Cardano, Girolamo, De subtilitate, 1663

Table of figures

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            <p type="main">
              <s id="s.011006">
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              partis eius ad eorum eandem altitudinem
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              & baſim habentem, eſt velut dimidij axis
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              cum reliquæ portionis axe, ad eandem re­
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              liquam axis portionem alterius partis ſphæ­
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              roidis. </s>
              <s id="s.011007">Ex quo patet quartum, quod cùm
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              ſphæroides, & portio ſphæræ altitudinem
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              & baſim eandem, aut æquales retinuerint,
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              ipſæ inuicem erunt æquales.
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                <arrow.to.target n="marg1546"/>
              </s>
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            <p type="margin">
              <s id="s.011008">
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              Sphæroidis
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              priuilegia
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              duo.</s>
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            <p type="margin">
              <s id="s.011009">
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              Cylindri pri­
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              uilegia duo.</s>
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            <p type="main">
              <s id="s.011010">Inde cylindri priuilegia duo. </s>
              <s id="s.011011">Cylindrus
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              cono triplus eſt, altitudinem eandem ac
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              baſim habenti. </s>
              <s id="s.011012">Sphæræ verò, cuius diameter
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              ſit æqualis altitudini ſuæ, & maior circulus
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              baſi Cylindri ipſe cylindrus ſeſquialter erit.
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              </s>
              <s id="s.011013">Hæ igitur ſunt ſexaginta proprietates, nobi­
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              litate, & pulchritudine, & admiratione
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              præſtantiores, Geometricarum figurarum
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              tam ſuperficialium, quàm corporearum:
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              quandoquidem non me præterit eas penè
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              eſſe infinitas, ſed cum his elegantia non
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              poſſunt conferri: vel quia nondum demon­
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              ſtratio earum inuenta eſt, vel quia non ex
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              ſola nominum dignotione poſſunt intellegi,
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              vel quia ad æqualitatem non referuntur, ſed
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              quaſi vagantur. </s>
              <s id="s.011014">Æqualitas enim Geometræ
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              quidam eſt ſcopus.
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              </s>
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            <p type="margin">
              <s id="s.011015">
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              Modi tres
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              inueniendi
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              Geometrica
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              theoremata.</s>
            </p>
            <p type="main">
              <s id="s.011016">Quòd ſi quid aliud demonſtret, vt ma­
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              ius, aut notum? </s>
              <s id="s.011017">maius quidem æqualis gra­
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              tia, notum cognito æquale. </s>
              <s id="s.011018">Tripliciter licet
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              hoc aſſequi, vocaturque Argumenti con­
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              cluſio, ſi directè procedat, aut per nega­
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              tionem, cùm ad inconueniens reſpondens
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              deducitur. </s>
              <s id="s.011019">Et per continuam potentiam, ve­
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              lut cùm paraboles, aut ſuperficiei ſphæræ
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              magnitudo ab Archimede demonſtratur:
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              eſtque hic modus, quo plerunque vtimur
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              in ſubtiliſſimis inuentis. </s>
              <s id="s.011020">Eſt verò duplex hic:
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              hic ſimplex, qui ex maioris ac minoris
                <expan abbr="cõſtat">conſtat</expan>
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              comparatione, vt in ſphæræ ſuperficiei ma­
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              gnitudine determinanda: alter ex propor­
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              tionibus, quæ fine carent, vt in area pa­
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              raboles. </s>
              <s id="s.011021">Nihil mirum igitur Geometriam
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              eſſe omnium ſcientiarum ſubtiliſſimam:
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              quæ cùm tamen à manifeſtiſſimis initium
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              ducat, meritò anſam præbuit, vt prima
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              omnium etiam pueris doceretur. </s>
              <s id="s.011022">Mirum eſt
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              quàm breui ex apertiſſimis paucis axioma­
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              tibus ad obſcuriſſima te trahat. </s>
              <s id="s.011023">Sic etiam
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              ex humillimis in altiſſima illicò aſſurgit.
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              </s>
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            <p type="margin">
              <s id="s.011024">
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              Geometria
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              ſcientiarum
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              ſubtiliſſima.</s>
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            <p type="margin">
              <s id="s.011025">
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              Reflexa pro­
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              portio quia
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              ſit.</s>
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            <p type="main">
              <s id="s.011026">Circa Mathematicas tamen contingunt
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              imperfectæ demonſtrationes, & quodam­
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              modò paralogiſmi. </s>
              <s id="s.011027">Imperfectæ autem de­
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              monſtrationes inueniuntur maximè in ge­
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              neribus proportionum non perfectæ natu­
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              ræ, qualis eſt reflexa proportio, quæ à no­
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              bis inuenta eſt. </s>
              <s id="s.011028">Et quia ſubtiliſſimæ & ipſa
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              eſt contemplationis, & omnibus figuris
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              æquilateris, quæ circulo inſcribuntur, com­
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              munis, ob id à nobis hîc erit demonſtran­
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              da, tum maximè quod illius auxilio ad la­
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              terum heptagoni inuentionem procedimus,
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              docemurque reſolutoria methodo vti. </s>
              <s id="s.011029">Ob
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              tot igitur cauſas, & tantas quamuis præ­
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              ter ordinem, demonſtratio huius propor­
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              tionis hic ſubiicitur. </s>
              <s id="s.011030">Cùm igitur fuerint tres
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              quantitates, quarum proportio aggregati
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              primæ, & ſecundæ ad tertiam fuerit, velut
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              tertiæ ad ſecundam, dicetur proportio hæc
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              reflexa: veluti in numeris capio, 9. 16. 20.
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              proportio 25. aggregati 9. & 16. ad 20. quæ
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              eſt qualis 20. ad 16. dicetur proportio re­
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              flexa. </s>
              <s id="s.011031">Nam 9. eſt prima quantitas, 16. ſecun­
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              da, 20. tertia. </s>
              <s id="s.011032">Quòd ſi proportio aggregati
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              primæ, & tertiæ præter hoc fuerit, qualis
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              ſecundæ ad primam, diceretur tunc reflexa
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              bis. </s>
              <s id="s.011033">Hæc autem in numeris exemplo decla­
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              rari non poteſt, ſed ab heptagono, vt do­
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              cebimus, ortum habet. </s>
              <s id="s.011034">Dico igitur, quod
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              ſimplex reflexa eſt inter duo latera conti­
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              nentia angulum duplum in aliquo triangu­
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              lo, & latus reſpiciens angulum duplum, &
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              latus reſpiciens angulum, qui eſt ſubduplus.
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              </s>
              <s id="s.011035">Sit igitur triangulus, ſeu triangulum (nihil
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              enim refert hæc curioſitas ) ABC, cuius B
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              angulus duplus ſit angulo A, dico propor­
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              tionem aggregati ex AB, BC ad latus AC,
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              quod angulum reſpicit B duplum, eſſe quale
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              AC ad B. </s>
              <s id="s.011036">C quod reſpicit A ſubduplum. </s>
              <s id="s.011037">Nam
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              ex nona primi elementorum diuido angu­
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              lum ABC per æqualia linea BD. </s>
              <s id="s.011038">In duobus
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              igitur triangulus ABC, & BCD angulus
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              C communis eſt, & A æqualis CBD, cum
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              vterque ſit medietas anguli B, & angulus
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              CDB ex trigeſima ſecunda primi elemento­
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              rum æqualis eſt angulo B: quare duo illi
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              trianguli erunt æqualium inuicem angulo­
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              rum. </s>
              <s id="s.011039">Et ideò per quartam ſexti elementorum
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              Euclidis
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              (
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              ſemper intellige
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              )
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              ratio A C ad
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              CB, eſt qualis CB ad CD Dupla igitur eſt
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              ratio AC ad CD, ei quæ eſt AC ad BC. </s>
              <s id="s.011040">At
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              quia angulus B per æqualia diuiſus eſt, erit
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              per tertiam ſexti elementorum ratio late­
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              rum vt partium baſis, ſcilicet. </s>
              <s id="s.011041">AB ad BC,
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              qualis AD ad DC: quare ex coniuncta pro­
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              portione propter decimaoctauamquinti ele­
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              mentorum ratio aggregati AC, & BC ad
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              BC, vt AC ad CD. </s>
              <s id="s.011042">At AC ad CD dupla
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              ei quæ eſt AC ad BC, dupla igitur eſt ra­
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              tio AB, & BC ad BC, ei quæ eſt A C ad
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              CB. </s>
              <s id="s.011043">Igitur ex definitione duplæ proportio­
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              nis ratio aggregati AB & B C ad A C, vt
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              AC ad BC, quod demonſtrandum fuit. </s>
              <s id="s.011044">Sit
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              igitur figuræ cuiuſuis æquilateræ in circu­
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              lo deſcriptæ, puta tredecim habentis late­
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              ra, latus vnum A B, & ſit A D ſubtenſa
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              duobus lateribus eiuſdem figuræ A C &
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              CD, & producatur B D quia ergo A B
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              eſt æqualis AC, & etiam eadem ratione
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              CD, erunt ſinguli arcus AC & CD æqua­
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              les A B arcui, quare totus arcus A D du­
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              plus arcui A B, ex demonſtratis in tertio
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              elementorum Euclidis, & vltima ſexti
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              eiuſdem angulus A B D duplus angulo
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                <figure id="id.016.01.245.2.jpg" xlink:href="016/01/245/2.jpg" number="103"/>
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              A D B duplus angulo ADB: quare ex nu­
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              per demonſtratis ratio aggregati laterum
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              AB, & B D ad latus AD, veluti lateris </s>
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