuum esse debebit, atque hic illius residuum. En itaque demonstrationem quartam huius gravissimi theorematis, cuius demonstrationem primam et secundam in Disquisitionibus Arithmeticis, tertiam nuper in commentatione peculiari tradidimus (Commentt. T. XVI): duas alias principiis rursus omnino diversis innitentes in posterum exponemus. Summopere sane est mirandum, quod hocce venustissimum theorema, quod primo omnes conatus tam pertinaciter eluserat, tot postea viis toto coelo inter se distantibus adiri potuerit.

34.

Etiam theoremata reliqua, quae quasi supplementum ad theorema fundamentale efficiunt, scilicet per quae dignoscuntur numeri primi, quorum residua vel non-residua sunt ${\textstyle -1}$, ${\textstyle +2}$ et ${\textstyle -2}$, ex iisdem principiis derivari possunt. Incipiemus a residuo ${\textstyle +2}$.

Statuendo ${\textstyle n=8a}$, ita ut ${\textstyle a}$ sit numerus primus, atque ${\textstyle k=1}$, per methodum art. 28, ${\textstyle W}$ erit productum e duobus factoribus, quorum alter erit ${\textstyle +\surd a}$, vel ${\textstyle +i\surd a}$, si 8, vel quod idem est 2, est residuum quadraticum ipsius ${\textstyle a}$; contra ${\textstyle -\surd a}$ vel ${\textstyle -i\surd a}$, si 2 est non-residuum ipsius ${\textstyle a}$. Factor secundus autem est $${\begin{aligned}&(1+i)\surd 8,{\text{ si }}a{\text{ est formae }}8\mu +1\\&(-1+i)\surd 8,{\text{ si }}a{\text{ est formae }}8\mu +3\\&(-1-i)\surd 8{\text{, si }}a{\text{ est formae }}8\mu +5\\&(1-i)\surd 8,{\text{ si }}a{\text{ est formae }}8\mu +7\end{aligned}}$$ Sed per art. 18 semper erit ${\textstyle W=(1+i)\surd n}$; dividendo hunc valorem per quatuor valores factoris secundi, patet, factorem primum fieri debere $${\begin{aligned}&+\surd a,{\text{ si }}a{\text{ est formae }}\quad 8\mu +1\\&-i\surd a,{\text{ si }}a{\text{ est formae }}8\mu +3\\&-\surd a,{\text{ si }}a{\text{ est formae }}8\mu +5\\&+i\surd a,{\text{ si }}a{\text{ est formae }}8\mu +7\end{aligned}}$$ Hinc sponte sequitur, in casu primo et quarto 2 esse debere residuum ipsius ${\textstyle a}$, in casu secundo et tertio autem non-residuum.

35.

Numeri primi, quorum residuum vel non-residuum est ${\textstyle -1}$, facile dignoscuntur adiumento theorematis sequentis, quod etiam per se ipsum satis memorabile est.

Theorema. Productum e duobus factoribus $${\begin{aligned}&W^{\prime }=1+r^{-1}+r^{-4}+{\text{etc.}}+r^{-(n-1)^{2}}\\&W=1+r+r^{4}+{\text{etc.}}+r^{(n-1)^{2}}\end{aligned}}$$ est ${\textstyle =n}$, si ${\textstyle n}$ est impar; vel ${\textstyle =0}$, si ${\textstyle n}$ est impariter par; vel ${\textstyle =2n}$, si ${\textstyle n}$ est pariter par.

Demonstr. Quum manifesto fiat $${\begin{aligned}W&=r+r^{4}+r^{9}+{\text{etc.}}+r^{nn}\\&=r^{4}+r^{9}+{\text{etc.}}+r^{(n+1)^{2}}\\&=r^{9}+{\text{etc.}}+r^{(n+2)^{2}}{\text{ etc.}}\end{aligned}}$$ productum ${\textstyle W{W}^{\prime }}$ ita quoque exhiberi poterit $${\begin{aligned}&{\phantom {+,}}1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}\\&+r^{-1}(r+r^{4}+r^{9}+r^{16}+{\text{etc.}}+r^{nn})\\&+r^{-4}(r^{4}+r^{9}+r^{16}+r^{25}+{\text{etc.}}+r^{(n+1)^{2}})\\&+r^{-9}(r^{9}+r^{16}+r^{25}+r^{36}+{\text{etc.}}+r^{(n+2)^{2}})\\&{\text{etc.}}\\&+r^{-(n-1)^{2}}(r^{(n-1)^{2}}+r^{nn}+r^{(n+1)^{2}}+r^{(n+2)^{2}}+{\text{etc.}}+r^{(2n-2)^{2}})\end{aligned}}$$ quod aggregatum verticaliter summatum producit $${\begin{aligned}&{\phantom {+}}n\\&+r(1+rr+r^{4}+r^{6}+{\text{etc.}}+r^{2n-2})\\&+r^{4}(1+r^{4}+r^{8}+r^{12}+{\text{etc.}}+r^{4n-4})\\&+r^{9}(1+r^{6}+r^{12}+r^{18}+{\text{etc.}}+r^{6n-6})\\&+{\text{etc.}}\\&+r^{(n-1)^{2}}(1+r^{2n-2}+r^{4n-4}+r^{6n-6}+{\text{etc.}}+r^{2(n-1)^{2}})\end{aligned}}$$ Iam si ${\textstyle n}$ impar est, singulae partes huius aggregati, praeter primam ${\textstyle n}$, erunt ${\textstyle =0}$; secunda enim manifesto fit ${\textstyle {\frac {r(1-r^{2n})}{1-rr}}}$, tertia ${\textstyle {\frac {r^{4}(1-r^{4n})}{1-r^{4}}}}$ etc. Quoties vero ${\textstyle n}$ par est, excipere insuper oportebit partem $$r^{{\frac {1}{4}}nn}(1+r^{n}+r^{2n}+r^{3n}+{\text{etc.}}+r^{nn-n})$$ quae fit ${\textstyle =nr^{{\frac {1}{4}}nn}}$. In casu priori itaque fit ${\textstyle WW^{\prime }=n}$, in posteriori autem ${\textstyle =n+nr^{{\frac {1}{4}}nn}}$; sed ${\textstyle r^{{\frac {1}{4}}nn}}$ fit ${\textstyle =+1}$, si ${\textstyle n}$ est pariter par, tunc itaque prodit ${\textstyle WW^{\prime }=2n}$; contra fit ${\textstyle r^{{\frac {1}{4}}nn}=-1}$, si ${\textstyle n}$ est impariter par, ubi itaque evadit ${\textstyle WW^{\prime }=0}$. Q. E. D.

36.

Iam per art. 22 constat, si ${\textstyle n}$ sit numerus primus impar, ${\textstyle {\frac {W^{\prime }}{W}}}$ fieri ${\textstyle =+1}$ vel ${\textstyle =-1}$, prout ${\textstyle -1}$ fuerit residuum vel non-residuum ipsius ${\textstyle n}$. Hinc in casu priori esse debebit ${\textstyle W^{2}=+n}$, in posteriori ${\textstyle W^{2}=-n}$; quamobrem per art. 13 concludimus, casum priorem tunc tantum locum habere posse, quando ${\textstyle n}$ sit formae ${\textstyle 4\mu +1}$, casumque posteriorem, quando ${\textstyle n}$ sit formae ${\textstyle 4\mu +3}$.

Denique e combinatione conditionum pro residuis ${\textstyle +2}$ et ${\textstyle -1}$ inventarum sponte sequitur, ${\textstyle -2}$ esse residuum cuiusvis numeri primi formae ${\textstyle 8\mu +1}$ vel ${\textstyle 8\mu +3}$, atque non-residuum cuiusvis numeri primi formae ${\textstyle 8\mu +5}$ vel ${\textstyle 8\mu +7}$.

1. Manifesto, quae illic erant ${\textstyle k}$ et ${\textstyle h}$, hic erunt ${\textstyle {\frac {n}{a}}}$ et ${\textstyle k}$ respectu factoris secundi, ${\textstyle {\frac {n}{b}}}$ et ${\textstyle k}$ respectu factoris tertii etc.