\( \def\ovr{\over\displaystyle} \def\dst{\displaystyle\strut} \def\tailfrac{\rlap{\dst \rlap{\phantom{\dst 1\ovr2}}\cdots}} \def\cofrac#1#2#3{\rlap{\dfrac{#1}{\phantom{#2+}}} \genfrac{}{}{0pt}{0}{}{#2+#3}} \)

Introductio in Analysin Infinitorum
Auctore Leonhardo Eulero
Tomus Primus

Caput XVIII
De Fractionibus Continuis

356. Quoniam in praecedentibus capitibus plura, cum de seriebus infinitis, tum de productis ex infinitis factoribus conflatis disserui, non incongruum fore visum est, si etiam nonnulla de tertio quodam expressionum infinitarum genere addidero, quod continuis fractionibus vel divisionibus continetur. Quanquam enim hoc genus parum adhuc est excultum, tamen non dubitamus, quin ex eo amplissimus usus in analysin infinitorum aliquando sit redundaturus. Exhibui enim iam aliquoties eiusmodi specimina, quibus haec expectatio non parum probabilis redditur. Imprimis vero ad ipsam arithmeticam et algebram communem non contemnenda subsidia affert ista speculatio, quae hoc capite breviter indicare atque exponere constitui.

357. Fractionem autem continuam voco eiusmodi fractionem, cuius denominator constat ex numero integro cum fractione, cuius denominator denuo est aggregatum ex integro et fractione, quae porro simili modo sit comparata, sive ista affectio in infinitum progrediatur sive alicubi sistatur. Huiusmodi ergo fractio continua erit sequens expressio \begin{equation} {a + {\cofrac 1 b {\cofrac 1 c {\cofrac 1 d {\cofrac 1 e {\cofrac 1 f \tailfrac }}}}}} \hbox{vel}\qquad\qquad {a + {\cofrac \alpha b {\cofrac \beta c {\cofrac \gamma d {\cofrac \delta e {\cofrac \varepsilon f \tailfrac }}}}}} \end{equation} in quarum forma priori omnes fractionum numeratores sunt unitates, quam potissimum hic contemplabor, in altera vero forma sunt numeratores numeri quicunque.

358. Exposita ergo fractionum harum continuarum forma, primum videndum est, quemadmodum earum significatio consueto more expressa inveniri queat. Quae ut facilius inveniri possit, progrediamur per gradus, abrumpendo illas fractiones primo in prima, tum in secunda, post in tertia et ita porro fractione; quo facto patebit fore \begin{aligned} a &= a\cr\cr {a + {1\over b}} &= { {ab + 1} \over {b}} \cr\cr {a + {\cofrac 1 b {\dfrac{1}{c}}}} &= { {abc + a + c} \over {bc + 1}} \cr\cr {a + {\cofrac 1 b {\cofrac 1 c {\dfrac{1}{d}}}}} &= { {abcd + ab + ad + cd + 1} \over {bcd + b + d}} \cr\cr {a + {\cofrac 1 b {\cofrac 1 c {\cofrac 1 d {\dfrac{1}{e}}}}}} &= { {abcde + abe + ade + cde + abc + a + c + e} \over {bcde + be + de + bc + 1}}\cr\cr &\hbox{etc.}\cr \end{aligned}

359. Etsi in his fractionibus ordinariis non facile lex, secundum quam numerator ac denominator ex litteris \(a\), \(b\), \(c\), \(d\) etc., componantur, perspicitur, tamen attendenti statim patebit, quemadmodum quaelibet fractio ex praecedentibus formari queat. Quilibet enim numerator est aggregatum ex numeratore ultimo per novam litteram multiplicato, et ex numeratore penultimo simplici; eademque lex in denominatoribus observatur. Scriptis ergo ordine litteris \(a\), \(b\), \(c\), \(d\) etc., ex iis fractiones inventae facile formabuntur hoc modo \begin{array}{ccccc} a\phantom{\,,} & b\phantom{\,,} & c\phantom{\,,} & d\phantom{\,,} & e \\ \displaystyle{1 \over 0}\,, & \displaystyle{a \over 1}\,, & \displaystyle{ {ab + 1}\over b}\,, & \displaystyle{ {abc + a + c}\over {bc + 1}}\,, & \displaystyle{ {abcd + ab + ad + cd + 1}\over {bcd + b + d}} \\ \end{array} ubi quilibet numerator invenitur, si praecedentium ultimus per indicem supra scriptum multiplicetur atque ad productum antepenultimus addatur; quae eadem lex pro denominatoribus valet. Quo autem hac lege ab ipso initio uti liceat, praefixi fractionem \(1\over 0\) quae, etiamsi e fractione continua non oriatur, tamen progressionis legem clariorem efficit. Quaelibet autem fractio exhibet valorem fractionis continuae usque ad eam litteram, quae antecedenti imminet, inclusive continuatae.

360. Simili modo altera fractionum continuarum forma \begin{equation} {a + {\cofrac \alpha b {\cofrac \beta c {\cofrac \gamma d {\cofrac \delta e {\cofrac \varepsilon f \tailfrac }}}}}} \end{equation} dabit, prout aliis aliisque locis abrumpitur, sequentes valores \begin{aligned} a &= a\cr\cr {a + {\alpha\over b}} &= { {ab + \alpha} \over {b}} \cr\cr {a + {\cofrac \alpha b {\dfrac{\beta}{c}}}} &= { {abc + \beta a + \alpha c} \over {bc + \beta}} \cr\cr {a + {\cofrac \alpha b {\cofrac \beta c {\dfrac{\gamma}{d}}}}} &= { {abcd + \beta ad + \alpha cd + \gamma ab + \alpha\gamma} \over {bcd + \beta d + \gamma b}} \cr\cr &\hbox{etc.}\cr \end{aligned} quarum fractionum quaeque ex binis praecedentibus sequentem in modum invenietur: \begin{array}{ccccc} a\phantom{\,,} & b\phantom{\,,} & c\phantom{\,,} & d\phantom{\,,} & e \\ \displaystyle{1 \over 0}\,, & \displaystyle{a \over 1}\,, & \displaystyle{ {ab + \alpha}\over b}\,, & \displaystyle{ {abc + \beta a + \alpha c}\over {bc + \beta}}\,, & \displaystyle{ {abcd + \beta ad + \alpha cd + \gamma ab + \alpha\gamma}\over {bcd + \beta d + \gamma b}} \\ \alpha\phantom{\,,} & \beta\phantom{\,,} & \gamma\phantom{\,,} & \delta\phantom{\,,} & \varepsilon \\ \end{array}

361. Fractionibus scilicet formandis supra inscribantur indices \(a\), \(b\), \(c\), \(d\), etc., infra autem subscribantur indices \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc. Prima fractio iterum constituatur \(1\over 0\), secunda \(a\over 1\), tum sequentium quaevis formabitur, si antecedentium ultimae numerator per indicem supra scriptum, penultimae vero numerator per indicem infra scriptum multiplicetur et ambo producta addantur, aggregatum erit numerator fractionis sequentis; simili modo eius denominator erit aggregatum ex ultimo denominatore per indicem supra scriptum et ex penultimo denominatore per indicem infra scriptum multiplicatis. Quaelibet vero fractio hoc modo inventa praebebit valorem fractionis continuae ad eum usque denominatorem, qui fractioni antecedenti est inscriptus, continuatae inclusive.

362. Quod si ergo hae fractiones eousque continuentur quoad fractio continua indices suppeditet, tum ultima fractio verum dabit valorem fractionis continuae. Praecedentes fractiones vero continuo propius ad hunc valorem accedent, ideoque perquam idoneam appropinquationem suggerent. Ponamus enim verum valorem fractionis continuae \begin{equation} {a + {\cofrac \alpha b {\cofrac \beta c {\cofrac \gamma d {\cofrac \delta e \tailfrac}}}}} \end{equation} esse \({}= x\) atque manifestum est fractionem primam \(1\over 0\) esse maiorem quam \(x\); secunda vero \(a\over 1\) minor erit quam \(x\); tertia \(a + {\alpha\over b}\) iterum vero valore erit maior; quarta denuo minor, atque ita porro hae fractiones alternatim erunt maiores et minores quam \(x\). Porro autem perspicuum est quamlibet fractionem propius accedere ad verum valorem \(x\) quam ulla praecedentium; unde hoc pacto citissime et commodissime valor ipsius \(x\) proxime obtinetur; etiamsi fractio continua in infinitum progrediatur, dummodo numeratores \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc. non nimis crescant; sin autem omnes isti numeratores fuerint unitates, tum appropinquatio nulli incommodo est obnoxia.

363. Quo ratio huius appropinquationis ad verum fractionis continuae valorem melius percipiatur, consideremus fractionum inventarum differentias. Ac, prima quidem \(1\over 0\) praetermissa, differentia inter secundam ac tertiam est \begin{equation} {} = {\alpha \over b}\,, \end{equation} quarta a tertia subtracta relinquit \begin{equation} {\alpha\beta \over b(bc+\beta)}\,, \end{equation} quarta a quinta subtracta relinquit \begin{equation} {\alpha\beta\gamma \over (bc + \beta)(bcd + \beta d + \gamma b)} \end{equation} etc. Hinc exprimetur valor fractionis continuae per seriem terminorum consuetam hoc modo, ut sit \begin{equation} x = a + {\alpha \over b} - {\alpha\beta \over b(bc+\beta)} + {\alpha\beta\gamma \over (bc + \beta)(bcd + \beta d + \gamma b)} - \cdots \end{equation} quae series toties abrumpitur quoties fractio continua non in infinitum progreditur.

364. Modum ergo invenimus fractionem continuam quamcunque in seriem terminorum, quorum signa alternantur, convertendi, si quidem prima littera \(a\) evanescat. Si enim fuerit \begin{equation} x = {\cofrac \alpha b {\cofrac \beta c {\cofrac \gamma d {\cofrac \delta e {\cofrac \varepsilon f \tailfrac }}}}} \end{equation} erit per ea, quae modo invenimus, \begin{equation} x = {\alpha \over b} - {\alpha\beta \over b(bc+\beta)} + {\alpha\beta\gamma \over (bc + \beta)(bcd + \beta d + \gamma b)} \\ - {\alpha\beta\gamma\delta \over (bcd + \beta d + \gamma b)(bcde + \beta d e + \gamma b e + \delta b c + \beta\delta)} + \cdots \end{equation} Unde, si \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc. fuerint numeri non crescentes, uti omnes unitates, denominatores vero \(a\), \(b\), \(c\), \(d\), etc. numeri integri quicunque affirmativi, valor fractionis continuae exprimetur per seriem terminorum maxime convergentem.

365. His probe consideratis, poterit vicissim series quaecunque terminorum alternantium in fractionem continuam converti, seu fractio continua inveniri cuius valor aequalis sit summae seriei propositae. Sit enim proposita haec series \begin{equation} x = A - B + C - D + E - F + \cdots \,, \end{equation} erit, singulis terminis cum serie ex fractione continua orta comparandis \begin{array}{llll} \displaystyle{A={\alpha\over b}}\,, &&\hbox{hincque} & \displaystyle{\alpha = A b}\,, \\ \displaystyle{ {B\over A} = {\beta\over{bc+\beta}}}\,, && \hbox{unde sit} & \displaystyle{\beta = {Bbc\over{A-B}}}\,, \\ \displaystyle { {C\over B} ={\gamma b \over bcd + \beta d + \gamma b}} \,, &&& \displaystyle \gamma = { {Cd(bc+\beta)}\over b(B-C)}\,, \\ \displaystyle { {D\over C} ={\delta (bc + \beta) \over bcde + \beta d e + \gamma b e + \delta b c + \beta\delta}} \,, &&& \displaystyle{\delta = {De(bcd + \beta d + \gamma b)\over (bc+\beta)(C-D)}} \\ &&\hbox{etc.} \end{array} At, cum sit \begin{equation} \beta = {Bbc\over A - B}\,, \end{equation} erit \begin{equation} bc + \beta = {Abc\over A-B}\,; \end{equation} unde \begin{equation} \gamma = {ACcd \over {(A-B)(B-C)}}\,. \end{equation} Porro fit \begin{aligned} bcd + \beta d + \gamma b &= (bc+\beta)d + \gamma b \cr &= {Abcd \over A - B} + {ACbcd \over (A-B)(B-C)} \cr &= {ABbcd \over(A-B)(B-C)} \,, \end{aligned} unde erit \begin{array}{ccc} \displaystyle{bcd + \beta d + \gamma b \over bc + \beta} = {Bd \over B-C} & \quad\hbox{et}\quad & \displaystyle\delta = {BDde \over (B-C)(C-D) }\,; \end{array} simili modo reperietur \begin{equation} \varepsilon = {CEef \over (C-D)(D-E)} \end{equation} et ita porro.

366. Quo ista lex clarius appareat, ponamus esse \begin{aligned} P &= b\,, \cr Q &= bc + \beta\,, \cr R &= bcd + \beta d + \gamma b\,, \cr S &= bcde + \beta de + \gamma be + \delta bc + \beta\delta\,, \cr T &= bcde\!f + \cdots\,, \cr V &= bcde\!fg + \cdots\,, \cr &\hbox{etc.}\cr \end{aligned} erit ex lege harum expressionum \begin{aligned} Q &= Pc + \beta \,,\cr R &= Qd + \gamma P \,,\cr S &= Re + \delta Q \,,\cr T &= Sf + \varepsilon R \,,\cr V &= Tg + \xi S \,,\cr &\hbox{etc.}\cr \end{aligned} Cum igitur his adhibendis litteris fit \begin{equation} x = {\alpha\over P} - {\alpha\beta\over PQ} + {\alpha\beta\gamma\over QR} - {\alpha\beta\gamma\delta\over RS} + {\alpha\beta\gamma\delta\varepsilon\over ST} - \cdots \,. \end{equation}

367. Quoniam ergo ponimus esse \begin{equation} x = A-B+C-D+E-F+\cdots\,, \end{equation} erit \begin{array}{ccc} \displaystyle{A={\alpha\over P}}\,, & \displaystyle{\alpha = AP} \,,\cr \displaystyle{ {B\over A} = {\beta\over{Q}}}\,, & \displaystyle{\beta = {BQ\over A}} \,,\cr \displaystyle { {C\over B} ={\gamma P \over R}}\,, & \displaystyle \gamma = {CR\over BP} \,,\cr \displaystyle { {D\over C} ={\delta Q \over S}}\,, & \displaystyle \delta = {DS\over CQ} \,,\cr \displaystyle { {E\over D} ={\varepsilon R \over T}}\,, & \displaystyle \varepsilon = {ET\over DR} \cr \hbox{etc.} & \hbox{etc.} \cr \end{array} Porro vero differentiis sumendis habebitur \begin{aligned} A-B &= {\alpha(Q-\beta)\over PQ} = {\alpha c \over Q} = {APc \over Q} \cr B-C &= {\alpha\beta(R-\gamma P)\over PQR} = {\alpha\beta d \over PR} = {BQd \over R} \cr C-D &= {\alpha\beta\gamma(S-\delta Q)\over QRS} = {\alpha\beta\gamma e \over QS} = {CRe \over S} \cr D-E &= {\alpha\beta\gamma\delta(T-\varepsilon R)\over RST} = {\alpha\beta\gamma\delta f \over RT} = {DSf \over T} \cr &\hbox{etc.}\cr \end{aligned} Si bini igitur in se invicem ducantur, fiet \begin{array}{ccc} \displaystyle{(A-B)(B-C)=ABcd\cdot{P\over R}} & \hbox{et} & \displaystyle{R\over P} = {ABcd \over (A-B)(B-C)} \cr \displaystyle{(B-C)(C-D)=BCde\cdot{Q\over S}} & \hbox{et} & \displaystyle{S\over Q} = {BCed \over (B-C)(C-D)} \cr \displaystyle{(C-D)(D-E)=CDef\cdot{R\over T}} & \hbox{et} & \displaystyle{T\over R} = {CDef \over (C-D)(D-E)} \cr \hbox{etc.}\cr \end{array} Unde, cum sit \begin{aligned} P&=b\,, \\ Q&={\alpha c\over A-B} = {Abc\over{A-B}}, \end{aligned} erit \begin{aligned} \alpha &= {Ab} \,,\cr \beta &= {Bbc \over A - B} \,,\cr \gamma &= {ACcd \over (A-B)(B-C) } \,,\cr \delta &= {BDde \over (B-C)(C-D) } \,,\cr \varepsilon &= {CEef \over (C-D)(D-E)} \cr &\hbox{etc.}\cr \end{aligned}

368. Inventis ergo valoribus numeratorum \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc., denominatores \(b\), \(c\), \(d\), \(e\), etc., arbitrio nostro relinquuntur; ita autem eos assumi convenit, ut, cum ipsi sint numeri integri, tum valores integros pro \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc., exhibeant. Hoc vero pendet quoque a natura numerorum \(A\), \(B\), \(C\), etc., utrum sint integri an fracti. Ponamus esse numeros integros, atque quaesito satisfiet statuendo \begin{array}{lll} \displaystyle{b = 1} & & \displaystyle{\alpha = A} \cr \displaystyle{c = A-B} & & \displaystyle{\beta = B} \cr \displaystyle{d = B-C} & \quad\hbox{unde fit}\quad & \displaystyle{\gamma = AC} \cr \displaystyle{e = C-D} & & \displaystyle{\delta = BD} \cr \displaystyle{f = D-E} & & \displaystyle{\varepsilon = CE} \cr \quad\hbox{etc.} & & \quad\hbox{etc.}\cr \end{array} Quocirca, si fuerit, \begin{equation} x = A-B+C-D+E-F+\cdots\,, \end{equation} idem ipsius \(x\) valor per fractionem continuam ita exprimi poterit, ut sit \begin{equation} x = {\cofrac A 1 {\cofrac B {A-B} {\cofrac {AC} {B-C} {\cofrac {BD} {C-D} {\cofrac {CE} {D-E} \tailfrac }}}}} \end{equation}

369. Sin autem omnes termini seriei sint numeri fracti, ita ut fuerit \begin{equation} x = {1\over A} - {1\over B} + {1\over C} - {1\over D} + {1\over E} - \cdots \end{equation} habebuntur pro \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), etc. sequentes valores \begin{aligned} \alpha &= {b\over A} \,,\cr \beta &= {Abc\over B-A} \,,\cr \gamma &= {B^2cd\over (B-A)(C-B)} \,,\cr \delta &= {C^2de\over (C-B)(D-C)} \,,\cr \varepsilon &= {D^2ef\over (D-C)(E-D)} \cr &\hbox{etc.}\cr \end{aligned} Ponatur ergo ut sequitur \begin{array}{lll} \displaystyle{b = A} & \quad\hbox{unde fit}\quad & \displaystyle{\alpha = 1} \,,\cr \displaystyle{c = B-A} & & \displaystyle{\beta = AA} \,,\cr \displaystyle{d = C-B} & & \displaystyle{\gamma = BB} \,,\cr \displaystyle{e = D-C} & & \displaystyle{\delta = CC} \cr &\quad\hbox{etc.} \cr \end{array} eritque per fractionem continuam \begin{equation} x = {\cofrac 1 A {\cofrac {AA} {B-A} {\cofrac {BB} {C-B} {\cofrac {CC} {D-C} \tailfrac}}}} \end{equation}

EXEMPLUM I.

Transformetur haec series infinita \begin{equation} x = 1 - {1\over 2} + {1\over 3} - {1\over 4} + {1\over 5} - \cdots \end{equation} in fractionem continuam.

Erit ergo \(A=1\), \(B=2\), \(C=3\), \(D=4\), etc., atque, cum seriei propositae valor sit \({}=\log 2\), erit \begin{equation} \log 2 = {\cofrac 1 1 {\cofrac 1 1 {\cofrac 4 1 {\cofrac 9 1 {\cofrac {16} 1 {\cofrac {25} 1 \tailfrac}}}}}} \end{equation}

EXEMPLUM II.

Transformetur haec series infinita \begin{equation} {\pi\over4} = 1 - {1\over 3} + {1\over 5} - {1\over 7} + {1\over 9} - \cdots \end{equation} ubi \(\pi\) denotat peripheriam circuli, cuius diameter \({}=1\), in fractionem continuam.

Substitutis \(A\), \(B\), \(C\), \(D\), etc. numeris \(1\), \(3\), \(5\), \(7\), etc. orietur \begin{equation} {\pi\over 4} = {\cofrac 1 1 {\cofrac 1 2 {\cofrac 9 2 {\cofrac {25} 2 {\cofrac {49} 2 \tailfrac}}}}} \end{equation} hincque, invertendo fractionem, erit \begin{equation} {4\over\pi} = 1 + {\cofrac 1 2 {\cofrac 9 2 {\cofrac {25} 2 {\cofrac {49} 2 \tailfrac}}}} \end{equation} quae est expressio, quam Brounckerus primum pro quadratura circuli protulit.

EXEMPLUM III.

Si proposita ista series infinita \begin{equation} x = {1\over m} - {1\over m+n} + {1\over m+2n} - {1\over m+3n} + \cdots \end{equation} quae, ob \begin{equation} A=m,\quad B=m+n,\quad C=m+2n,\quad \hbox{etc.}, \end{equation} in hanc fractionem continuam mutatur \begin{equation} x = {\cofrac 1 m {\cofrac {mm} {\phantom{n}n} {\cofrac {(m+n)^2} {\phantom{m+n}n} {\cofrac {(m+2n)^2} {\phantom{m+2n}n} {\cofrac {(m+3n)^2} {\phantom{nn}n} \tailfrac}}}}} \end{equation} ex qua sit, invertendo, \begin{equation} {1\over x} - m = {\cofrac {mm} {\phantom{n}n} {\cofrac {(m+n)^2} {\phantom{m+n}n} {\cofrac {(m+2n)^2} {\phantom{mm+n}n} {\cofrac {(m+3n)^2} {\phantom{nn}n} \tailfrac } } } } \end{equation}

EXEMPLUM IV.

Quoniam supra (§ 178) invenimus esse \begin{equation} {\pi\cos{m\pi\over n} \over n\sin{m\pi\over n} } = {1\over m} - {1\over n-m} + {1\over n+m} - {1\over 2n-m} + {1\over 2n+m} - \cdots \end{equation} erit, pro fractione continuanda, \begin{equation} A=m,\quad B=n-m,\quad C=n+m,\quad D=2n-m,\quad \cdots \end{equation} unde fiet \begin{equation} {\pi\cos{m\pi\over n} \over n\sin{m\pi\over n} } = {\cofrac 1 m {\cofrac {mm} {n-2m} {\cofrac {(n-m)^2} {\phantom{2m+}2m} {\cofrac {(n+m)^2} {n-2m} {\cofrac {(2n-m)^2} {\phantom{2m+}2m} {\cofrac {(2n+m)^2} {n-2m} \tailfrac}}}}}} \end{equation}

370. Si series proposita per continuos factores progrediatur, ut sit \begin{equation} x={1\over A} - {1\over AB} + {1\over ABC} - {1\over ABCD} + {1\over ABCDE} - \cdots\, \end{equation} tum prodibunt sequentes determinationes \begin{aligned} \alpha &= {b\over A} \,,\cr \beta &= {bc\over{B-1}} \,,\cr \gamma &= {Bcd\over{(B-1)(C-1)}} \,,\cr \delta &= {Cde\over{(C-1)(D-1)}} \,,\cr \varepsilon &= {Def\over{(D-1)(E-1)}} \cr & \hbox{etc.,} \end{aligned} fiat ergo ut sequitur, \begin{array}{lll} \displaystyle{b = A} & \quad\hbox{unde fit}\quad & \displaystyle{\alpha = 1} \,,\cr \displaystyle{c = B-1} & & \displaystyle{\beta = A} \,,\cr \displaystyle{d = C-1} & & \displaystyle{\gamma = B} \,,\cr \displaystyle{e = D-1} & & \displaystyle{\delta = C} \cr \displaystyle{f = E-1} & & \displaystyle{\varepsilon = D} \cr &\quad\hbox{etc.} \cr \end{array} unde consequenter fiet \begin{equation} x = {\cofrac 1 A {\cofrac {A} {B-1} {\cofrac {B} {C-1} {\cofrac {C} {D-1} {\cofrac {D} {E-1} \tailfrac}}}}} \end{equation}

EXEMPLUM I.

Quoniam, posito \(e\) numero cuius logarithmus est \({}=1\), supra invenimus esse \begin{equation} {1\over e} = 1 - {1\over 1} + {1\over 1\cdot 2} - {1\over 1\cdot 2\cdot 3} + {1\over 1\cdot 2\cdot 3\cdot 4} - \cdots \end{equation} seu \begin{equation} 1 - {1\over e} = {1\over 1} - {1\over 1\cdot 2} + {1\over 1\cdot 2\cdot 3} - {1\over 1\cdot 2\cdot 3\cdot 4} + \cdots \end{equation} haec series in fractionem continuam convertetur ponendo \begin{equation} A=1,\quad B=2,\quad C=3,\quad D=4,\quad \cdots \end{equation} quo ergo facto habebitur \begin{equation} 1 - {1\over e} = {\cofrac 1 1 {\cofrac 1 1 {\cofrac 2 2 {\cofrac 3 3 {\cofrac 4 4 {\cofrac 5 5 \tailfrac}}}}}} \end{equation} unde, asymmetria initio reiecta, erit \begin{equation} {1\over {e-1}} = {\cofrac 1 1 {\cofrac 2 2 {\cofrac 3 3 {\cofrac 4 4 {\cofrac 4 4 {\cofrac 5 5 \tailfrac}}}}}} \end{equation}

EXEMPLUM II.

Invenimus quoque arcus, qui radio aequalis sumitur, cosinum esse \begin{equation} {} = 1 - {1\over 2} + {1\over 2\cdot 12} - {1\over 2\cdot 12\cdot 30} + {1\over 2\cdot 12\cdot 30\cdot 56} - \cdots \end{equation} Si ergo fiat \begin{equation} A=1,\quad B=2,\quad C=12,\quad D=30,\quad E=56, \quad \cdots \end{equation} atque cosinus arcus qui radio aequatur, ponatur \({}=x\), erit \begin{equation} x = {\cofrac 1 1 {\cofrac 1 1 {\cofrac 2 {11} {\cofrac {12} {29} {\cofrac {30} {55} \tailfrac}}}}} \end{equation} seu \begin{equation} { {1\over x} - 1} = {\cofrac 1 1 {\cofrac 2 {11} {\cofrac {12} {29} {\cofrac {30} {55} \tailfrac}}}} \end{equation}

371. Sit series insuper cum geometrica coniuncta, scilicet \begin{equation} x = A - Bz + Cz^2 - Dz^3 + Ez^4 - Fz^5 + \cdots \,, \end{equation} erit \begin{aligned} \alpha &= {Ab} \,,\cr \beta &= {Bbcz \over A - Bz} \,,\cr \gamma &= {ACcdz \over (A-Bz)(B-Cz) } \,,\cr \delta &= {BDdez \over (B-Cz)(C-Dz) } \,,\cr \varepsilon &= {CEefz \over (C-Dz)(D-Ez)} \cr &\hbox{etc.}\cr \end{aligned} Ponatur nunc \begin{array}{lll} \displaystyle{b = 1} & \quad\hbox{erit}\quad & \displaystyle{\alpha = A} \,,\cr \displaystyle{c = A-Bz} & & \displaystyle{\beta = Bz} \,,\cr \displaystyle{d = B-Cz} & & \displaystyle{\gamma = ACz} \,,\cr \displaystyle{e = C-Dz} & & \displaystyle{\delta = BDz} \cr &\quad\hbox{etc.} \cr \end{array} unde fiet \begin{equation} x = {\cofrac {A} {1} {\cofrac {Bz} {A-Bz} {\cofrac {ACz} {B-Cz} {\cofrac {BDz} {C-Dz} \tailfrac}}}} \end{equation}

372. Quo autem hoc negotium generalius absolvamus, ponamus esse \begin{equation} x = {A\over L} - {By\over Mz} + {Cy^2\over Nz^2} - {Dy^3\over Oz^3} + {Ey^4\over Pz^4} - \cdots \end{equation} fietque, comparatione instituta, \begin{aligned} \alpha &= {Ab\over L} \,,\cr \beta &= {BLbcy \over AMz - BLy} \,,\cr \gamma &= {ACM^2cdyz \over (AMz-BLy)(BNz-CMy) } \,,\cr \delta &= {BDN^2deyz \over (BNz-CMy)(COz-DNy) } \,,\cr &\hbox{etc.,}\cr \end{aligned} statuantur valores \(b\), \(c\), \(d\), etc., sequenti modo \begin{array}{lll} \displaystyle{b = L} & \quad\hbox{erit}\quad & \displaystyle{\alpha = A} \,,\cr \displaystyle{c = AMz-BLy} & & \displaystyle{\beta = BLLy} \,,\cr \displaystyle{d = BNz-CMy} & & \displaystyle{\gamma = ACM^2yz} \,,\cr \displaystyle{e = COz-DNy} & & \displaystyle{\delta = BDN^2yz} \,,\cr \displaystyle{f = DPz-EOy} & & \displaystyle{\varepsilon = CDO^2yz} \cr &\quad\hbox{etc.} \cr \end{array} unde series proposita per sequentem factionem continuam exprimetur \begin{equation} x = {\cofrac {A} {L} {\cofrac {BLLy} {AMz-BLy} {\cofrac {ACMMyz} {BNz-CMy} {\cofrac {BDNNyz} {COz-DNy} \tailfrac}}}} \end{equation}

373. Habeat denique series proposita huiusmodi formam \begin{equation} x = {A\over L} - {ABy\over LMz} + {ABCy^2\over LMNz^2} - {ABCDy^3\over LMNOz^3} + \cdots \end{equation} atque sequentes valores prodibunt \begin{aligned} \alpha &= {Ab\over L} \,,\cr \beta &= {Bbcy \over Mz - By} \,,\cr \gamma &= {CMcdyz \over (Mz-By)(Nz-Cy) } \,,\cr \delta &= {DNdeyz \over (Nz-Cy)(Oz-Dy) } \,,\cr \varepsilon &= {EOefyz \over (Oz-Dy)(Pz-Ey) } \,,\cr &\hbox{etc.,}\cr \end{aligned} ad valores ergo integros inveniendos fiat \begin{array}{lll} \displaystyle{b = Lz} & \quad\hbox{erit}\quad & \displaystyle{\alpha = Az} \,,\cr \displaystyle{c = Mz-By} & & \displaystyle{\beta = BLyz} \,,\cr \displaystyle{d = Nz-Cy} & & \displaystyle{\gamma = CMyz} \,,\cr \displaystyle{e = Oz-Dy} & & \displaystyle{\delta = DNyz} \,,\cr \displaystyle{f = Pz-Ey} & & \displaystyle{\varepsilon = EOyz} \cr &\quad\hbox{etc.} \cr \end{array} Unde valor seriei propositae ita exprimetur, ut sit \begin{equation} x = {\cofrac {Az} {Lz} {\cofrac {BLyz} {Mz-By} {\cofrac {CMyz} {Nz-Cy} {\cofrac {DNyz} {Oz-Dy} \tailfrac}}}} \end{equation} Vel, ut lex progressionis statim a principio fiat manifesta, erit \begin{equation} {Az\over x} - Ay = Lz - Ay + \cofrac {BLyz} {Mz-By} {\cofrac {CMyz} {Nz-Cy} {\cofrac {DNyz} {Oz-Dy} \tailfrac}} \end{equation}

374. Hoc modo innumerabiles inveniri poterunt fractiones continuae in infinitum progredientes, quarum valor verus exhiberi queat. Cum enim, ex supra traditis, infinitae series, quarum summae constent, ad hoc negotium accommodari queant, unaquaeque transformari poterit in fractionem continuam, cuius adeo valor summae illius seriei est aequalis. Exempla, quae iam hic sunt allata, sufficiunt ad hunc usum ostendendum; verumtamen optandum esset, ut methodus detegeretur, cuius beneficio, si proposita fuerit fractio continua quaecunque, eius valor immediate inveniri posset. Quanquam enim fractio continua transmutari potest in seriem infinitam, cuius summa per methodos cognitas investigari queat, tamen plerumque istae series tantopere fiunt intricatae, ut earum summa, etiamsi sit satis simplex, vix ac ne vix quidem obtineri possit.

375. Quo autem clarius perspiciatur, dari eiusmodi fractiones continuas, quarum valor aliunde facile assignari queat, etiamsi ex seriebus infinitis, in quas convertuntur, nihil admodum colligere liceat, consideremus hanc fractionem continuam \begin{equation} x = {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 \tailfrac}}}} \end{equation} cuius omnes denominatores sunt inter se aequales; si enim hinc modo supra exposito, fractiones formemus \begin{array}{cccccccc} 0\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} \\ \displaystyle{1 \over 0}\,, & \displaystyle{0 \over 1}\,, & \displaystyle{1\over 2}\,, & \displaystyle{2\over 5}\,, & \displaystyle{5\over 12}\,, & \displaystyle{12\over 29}\,, & \displaystyle{29\over 70}\,, & \hbox{etc.} \end{array} Hinc autem porro oritur haec series \begin{equation} x = 0 + {1\over 2} - {1\over 2\cdot 5} + {1\over 5\cdot 12} - {1\over 12\cdot 29} + {1\over 29\cdot 70} - \cdots \end{equation} vel, si bini termini coniungantur, erit \begin{equation} x = {2\over 1\cdot 5} + {2\over 5\cdot 29} + {2\over 29\cdot 169} + \cdots \end{equation} vel \begin{equation} x = {1\over 2} - {2\over 2\cdot 12} - {2\over 12\cdot 70} - \ldots \end{equation} Quin etiam, cum sit \begin{align} x &= {1\over 4} - {1\over 2\cdot 2\cdot 5} + {1\over 2\cdot 5\cdot 12} - {1\over 2\cdot 12\cdot 29} + \ldots \\ &+ {1\over 4} - {1\over 2\cdot 2\cdot 5} + {1\over 2\cdot 5\cdot 12} - {1\over 2\cdot 12\cdot 29} + \ldots \end{align} erit \begin{equation} x = {1\over 4} + {1\over 1\cdot 5} - {1\over 2\cdot 12} + {1\over 5\cdot 29} - {1\over 12\cdot 70} + \ldots \end{equation} quae series etiamsi vehementer convergant, tamen vera earum summa ex earum forma colligi nequit.

376. Pro huiusmodi autem fractionibus continuis, in quibus denominatores omnes vel sunt aequales, vel iidem revertuntur; ita ut ea fractio, si ab initio aliquot terminis truncetur, toti adhuc sit aequalis, facilis habetur modus earum summas explorandi. In exemplo enim proposito, cum sit \begin{equation} x = {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 \tailfrac}}}} \end{equation} erit \begin{equation} x = {1\over 2+x}\,, \end{equation} ideoque \begin{equation} xx + 2x = 1 \,, \end{equation} et \begin{equation} x + 1 = \sqrt 2 \,; \end{equation} ita ut valor huius fractionis continuae sit \begin{equation} {} = \sqrt 2 - 1\,. \end{equation} Fractiones vero ex fractione continua ante erutae, continuo propius ad hunc valorem accedunt, idque tam cito, ut vix promptior modus ad valorem hunc irrationalem per numeros rationales proxime exprimendum, inveniri queat. Est enim \(\sqrt{2}-1\) tam prope \({}={29\over 70}\), ut error sit insensibilis : namque, radicem extrahendo, erit \begin{equation} \sqrt{2}-1 = 0.41421356236\,, \end{equation} atque \begin{equation} {29\over 70} = 0.41428571428\,, \end{equation} ita ut error tantum in partibus centesimis millesimis consistat.

377. Quemadmodum ergo fractiones continuae commodissimum suppeditant modum ad valorem \(\sqrt 2\) appropinquandi, ita indidem facillima via aperitur ad radices aliorum numerorum proxime investigandas. Ponamus hunc in finem \begin{equation} x = {\cofrac 1 a {\cofrac 1 a {\cofrac 1 a {\cofrac 1 a {\cofrac 1 a \tailfrac}}}}} \end{equation} erit \begin{equation} x = {1\over a+x} \end{equation} et \begin{equation} xx + ax = 1 \,, \end{equation} unde fit \begin{equation} x = -{1\over 2}a + \sqrt{1+{1\over 4}aa} = {\sqrt{aa+4}-a\over 2} \,. \end{equation} Haec ergo fractio continua inserviet valori radicis quadratae ex numero \(aa+4\) inveniendo. Hincque adeo substituendo loco \(a\) successive numeros \(1\), \(2\), \(3\), \(4\), etc., reperientur \(\sqrt 5\), \(\sqrt 2\), \(\sqrt{13}\), \(\sqrt 5\), \(\sqrt{29}\), \(\sqrt{10}\), \(\sqrt{53}\), etc., perductis scilicet his radicibus ad formam simplicissimam. Erit ergo \begin{array}{cccccc} 1\phantom{\,,} & 1\phantom{\,,} & 1\phantom{\,,} & 1\phantom{\,,} & 1\phantom{\,,} & 1\phantom{\,,} \\ \displaystyle{0\over 1}\,, & \displaystyle{1\over 1}\,, & \displaystyle{1\over 2}\,, & \displaystyle{2\over 3}\,, & \displaystyle{3\over 5}\,, & \displaystyle{5\over 8}\,, & \hbox{etc.} = \displaystyle{\sqrt 5 - 1\over 2} \,, \\ \\ 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} & 2\phantom{\,,} \\ \displaystyle{0\over 1}\,, & \displaystyle{1\over 2}\,, & \displaystyle{2\over 5}\,, & \displaystyle{5\over 12}\,, & \displaystyle{12\over 29}\,, & \displaystyle{29\over 70}\,, & \hbox{etc.} = \displaystyle{\sqrt 2 - 1} \,, \\ \\ 3\phantom{\,,} & 3\phantom{\,,} & 3\phantom{\,,} & 3\phantom{\,,} & 3\phantom{\,,} & 3\phantom{\,,} \\ \displaystyle{0\over 1}\,, & \displaystyle{1\over 3}\,, & \displaystyle{3\over 10}\,, & \displaystyle{10\over 33}\,, & \displaystyle{33\over 109}\,, & \displaystyle{109\over 360}\,, & \hbox{etc.} = \displaystyle{\sqrt{13} - 3\over 2} \,, \\ \\ 4\phantom{\,,} & 4\phantom{\,,} & 4\phantom{\,,} & 4\phantom{\,,} & 4\phantom{\,,} & 4\phantom{\,,} \\ \displaystyle{0\over 1}\,, & \displaystyle{1\over 4}\,, & \displaystyle{4\over 17}\,, & \displaystyle{17\over 72}\,, & \displaystyle{72\over 305}\,, & \displaystyle{305\over 1292}\,, & \hbox{etc.} = \displaystyle{\sqrt 5 - 2} \,, \\ &&&\quad\hbox{etc.}\\ \end{array} Notandum autem eo promptiorem esse approximationem, quo maior fuerit numeris \(a\). Sic in ultimo exemplo erit \begin{equation} \sqrt 5 = 2 {305\over 1292}\,, \end{equation} ut error minor sit quam \(1\over 1292\cdot 5473\), ubi \(5473\) est denominator sequentis fractionis \({1292\over 5473}\).

378. Hoc vero modo aliorum numerorum radices exhiberi nequeunt, nisi qui sint summa duorum quadratorum. Ut igitur haec approximatio ad alios numeros extendatur, ponamus esse \begin{equation} x = {\cofrac 1 a {\cofrac 1 b {\cofrac 1 a {\cofrac 1 b {\cofrac 1 a {\cofrac 1 b \tailfrac}}}}}} \end{equation} erit \begin{equation} x = {\cofrac 1 a {\displaystyle{1\over b+x}}} = {b+x\over ab+1+ax}\,; \end{equation} ideoque \begin{equation} axx+abx=b \end{equation} et \begin{equation} x = -{1\over 2} b\pm \sqrt{ {1\over 4}bb + {b\over a}} = {-ab+\sqrt{aabb+4ab}\over 2a} \,. \end{equation}

Unde iam omnium numerorum radices inveniri poterunt. Sit, verbi gratia \(a=2\), \(b=7\); erit \begin{equation} x = {-14+\sqrt{14\cdot 18}\over 4} = {-7+3\sqrt 7 \over 2}\,; \end{equation} at valorem ipsius \(x\) proxime exhibebunt sequentes fractiones \begin{array}{cccccc} 2\phantom{\,,} & 7\phantom{\,,} & 2\phantom{\,,} & 7\phantom{\,,} & 2\phantom{\,,} & 7\phantom{\,,} \\ \displaystyle{0\over 1}\,, & \displaystyle{1\over 2}\,, & \displaystyle{7\over 15}\,, & \displaystyle{15\over 32}\,, & \displaystyle{112\over 239}\,, & \displaystyle{239\over 510}\,, & \hbox{etc.} \end{array} Erit ergo proxime \begin{equation} {-7+3\sqrt 7\over 2}={239\over 510} \end{equation} et \begin{equation} \sqrt 7 = {2024\over 765} = 2.645\,751\,6 \,; \end{equation} at revera est \begin{equation} \sqrt 7 = 2.645\,751\,31 \,; \end{equation} ita ut error minor sit quam \(3\over 10\,000\,000\).

379. Progrediamur autem ulterius ponendo \begin{equation} x = {\cofrac 1 a {\cofrac 1 b {\cofrac 1 c {\cofrac 1 a {\cofrac 1 b {\cofrac 1 c {\cofrac 1 a \tailfrac}}}}}}} \end{equation} erit \begin{equation} x = {\cofrac 1 a {\cofrac 1 b {\displaystyle{1\over c+x}}}} = {\cofrac 1 a {\displaystyle {c+x\over bx+bc+1}} } = {bx+bc+1 \over (ab+1)x+abc+a+c}\,, \end{equation} unde \begin{equation} (ab+1)xx + (abc+a-b+c)x = bc+1 \,, \end{equation} atque \begin{equation} x = {-abc-a+b-c+\sqrt{(abc+a+b+c)^2+4} \over 2(ab+1)} \,; \end{equation} ubi quantitas post signum radicale posita iterum est summa duorum quadratorum, neque ergo haec forma radicibus ex aliis numeris extrahendis inservit, nisi ad quos prima forma iam suffecerat. Simili modo si quatuor litterae \(a\), \(b\), \(c\), \(d\), continuo repetitae denominatores fractionis continuae constituant, tum ea plus non inserviet quam secunda, quae duas tantum litteras continebat, et ita porro.

380. Cum igitur fractiones continuae tam utiliter ad extractionem radicis quadratae adhiberi queant, simul inservient aequationibus quadratis resolvendis; quod quidem ex ipso calculo est manifestum, dum \(x\) per aequationem quadraticam affectam determinatur. Potest autem vicissim facile cuiusque aequationis quadratae radix per fractionem continuam hoc modo exprimi. Sit proposita ista aequatio \begin{equation} xx=ax+b\,, \end{equation} ex qua, cum sit \begin{equation} x=a+{b\over x}\,, \end{equation} substituatur in ultimo termino loco \(x\) valor idem iam inventus, eritque \begin{equation} x=a+{\cofrac b a {\dfrac{b}{x}}} \end{equation} simili ergo modo procedendo, erit per fractionem continuam infinitam \begin{equation} x = a+{\cofrac b a {\cofrac b a {\cofrac b a \ldots}} } \end{equation} quae autem, cum numeratores \(b\) non sint unitates, non tam commode adhiberi potest.

381. Ut autem usus in arithmetica ostendatur, primum notandum est omnem fractionem ordinariam in fractionem continuam converti posse. Sit enim proposita fractio \begin{equation} x={A\over B}\,, \end{equation} in qua sit \(A > B\); dividatur \(A\) per \(B\), sitque quotus \({}=a\) et residuum \(C\); tum per hoc residuum \(C\) dividatur praecedens divisor \(B\), prodeatque quotus \(b\) et relinquatur residuum \(D\), per quod denuo praecedens divisor \(C\) dividatur; sicque haec operatio, quae vulgo ad maximum communem divisorem numerorum \(A\) et \(B\) investigandum usurpari solet, continuetur, donec ipsa finiatur; sequenti modo

B) A (a
   —
   C) B (b
      —
      D) C (c
         —
         E) D (d
            —
            F  etc.

eritque per naturam divisionis \begin{array}{lll} \displaystyle{A=\alpha B+C}\,, & \quad\hbox{unde}\quad & \displaystyle{ {A\over B}=a+{C\over B }}\,, \phantom{\,,\quad \displaystyle{ {C\over B}= {1\over b+{D\over C}}}}\cr \displaystyle{B=bC+D}\,, & & \displaystyle{ {B\over C}=b+{D\over C}}\,,\quad \displaystyle{ {C\over B}= {1\over b+{D\over C}}} \,, \cr \displaystyle{C=cD+E}\,, & & \displaystyle{ {C\over D}=c+{E\over D}}\,,\quad \displaystyle{ {D\over C}= {1\over c+{E\over D}}} \,, \cr \displaystyle{D=dE+F}\,, & & \displaystyle{ {D\over E}=d+{F\over E}}\,,\quad \displaystyle{ {E\over D}= {1\over d+{F\over E}}} \,, \cr &\quad\hbox{etc.} \cr \end{array}

hinc, sequentes valores in praecedentibus substituendo, erit \begin{equation} x = {A\over B} = a + {C\over B} = a + {\cofrac 1 b { {D\over C}}} = a + {\cofrac 1 b {\cofrac 1 c { {E\over D}}}} \end{equation} unde tandem \(x\) per meros quotos inventos \(a\), \(b\), \(c\), \(d\), etc., sequentem in modum exprimetur, ut sit \begin{equation} x = {a + {\cofrac 1 b {\cofrac 1 c {\cofrac 1 d {\cofrac 1 e {\cofrac 1 f \tailfrac }}}}}} \end{equation}

EXEMPLUM I.

Sit proposita ista fractio \(1461\over 59\), quae sequenti modo in fractionem continuam transmutabitur, cuius omnes numeratores erunt unitates. Instituatur scilicet eadem operatio, qua maximus communis divisor numerorum \(59\) et \(1461\) quaeri solet.

59) 1461 (24
    118
    ————
     281
     236
     ———
      45) 59 (1
          45
          ——
          14) 45 (3
              42
              ——
               3) 14 (4
                  12
                  ——
                   2) 3 (1
                      2
                      —
                      1) 2 (2
                         2
                         —
                         0

Hinc ergo ex quotis fiet \begin{equation} {1461\over 59} = {24 + {\cofrac 1 1 {\cofrac 1 3 {\cofrac 1 4 {\cofrac 1 1 {\displaystyle {1\over 2}} }}}}} \end{equation}

EXEMPLUM II.

Fractiones quoque decimales eodem modo transmutari poterunt; sit enim proposita \begin{equation} \sqrt 2 = 1.41421356 = {141\,421\,356\over 100\,000\,000}\,, \end{equation} unde haec operatio instituatur \begin{array}{r|r|r} 100\,000\,000 & 141\,421\,356 & 1 \cr 82\,842\,712 & 100\,000\,000 & 2 \cr \hline 17\,157\,288 & 41\,421\,356 & 2 \cr 14\,213\,560 & 34\,314\,576 & 2 \cr \hline 2\,943\,728 & 7\,106\,780 & 2 \cr 2\,438\,648 & 5\,887\,456 & 2 \cr \hline 505\,080 & 1\,219\,324 & 2 \cr 418\,728 & 1\,010\,160 & 2\cr \hline \hbox{etc.}\;\; & 209\,364 \end{array} Ex qua operatione iam patet omnes denominatores esse \(2\), atque adeo esse \begin{equation} \sqrt 2 = {1 + {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 {\cofrac 1 2 \tailfrac }}}}} \end{equation} cuius expressionis ratio iam ex superioribus patet.

EXEMPLUM III.

Imprimis vero etiam hic attentione dignus est numerus \(e\), cuius logarithmus est \({}= 1\), qui est \begin{equation} e=2.718281828459 \,, \end{equation} unde oritur \begin{equation} {e-1\over2} = 0.8591409142295 \,, \end{equation} quae fractio decimalis, si superiori modo tractetur, dabit quotos sequentes \begin{array}{r|r|r} 8\,591\,409\,142\,295 & 10\,000\,000\,000\,000 & 1 \cr 8\,451\,545\,146\,224 & 8\,591\,409\,142\,295 & 6 \cr \hline 139\,863\,996\,071 & 1\,408\,590\,857\,704 & 10 \cr 139\,312\,557\,916 & 1\,398\,639\,960\,710 & 14 \cr \hline 551\,438\,155 & 9\,950\,896\,994 & 18 \cr 550\,224\,488 & 9\,925\,886\,790 & 22 \cr \hline 1\,213\,667 & 25\,010\,204 & \quad\hbox{etc.}\cr \end{array} si iste calculus exactius adhuc, assumpto valore ipsius \(e\), ulterius continuetur, tum prodibunt isti quoti \begin{equation} 1,\quad 6,\quad 10,\quad 14,\quad 18,\quad 22,\quad 26,\quad 30,\quad 34,\quad \cdots \end{equation} qui, dempto primo, progressionem arithmeticam constituunt, unde patet fore \begin{equation} {e-1\over 2} = {\cofrac 1 1 {\cofrac 1 6 {\cofrac 1 {10} {\cofrac 1 {14} {\cofrac 1 {18} {\cofrac 1 {22} \tailfrac }}}}}} \end{equation} cuius fractionis ratio ex calculo infinitesimale dari potest.

382. Cum igitur ex huiusmodi expressionibus fractiones erui queant, quae quam citissime ad verum valorem expressionis deducant, haec methodus adhiberi poterit ad fractiones decimales per ordinarias fractiones, quae ad ipsas proxime accedant, exprimendas. Quin etiam, si fractio fuerit proposita, cuius numerator et denominator sint numeri valde magni, fractiones ex minoribus numeris constantes inveniri poterunt quae, etiamsi propositae non sint penitus aequales, tamen ab ea quam minime discrepent. Hincque problema a Wallisio olim tractatum facile resolvi potest, quo quaeruntur fractiones minoribus numeris expressae, quae tam prope exhauriant valorem fractionis cuiuspiam in numeris maioribus propositae, quantum fieri poterit numeris non maioribus. Fractiones autem nostra hac methodo ortae tam prope ad valorem fractionis continuae, ex qua eliciuntur, accedunt, ut nullae numeris non maioribus constantes dentur, quae propius accedant.

EXEMPLUM I.

Exprimatur ratio diametri ad peripheriam numeris tam exiguis, ut accuratior exhiberi nequeat, nisi numeri maiores adhibeantur. Si fractio decimalis cognita \begin{equation} 3.1415926535\cdots \end{equation} modo exposito per divisionem continuam evolvatur, reperientur sequentes quoti \begin{equation} 3,\quad 7,\quad 15,\quad 1,\quad 292,\quad 1,\quad 1,\quad \cdots \,, \end{equation} ex quibus sequentes fractiones formabuntur \begin{equation} {1\over 0},\quad {3\over 1},\quad {22\over 7},\quad {333\over 106},\quad {355\over 113},\quad {103993\over 33102},\quad \cdots \,, \end{equation} secunda fractio iam ostendit esse diametrum ad peripheriam ut \(1:3\), neque certe numeris non maioribus accuratius dari poterit. Tertia fractio dat rationem Archimedeam \(7:22\), at quinta Metianam, quae ad verum tam prope accedit, ut error minor sit parte \({1\over 113\cdot 33102}\). Ceterum hae fractiones alternatim vero sunt maiores minoresque.

EXEMPLUM II.

Exprimatur ratio diei ad annum solarem medium in numeris minimis proxime. Cum annus iste sit \(\,365^d\,5^h\,48'\,55''\), continebit in fractione annus unus \begin{equation} 365 {\textstyle{20935\over 86400}} \end{equation} dies. Tantum ergo opus est ut haec fractio evolvatur, quae dabit sequentes quotos \begin{equation} 4,\quad 7,\quad 1,\quad 6,\quad 1,\quad 2,\quad 2,\quad 4 \end{equation} unde istae eliciuntur fractiones \begin{equation} {0\over 1},\quad {1\over 4},\quad {7\over 29},\quad {8\over 33},\quad {55\over 227},\quad {63\over 260},\quad {181\over 747},\quad \cdots \end{equation} Horae ergo cum minutis primis et secundis, quae supra \(365\) dies adsunt, quatuor annis unum diem circiter faciunt, unde calendarium Julianum originem habet. Exactius autem \(33\) annis \(8\) dies implentur, vel \(747\) annis \(181\) dies; unde sequitur quadringentis annis abundare \(97\) dies. Quare, cum hoc intervallo calendarium Julianum inserat \(100\) dies, Gregorianum quaternis seculis tres annos bissextiles in communes convertit.