Cardano, Girolamo, De subtilitate, 1663

Table of figures

< >
< >
page |< < of 403 > >|
    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="s.011072">
                <pb pagenum="600" xlink:href="016/01/247.jpg"/>
              generis quantitatibus. </s>
              <s id="s.011073">Indicio autem pa­
                <lb/>
              ralogiſmi eſt falſi experimentum in conclu­
                <lb/>
              ſione vel mediis, aut deprehenſio alicuius de­
                <lb/>
              fectus ex his dictis. </s>
              <s id="s.011074">Sed maior parologiſmus
                <lb/>
              ac difficilior oritur in diuerſo genere: velut
                <lb/>
              capio circulum ABC, & ducta CDA per
                <lb/>
              centrum, & CE CK, CB æqualiter altera ab
                <lb/>
              altera diſtantibus,
                <expan abbr="itemq;">itemque</expan>
              CG ab altera parte
                <lb/>
              tunc anguli CBL, CKL, CEL, CAL perife­
                <lb/>
              ria, & recta contenti ſunt acuti: quod facilè
                <lb/>
              demonſtratur ductis gratia exempli D E ex
                <lb/>
              centro, & EF contingente, tunc FED rectus
                <lb/>
              eſt: ſed CEL minor DEF, angulo DEC, &
                <lb/>
              angulo contactus FEK, igitur CEL eſt acu­
                <lb/>
              tus: ſimiliter CAL acutus eſt, ex demonſtra­
                <lb/>
              tis ab Euclide in 3 libro, & tamen maior
                <lb/>
              CEL, vt CEL maior CKL, & CKL eſt ma­
                <lb/>
              ior CBL: quod patet, quia deficiunt omnes
                <lb/>
              angulis contactus, qui ſunt æquales, vt à no­
                <lb/>
              bis demonſtratum eſt in tertio noſtrorum
                <lb/>
              Elementorum: & deficiunt, etiam angulo re­
                <lb/>
              ctilineo contento à ſemidiametro, & lineis
                <lb/>
              CB, CK, CE, qui vt magis remouentur di­
                <lb/>
              ctæ lineæ à linea CA, ſemper ſunt maiores:
                <lb/>
              ergo ex communi animi ſententia angulus
                <lb/>
              CBL minor eſt angulo CKL, & angulus
                <lb/>
              CKL angulo CEL, & CEL angulo CAL.
                <lb/>
              </s>
              <s id="s.011075">Sed angulus CGA maior eſt recto: ducta
                <lb/>
              enim GH contangente, & DG ſemidiame­
                <lb/>
              tro, fiet DGH rectus ex demonſtratis ab
                <lb/>
              Euclide (vt dixi) in tertio Elementorum, &
                <lb/>
              angulus DGC eſt maior angulo contactus,
                <lb/>
              vt ibi infertur pro corollario: igitur detra­
                <lb/>
              cto ab angulo DGH recto angulo contactus
                <lb/>
                <figure id="id.016.01.247.1.jpg" xlink:href="016/01/247/1.jpg" number="107"/>
                <lb/>
              & addito DGC, cum fiat angulus CGL
                <lb/>
              erit ex communi animi ſententia CGL ob­
                <lb/>
              tuſus. </s>
              <s id="s.011076">Igitur linea CB tranſeunte ſenſim ex
                <lb/>
              B vſque ad G, anguli periferia, & recta con­
                <lb/>
              tenti ſemper augebuntur, & ſenſim & per
                <lb/>
              omne genus magnitudinis vſque ad obtu­
                <lb/>
              ſum, vt patet: & tamen nunquam fiet rectus,
                <lb/>
              vt demonſtratum eſt, quia in A, & ante A
                <lb/>
              ſemper eſt acutus, poſt A obtuſus: igitur pa­
                <lb/>
                <arrow.to.target n="marg1553"/>
                <lb/>
              tet intentum. </s>
              <s id="s.011077">Maior hæc fit paralogiſmus in
                <lb/>
              diuerſo genere, & talis eſt. </s>
              <s id="s.011078">Aliqua quantitas
                <lb/>
              continuè augetur plus, quàm ad duplum, vel
                <lb/>
              ſaltem ad duplum, donec perueniat ad lon­
                <lb/>
              gè maiorem quantitatem alia, vtpote cen­
                <lb/>
              tuplo maiorem, & tamen antequam perue­
                <lb/>
              niat ad illam extremam quantitatem, nun­
                <lb/>
              quam fit æqualis, aut maior illa minore
                <lb/>
              quantitate. </s>
              <s id="s.011079">Et hoc videtur impoſſibile dua­
                <lb/>
              bus de cauſis. </s>
              <s id="s.011080">Prima, quoniam oporteret, vt
                <lb/>
              in vltimo argumento augeretur non æquali­
                <lb/>
              ter, id eſt ad duplum, ſed magis, quàm cen­
                <lb/>
              tuplo. </s>
              <s id="s.011081">Secunda, quia cùm quantitas illa mi­
                <lb/>
              nor non poſſet excedere minimam illam
                <expan abbr="aliã">aliam</expan>
                <lb/>
              maiore magnitudine, quàm ipſa ſit, oportet
                <lb/>
              vt illa minima ad duplum creſcente tandem
                <lb/>
              ſuperet hanc quantitatem, & tamen non ſu­
                <lb/>
              perat. </s>
              <s id="s.011082">Imò ſequitur maius miraculum, & eſt
                <lb/>
              quòd accipio duas quantitates, quæ parum
                <lb/>
              magnitudine diſtant, & tamen maiore per­
                <lb/>
              petuò ad duplum aucta vſque infinitum, &
                <lb/>
              maiore ſemper diuiſa per medium in inſini­
                <lb/>
              tum, illa minor aucta nunquam excedet ali­
                <lb/>
              quam partem huius maioris per medium di­
                <lb/>
              uiſæ. </s>
              <s id="s.011083">Oſtendo autem omnia hæc demonſtra­
                <lb/>
              tione vna. </s>
              <s id="s.011084">Capio exiguum aliquem angu­
                <lb/>
              lum, qui ſit K, rectilineum tamen, quem con­
                <lb/>
              ſtat in infinitum per æqualia diuidi poſſe, &
                <lb/>
              hoc facilè fit ſemper magis producendo la­
                <lb/>
              tera, vt acutior per diuiſionem continuam
                <lb/>
              fiat angulus: nam baſis eo ſemper maior fiet,
                <lb/>
              ideòque baſes angulorum poterunt ad ean­
                <lb/>
              dem peruenire magnitudinem: & tunc ducta
                <lb/>
              linea ex loco diuiſionis baſis ad angulum, ſi
                <lb/>
              baſis erit per æqualia diuiſa, erit etiam an­
                <lb/>
              gulus. </s>
              <s id="s.011085">Inde capio tres circulos AB, AC, AD
                <lb/>
              in continua proportione quacunque volue­
                <lb/>
              ris, ſe contangentes in puncto A, & ex de­
                <lb/>
              monſtratis in 3. Element. </s>
              <s id="s.011086">ab Euclide illo­
                <lb/>
              rum centra erunt in vna diametro, quæ ſit
                <lb/>
              AEFG, & tunc certum eſt, quòd angulus B
                <lb/>
              AD eſt maior angulo BAC & CAD ſeor­
                <lb/>
              ſum ſumpto: nam totum eſt maius ſua parte:
                <lb/>
              vel igitur angulus BAC eſt æqualis angulo
                <lb/>
              CAD, & tunc angulus BAD erit duplus an­
                <lb/>
              gulo B A C: vel angulus B A C eſt maior
                <lb/>
              CAD: eritque angulus BAD plus, quam du­
                <lb/>
              plua angulo CAD. </s>
              <s id="s.011087">Vel ſi ponatur angulus
                <lb/>
              C A D, maior angulo BAC, erit angulus
                <lb/>
              BAD, maior duplo anguli B A C. </s>
              <s id="s.011088">Conſtat
                <lb/>
              igitur quod neceſſarium eſt, quòd angulus
                <lb/>
              BAD ſit duplus, aut duplo maior altero an­
                <lb/>
              gulorum BAC vel CAD. </s>
              <s id="s.011089">Sit igitur duplus
                <lb/>
              vel maior duplo, gratia exempli, angulo
                <lb/>
              BAC (nam hoc eſt verum ) tunc capio duos
                <lb/>
              angulos BAC & K: dico igitur quòd ſemper
                <lb/>
              duplicato angulo BAC, & diuiſo angulo K
                <lb/>
              quouſque velis, etiamſi in infinitum proce­
                <lb/>
              das, nunquam tantùm BAC excreſcere po­
                <lb/>
              terit, vt minimam partem anguli K vel
                <lb/>
              æquet, vel ſuperet, cum tamen differentia il­
                <lb/>
              lorum angulorum minima ſit cùm iam an­
                <lb/>
              guli ipſi minimi ſint, vtpote pars milleſima
                <lb/>
              K. </s>
              <s id="s.011090">Nam inſcriptis circulis ſemper eadem ra­
                <lb/>
              tione continua minoribus, quam DA ſe ha­
                <lb/>
              bet ad BA, duplicabitur interior angulus,
                <lb/>
              qui fiet ex circumferentiæ parte conuexa in­
                <lb/>
              terioris cum concaua circuli AB periferiæ,
                <lb/>
              & hoc donec perueniat ad anguli eius ma­
                <lb/>
              gnitudinem, qui periferia continetur duo­
                <lb/>
              bus rectis, ſolùm eò minorem, quò ſunt duo
                <lb/>
              anguli contactus: augeatur enim quantum
                <lb/>
              libet circuli interioris paruitate, & ducatur
                <lb/>
              contingens A H maiorem circulum, quæ
                <lb/>
              etiam neceſſariò continget minorem, quia
                <lb/>
              vt demonſtratum eſt, diameter circuli maio­
                <lb/>
              ris eſt idem cum diametro minoris. </s>
              <s id="s.011091">Si igitur
                <lb/>
              fingamus AH eſſe latus partis vnius anguli
                <lb/>
              K quantumcunque minimæ, reliquum latus
                <lb/>
              neceſſariò cadet infra periferiam circuli mi­
                <lb/>
              noris, aliter inter contingentem AH, & cir­
                <lb/>
              culum minorem recta cadere poſſet, contra
                <lb/>
              demonſtrata ab Euclide in 3. libro. </s>
              <s id="s.011092">Igitur ſi
                <lb/>
              recta cadit infra periferiam circuli minoris
                <lb/>
              fiet angulus contactus circulorum pars an­
                <lb/>
              anguli à rectis contenti: ergo cùm pars </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum