Blancanus, Josephus, Sphaera mvndi, sev cosmographia demonstratiua , ac facile methodo tradita : in qua totius Mundi fabrica, vna cum nouis, Tychonis, Kepleri, Galilaei, aliorumq' ; Astronomorum adinuentis continentur ; Accessere I. Breuis introductio ad geographiam. II. Apparatus ad mathematicarum studium. III. Echometria, idest Geometrica tractatio de Echo. IV. Nouum instrumentum ad Horologia

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        <div xml:id="echoid-div463" type="section" level="1" n="351">
          <p>
            <s xml:id="echoid-s18403" xml:space="preserve">
              <pb o="215" file="0237" n="242" rhead="AD MATHEMATICAS."/>
            in ſcenar@m picturis redderent ſpeciem, & </s>
            <s xml:id="echoid-s18404" xml:space="preserve">quæ in directio planiſq; </s>
            <s xml:id="echoid-s18405" xml:space="preserve">frontibu@ fint figuratæ, alia abſcenden-
              <lb/>
            tia, aiia prominentia eſſe videantur. </s>
            <s xml:id="echoid-s18406" xml:space="preserve">horum doctrinam, videtur innouaſſe Marchio Guiduſubaldus in ſua
              <lb/>
            Perſpectiua. </s>
            <s xml:id="echoid-s18407" xml:space="preserve">Federicus etiam Comandinus putat veteres de centro grauitatis ſolidorum ſcripſiſſe, cum Ar-
              <lb/>
            chimedes deinſidentibus aquæ centri grauitatis conoidis fecerit mentionem. </s>
            <s xml:id="echoid-s18408" xml:space="preserve">quam partem ipſe conatus eſt
              <lb/>
            renou
              <unsure/>
            are, ſedeam Lucas Valerius multo magis ampliauit. </s>
            <s xml:id="echoid-s18409" xml:space="preserve">Hæc ſunt igitur diuina illa veterum monumen-
              <lb/>
            ta, quæ obſæculorum barbariẽ intercidiſſe dolemus:</s>
            <s xml:id="echoid-s18410" xml:space="preserve">
              <unsure/>
            quæ fortèapud Arabes, aut alias nationes ſub alio idio-
              <lb/>
            mate latitant, donec Principum noſtrorum induftria ea requiſierit.</s>
            <s xml:id="echoid-s18411" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div464" type="section" level="1" n="352">
          <head xml:id="echoid-head371" style="it" xml:space="preserve">De Geometriæ promotione, ex arte Geometricè demonſtr andi, vbi de Reſolutione.</head>
          <p>
            <s xml:id="echoid-s18412" xml:space="preserve">HOcloc
              <unsure/>
            o mei muneris eſſe animaduerti nonnulla de arte Geometricè demonſſrandi in medium aſſerre;
              <lb/>
            </s>
            <s xml:id="echoid-s18413" xml:space="preserve">quandoquidem ea eſt quæ cæteris omnibus Mathem. </s>
            <s xml:id="echoid-s18414" xml:space="preserve">ſpiritum ac vitam quodammodo infundit, & </s>
            <s xml:id="echoid-s18415" xml:space="preserve">qua
              <unsure/>
              <lb/>
            reliquæ deſtitutæ ſcientiæ, ac Philoſophiæ nomine prorſus indignę videantur. </s>
            <s xml:id="echoid-s18416" xml:space="preserve">præterea quoiure quiſpiam
              <lb/>
            fibi Mathematici nomen arrogare audeat, qui nec ſua rectè demonſtrar
              <unsure/>
            e, nec de alienis rectè iudicare queat. </s>
            <s xml:id="echoid-s18417" xml:space="preserve">
              <lb/>
            hac veteres magni illi Geometræ ſuffulti mirabiles illas demonſtrationes@,|quæ noſtris ingenijs impoſſibi-
              <lb/>
            les videtur, fęliciter excogitarunt. </s>
            <s xml:id="echoid-s18418" xml:space="preserve">Vtinam autem extarent ea quæ de ea Euclides, Apollonius, & </s>
            <s xml:id="echoid-s18419" xml:space="preserve">Ari-
              <lb/>
            ftæus conſcripſerunt; </s>
            <s xml:id="echoid-s18420" xml:space="preserve">non enim opus nunc eſſet nos in ea vtcunque adumbranda laborare. </s>
            <s xml:id="echoid-s18421" xml:space="preserve">Quamuis au-
              <lb/>
            tem hanc artem, vt bene ait Petrus Nonnius cap. </s>
            <s xml:id="echoid-s18422" xml:space="preserve">4. </s>
            <s xml:id="echoid-s18423" xml:space="preserve">de err. </s>
            <s xml:id="echoid-s18424" xml:space="preserve">Orontij, ex quotidiano librorum Euclidis, & </s>
            <s xml:id="echoid-s18425" xml:space="preserve">
              <lb/>
            aliorum Geometrarum ſtudio, & </s>
            <s xml:id="echoid-s18426" xml:space="preserve">imitatione conſequi poſſimus, facilius tamen additis ſequentibus anno-
              <lb/>
            tationibus, eam conſequemur.</s>
            <s xml:id="echoid-s18427" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div465" type="section" level="1" n="353">
          <head xml:id="echoid-head372" style="it" xml:space="preserve">Quid ſit Geometrica demonstratio.</head>
          <p>
            <s xml:id="echoid-s18428" xml:space="preserve">DEmonf
              <unsure/>
            tratio Geometrica eſt diſeurſus certus, & </s>
            <s xml:id="echoid-s18429" xml:space="preserve">euidens ex veris, & </s>
            <s xml:id="echoid-s18430" xml:space="preserve">proprijs Geometrię principijs per
              <lb/>
            Enthymemata ad concluſionem procedens. </s>
            <s xml:id="echoid-s18431" xml:space="preserve">vt autem bene intelligatur quid ſit veritas conclufionis
              <lb/>
            Geometricæ, & </s>
            <s xml:id="echoid-s18432" xml:space="preserve">alia huc ſpectantia, lege tractatum de natura Mathematicarum in fine operis noſtri de locis
              <lb/>
            Mathem. </s>
            <s xml:id="echoid-s18433" xml:space="preserve">vbi dictum eſt quid ſit materia intelligibilis, quæ ſola capax eſt Geometricæ veritatis, & </s>
            <s xml:id="echoid-s18434" xml:space="preserve">perfectio-
              <lb/>
            nis: </s>
            <s xml:id="echoid-s18435" xml:space="preserve">ea autem eſt quantitas abſtracta, &</s>
            <s xml:id="echoid-s18436" xml:space="preserve">c. </s>
            <s xml:id="echoid-s18437" xml:space="preserve">ſic vera, & </s>
            <s xml:id="echoid-s18438" xml:space="preserve">Geometrica æqualitas ea eſt, quæ duæ, v.</s>
            <s xml:id="echoid-s18439" xml:space="preserve">g. </s>
            <s xml:id="echoid-s18440" xml:space="preserve">lineæ ita ſunt
              <lb/>
            æquales, vt nullum omnino diſcrimen interſit, non ſolum ſenſibile, ſed nec intelligibile. </s>
            <s xml:id="echoid-s18441" xml:space="preserve">quædam enim ad
              <lb/>
            ſenſum videri poſſunt æqualia, quæ tamen Geometricè, & </s>
            <s xml:id="echoid-s18442" xml:space="preserve">verè non ſunt æqualia. </s>
            <s xml:id="echoid-s18443" xml:space="preserve">vbi notandum eſt Geo-
              <lb/>
            metram, dum demonſtrat, ſupponere ſe habere hanc materiam intelligibilem præſentem’, atque in ipſa
              <lb/>
            poſſe ſe operari, ideſt, ducere in eas lineas, angulos, tria@gula, &</s>
            <s xml:id="echoid-s18444" xml:space="preserve">c. </s>
            <s xml:id="echoid-s18445" xml:space="preserve">quamuis in ſuo Abaco delineet lineas,
              <lb/>
            & </s>
            <s xml:id="echoid-s18446" xml:space="preserve">figuras ſenfibiles, non tamen propterea (vt ait Ariſt. </s>
            <s xml:id="echoid-s18447" xml:space="preserve">text 25. </s>
            <s xml:id="echoid-s18448" xml:space="preserve">primi poſter.) </s>
            <s xml:id="echoid-s18449" xml:space="preserve">falſum ſupponit. </s>
            <s xml:id="echoid-s18450" xml:space="preserve">quia deli-
              <lb/>
            m
              <unsure/>
            eationesillas ſenſibiles pro intelligibilibus ſupponit, vt melius intelligatur. </s>
            <s xml:id="echoid-s18451" xml:space="preserve">& </s>
            <s xml:id="echoid-s18452" xml:space="preserve">vt ait Ariſtoteles Geome-
              <lb/>
            tra nihil concludit eò quod hæc eſt linea ſenfibilis, quam ipſe exponit, ſed virtute illius intelligibilis, quæ
              <lb/>
            per ſenfibilem oſtenditur. </s>
            <s xml:id="echoid-s18453" xml:space="preserve">& </s>
            <s xml:id="echoid-s18454" xml:space="preserve">quamuis hæc materia intelligibilis nulla nunc extaret, ſatis eſt ſi poſſit extare,
              <lb/>
            ſcientia enim abſtrahit ab exiſtentia ſui ſub@ecti.</s>
            <s xml:id="echoid-s18455" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div466" type="section" level="1" n="354">
          <head xml:id="echoid-head373" style="it" xml:space="preserve">Form
            <unsure/>
          a
            <unsure/>
          Geometricæ Demonſtrationis.</head>
          <p>
            <s xml:id="echoid-s18456" xml:space="preserve">HAhe debemus elicere ex Euclidis, & </s>
            <s xml:id="echoid-s18457" xml:space="preserve">aliorum demonſtrationibus qui Primo loco ponit Propoſitionem,
              <lb/>
            quæ ſcilicet proponitur vt probetur, vel vt effic
              <unsure/>
            iatur; </s>
            <s xml:id="echoid-s18458" xml:space="preserve">illud dicitur Theorema, hoc Problema. </s>
            <s xml:id="echoid-s18459" xml:space="preserve">Secun-
              <lb/>
            do Propoſitionem explie
              <unsure/>
            at appoſita figura, quæ in problemate continet quædam Data, dantur enim vel
              <lb/>
            puncta, vel lineæ, vel anguli, &</s>
            <s xml:id="echoid-s18460" xml:space="preserve">c. </s>
            <s xml:id="echoid-s18461" xml:space="preserve">ſic in prima Euclidis, datur linea vna, in ſecunda datur linea, & </s>
            <s xml:id="echoid-s18462" xml:space="preserve">punctum.
              <lb/>
            </s>
            <s xml:id="echoid-s18463" xml:space="preserve">in Theoremate exibetur figura de qua paſſio demonſtranda eſt, ideſt, quæ eſt ſubiectum demonſtrationis: </s>
            <s xml:id="echoid-s18464" xml:space="preserve">
              <lb/>
            ſic in quarta exibentur duo triangula, de quibus demonſtrandæ ſunt aliquot æqualitates, & </s>
            <s xml:id="echoid-s18465" xml:space="preserve">in ijs explicatur
              <lb/>
            propoſitio. </s>
            <s xml:id="echoid-s18466" xml:space="preserve">Tertio, ſequitur Conſtructio, vt plurimum enim præter data, & </s>
            <s xml:id="echoid-s18467" xml:space="preserve">ſubiectum neceſſe eſt ad de-
              <lb/>
            monſtrandum conſtruere alias lin
              <unsure/>
            e
              <unsure/>
            as, vel angulos, vel circulos, &</s>
            <s xml:id="echoid-s18468" xml:space="preserve">c. </s>
            <s xml:id="echoid-s18469" xml:space="preserve">ſic in Prima Euclidis conſtruuntur duo
              <lb/>
            circuli, & </s>
            <s xml:id="echoid-s18470" xml:space="preserve">duæ lineæ. </s>
            <s xml:id="echoid-s18471" xml:space="preserve">in omni problemate necceſſaria eſt conſtructio ſaltem ipſius problematis. </s>
            <s xml:id="echoid-s18472" xml:space="preserve">in Theore-
              <lb/>
            mate, nulla aliquando opus eſt conſtructione, vt patet in 15. </s>
            <s xml:id="echoid-s18473" xml:space="preserve">primi. </s>
            <s xml:id="echoid-s18474" xml:space="preserve">Quarto, ſequitur diſcurſus circa ſigu-
              <lb/>
            ram conftructam, qui propriè eſt ipſa Demonſtratio procedens per enthymemata, quæ probat aut factum
              <lb/>
            eſſe, aut verum eſſe, quod proponebatur. </s>
            <s xml:id="echoid-s18475" xml:space="preserve">hi autem diſcurſus geometrici debent eſſe breues, & </s>
            <s xml:id="echoid-s18476" xml:space="preserve">ſimplices, & </s>
            <s xml:id="echoid-s18477" xml:space="preserve">
              <lb/>
            propterea n@hil in eis reperitur, quod ex præcedentibus non ſit iam manifeſtum, & </s>
            <s xml:id="echoid-s18478" xml:space="preserve">ideo procedit enthyme-
              <lb/>
            maticè non ſyllogiſticè; </s>
            <s xml:id="echoid-s18479" xml:space="preserve">quamuis poſſit ad formam ſyllogiſticam reduci, vt patet in ſcholio P. </s>
            <s xml:id="echoid-s18480" xml:space="preserve">Claudij ad
              <lb/>
            primam primi, ſed id eſſet longum, & </s>
            <s xml:id="echoid-s18481" xml:space="preserve">tædioſum ac minus perſpicuum, & </s>
            <s xml:id="echoid-s18482" xml:space="preserve">multa eſſent ſępius repetenda, & </s>
            <s xml:id="echoid-s18483" xml:space="preserve">
              <lb/>
            ſuperuacanea. </s>
            <s xml:id="echoid-s18484" xml:space="preserve">demonſtratio porrò quo b
              <unsure/>
            reuior, ac ſimplicior, eo melior. </s>
            <s xml:id="echoid-s18485" xml:space="preserve">Eſt autem omn s demonſtratio aut
              <lb/>
            ad impoſſibile. </s>
            <s xml:id="echoid-s18486" xml:space="preserve">Oſtenſiua oſtendit per cauſam materialem, aut formalem, aut à ſigno: </s>
            <s xml:id="echoid-s18487" xml:space="preserve">Quæ ad impoſſibile
              <lb/>
            eſt, vel deducit contra principia, vel contra demonſtrata, vel contra hypoteſim, ſeu ſuppoſitionem. </s>
            <s xml:id="echoid-s18488" xml:space="preserve">Sexta
              <lb/>
            primi repugnant principio illi totum eſt maius ſua parte. </s>
            <s xml:id="echoid-s18489" xml:space="preserve">vij. </s>
            <s xml:id="echoid-s18490" xml:space="preserve">eſt contra v. </s>
            <s xml:id="echoid-s18491" xml:space="preserve">xxv. </s>
            <s xml:id="echoid-s18492" xml:space="preserve">eſt contra hypotheſim. </s>
            <s xml:id="echoid-s18493" xml:space="preserve">Quin-
              <lb/>
            tò. </s>
            <s xml:id="echoid-s18494" xml:space="preserve">Tandem vltima pars huius diſcurſus eſt concluſio, quæ eſt ipſa propoſitio iam demonſtrata, cui in P
              <unsure/>
            </s>
          </p>
        </div>
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