Cardano, Girolamo
,
De subtilitate
,
1663
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generis quantitatibus. </
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<
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">Indicio autem pa
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ralogiſmi eſt falſi experimentum in conclu
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ſione vel mediis, aut deprehenſio alicuius de
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fectus ex his dictis. </
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<
s
id
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">Sed maior parologiſmus
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ac difficilior oritur in diuerſo genere: velut
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capio circulum ABC, & ducta CDA per
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centrum, & CE CK, CB æqualiter altera ab
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altera diſtantibus,
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itemq;
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CG ab altera parte
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tunc anguli CBL, CKL, CEL, CAL perife
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ria, & recta contenti ſunt acuti: quod facilè
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demonſtratur ductis gratia exempli D E ex
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centro, & EF contingente, tunc FED rectus
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eſt: ſed CEL minor DEF, angulo DEC, &
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angulo contactus FEK, igitur CEL eſt acu
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tus: ſimiliter CAL acutus eſt, ex demonſtra
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tis ab Euclide in 3 libro, & tamen maior
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CEL, vt CEL maior CKL, & CKL eſt ma
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ior CBL: quod patet, quia deficiunt omnes
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angulis contactus, qui ſunt æquales, vt à no
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bis demonſtratum eſt in tertio noſtrorum
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Elementorum: & deficiunt, etiam angulo re
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ctilineo contento à ſemidiametro, & lineis
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CB, CK, CE, qui vt magis remouentur di
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ctæ lineæ à linea CA, ſemper ſunt maiores:
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ergo ex communi animi ſententia angulus
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CBL minor eſt angulo CKL, & angulus
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CKL angulo CEL, & CEL angulo CAL.
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</
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<
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id
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">Sed angulus CGA maior eſt recto: ducta
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enim GH contangente, & DG ſemidiame
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tro, fiet DGH rectus ex demonſtratis ab
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Euclide (vt dixi) in tertio Elementorum, &
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angulus DGC eſt maior angulo contactus,
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vt ibi infertur pro corollario: igitur detra
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cto ab angulo DGH recto angulo contactus
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& addito DGC, cum fiat angulus CGL
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erit ex communi animi ſententia CGL ob
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tuſus. </
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<
s
id
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s.011076
">Igitur linea CB tranſeunte ſenſim ex
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B vſque ad G, anguli periferia, & recta con
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tenti ſemper augebuntur, & ſenſim & per
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omne genus magnitudinis vſque ad obtu
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ſum, vt patet: & tamen nunquam fiet rectus,
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vt demonſtratum eſt, quia in A, & ante A
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ſemper eſt acutus, poſt A obtuſus: igitur pa
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tet intentum. </
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<
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id
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s.011077
">Maior hæc fit paralogiſmus in
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diuerſo genere, & talis eſt. </
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>
<
s
id
="
s.011078
">Aliqua quantitas
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continuè augetur plus, quàm ad duplum, vel
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ſaltem ad duplum, donec perueniat ad lon
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gè maiorem quantitatem alia, vtpote cen
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tuplo maiorem, & tamen antequam perue
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niat ad illam extremam quantitatem, nun
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quam fit æqualis, aut maior illa minore
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quantitate. </
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>
<
s
id
="
s.011079
">Et hoc videtur impoſſibile dua
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bus de cauſis. </
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<
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id
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">Prima, quoniam oporteret, vt
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in vltimo argumento augeretur non æquali
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ter, id eſt ad duplum, ſed magis, quàm cen
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tuplo. </
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<
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id
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">Secunda, quia cùm quantitas illa mi
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nor non poſſet excedere minimam illam
<
expan
abbr
="
aliã
">aliam</
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maiore magnitudine, quàm ipſa ſit, oportet
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vt illa minima ad duplum creſcente tandem
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ſuperet hanc quantitatem, & tamen non ſu
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perat. </
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>
<
s
id
="
s.011082
">Imò ſequitur maius miraculum, & eſt
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quòd accipio duas quantitates, quæ parum
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magnitudine diſtant, & tamen maiore per
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petuò ad duplum aucta vſque infinitum, &
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maiore ſemper diuiſa per medium in inſini
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tum, illa minor aucta nunquam excedet ali
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quam partem huius maioris per medium di
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uiſæ. </
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<
s
id
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s.011083
">Oſtendo autem omnia hæc demonſtra
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tione vna. </
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<
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id
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">Capio exiguum aliquem angu
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lum, qui ſit K, rectilineum tamen, quem con
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ſtat in infinitum per æqualia diuidi poſſe, &
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hoc facilè fit ſemper magis producendo la
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tera, vt acutior per diuiſionem continuam
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fiat angulus: nam baſis eo ſemper maior fiet,
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ideòque baſes angulorum poterunt ad ean
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dem peruenire magnitudinem: & tunc ducta
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linea ex loco diuiſionis baſis ad angulum, ſi
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baſis erit per æqualia diuiſa, erit etiam an
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gulus. </
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<
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id
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">Inde capio tres circulos AB, AC, AD
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in continua proportione quacunque volue
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ris, ſe contangentes in puncto A, & ex de
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monſtratis in 3. Element. </
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<
s
id
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">ab Euclide illo
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rum centra erunt in vna diametro, quæ ſit
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AEFG, & tunc certum eſt, quòd angulus B
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AD eſt maior angulo BAC & CAD ſeor
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ſum ſumpto: nam totum eſt maius ſua parte:
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vel igitur angulus BAC eſt æqualis angulo
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CAD, & tunc angulus BAD erit duplus an
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gulo B A C: vel angulus B A C eſt maior
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CAD: eritque angulus BAD plus, quam du
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plua angulo CAD. </
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<
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id
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">Vel ſi ponatur angulus
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C A D, maior angulo BAC, erit angulus
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BAD, maior duplo anguli B A C. </
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<
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id
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">Conſtat
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igitur quod neceſſarium eſt, quòd angulus
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BAD ſit duplus, aut duplo maior altero an
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gulorum BAC vel CAD. </
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<
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id
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">Sit igitur duplus
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vel maior duplo, gratia exempli, angulo
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BAC (nam hoc eſt verum ) tunc capio duos
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angulos BAC & K: dico igitur quòd ſemper
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duplicato angulo BAC, & diuiſo angulo K
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quouſque velis, etiamſi in infinitum proce
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das, nunquam tantùm BAC excreſcere po
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terit, vt minimam partem anguli K vel
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æquet, vel ſuperet, cum tamen differentia il
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lorum angulorum minima ſit cùm iam an
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guli ipſi minimi ſint, vtpote pars milleſima
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K. </
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<
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">Nam inſcriptis circulis ſemper eadem ra
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tione continua minoribus, quam DA ſe ha
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bet ad BA, duplicabitur interior angulus,
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qui fiet ex circumferentiæ parte conuexa in
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terioris cum concaua circuli AB periferiæ,
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& hoc donec perueniat ad anguli eius ma
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gnitudinem, qui periferia continetur duo
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bus rectis, ſolùm eò minorem, quò ſunt duo
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anguli contactus: augeatur enim quantum
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libet circuli interioris paruitate, & ducatur
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contingens A H maiorem circulum, quæ
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etiam neceſſariò continget minorem, quia
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vt demonſtratum eſt, diameter circuli maio
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ris eſt idem cum diametro minoris. </
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<
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id
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">Si igitur
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fingamus AH eſſe latus partis vnius anguli
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K quantumcunque minimæ, reliquum latus
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neceſſariò cadet infra periferiam circuli mi
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noris, aliter inter contingentem AH, & cir
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culum minorem recta cadere poſſet, contra
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demonſtrata ab Euclide in 3. libro. </
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<
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id
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">Igitur ſi
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recta cadit infra periferiam circuli minoris
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fiet angulus contactus circulorum pars an
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anguli à rectis contenti: ergo cùm pars </
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