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-div467" type="section" level="1" n="355">
          <p>
            <s xml:id="echoid-s18590" xml:space="preserve">
              <pb o="217" file="0239" n="244" rhead="AD MATHEMATICAS."/>
            ſtrationũ, ſic aliquando probant angulos ad baſim eſſe æquales in 5. </s>
            <s xml:id="echoid-s18591" xml:space="preserve">per 4. </s>
            <s xml:id="echoid-s18592" xml:space="preserve">nam illi anguli opponuntur æqua-
              <lb/>
            libus lateribus, ergo inferunt per 4. </s>
            <s xml:id="echoid-s18593" xml:space="preserve">ſunt æquales. </s>
            <s xml:id="echoid-s18594" xml:space="preserve">error eſt quia in 4. </s>
            <s xml:id="echoid-s18595" xml:space="preserve">id probatum eſt in duobus diuerſis trian-
              <lb/>
            gulis, in 5. </s>
            <s xml:id="echoid-s18596" xml:space="preserve">autem probandum eſt in vno tant
              <unsure/>
            um.</s>
            <s xml:id="echoid-s18597" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18598" xml:space="preserve">Nona dicitur fallacia vniuerſalis, debet enim propoſicio eſſe talis, vt omnibus ſubiectis ſpeciebus, & </s>
            <s xml:id="echoid-s18599" xml:space="preserve">om-
              <lb/>
            nibus caſibus aptari poſſit: </s>
            <s xml:id="echoid-s18600" xml:space="preserve">ſi quis vellet oſtendere, in omni triangulo paralellam baſi, eſſe baſi minorem, in
              <lb/>
            demonſtratione vero id ſolum oſtẽderet in figura Iſoſcelis, & </s>
            <s xml:id="echoid-s18601" xml:space="preserve">ex natura eius, eſſet pſeudographus. </s>
            <s xml:id="echoid-s18602" xml:space="preserve">ſic etiam
              <lb/>
            quando habet varios caſus; </s>
            <s xml:id="echoid-s18603" xml:space="preserve">conſidera ſecundum primi, quæ varios habet eaſus pro varietate locorum puncti
              <lb/>
            adati, & </s>
            <s xml:id="echoid-s18604" xml:space="preserve">tamen demonſtrationem Euclidis conuenit omnibus illis caſibus.</s>
            <s xml:id="echoid-s18605" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18606" xml:space="preserve">Decima, eſt contra C
              <unsure/>
            eometriæ principia; </s>
            <s xml:id="echoid-s18607" xml:space="preserve">ſic errabat Antiphon dum ad quadrandum circulum aſſumebat
              <lb/>
            lineam curuam conſtare ex minimis rectis lineolis, quod ſalſum eſt. </s>
            <s xml:id="echoid-s18608" xml:space="preserve">vide Ariſt. </s>
            <s xml:id="echoid-s18609" xml:space="preserve">vlt. </s>
            <s xml:id="echoid-s18610" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s18611" xml:space="preserve">Elench. </s>
            <s xml:id="echoid-s18612" xml:space="preserve">cum noſtra
              <lb/>
            explicatione.</s>
            <s xml:id="echoid-s18613" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18614" xml:space="preserve">Vndecima, cum aliquod falſum vel impoſſibile aſſumitur. </s>
            <s xml:id="echoid-s18615" xml:space="preserve">ſic fallit Hippocrates in ſua circuli quadratio-
              <lb/>
            ne, qui cum lunn
              <unsure/>
            lam quadraſſet, aſſumit poſtea ex quodam trapezin accipi poſſe tria triangula æqualia tri-
              <lb/>
            bus lunulis, eo modo quo antea ex quadrato quodam triangulum æquale lunulæ acceperat; </s>
            <s xml:id="echoid-s18616" xml:space="preserve">quod falſum eſt.
              <lb/>
            </s>
            <s xml:id="echoid-s18617" xml:space="preserve">vide cap. </s>
            <s xml:id="echoid-s18618" xml:space="preserve">31. </s>
            <s xml:id="echoid-s18619" xml:space="preserve">ſecundi priorum Ariſt. </s>
            <s xml:id="echoid-s18620" xml:space="preserve">cum noſtra explicatione.</s>
            <s xml:id="echoid-s18621" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18622" xml:space="preserve">Duodecima, cum aliquid aſſumitur, quod non ſit euidens, aut quod non ſit demonſtratum, etiamſi verum
              <lb/>
            ſit. </s>
            <s xml:id="echoid-s18623" xml:space="preserve">Obijcies fortè, Euclides in prima aſſumit duos illos circulos ſe mutuo ſecare, quod non probat. </s>
            <s xml:id="echoid-s18624" xml:space="preserve">Reſpon-
              <lb/>
            deo id eſſe euidens, quid enim euidentius eſt, quam ſi vnus circulus habebt centrum in circumferentia alte-
              <lb/>
            rius, ſimulq; </s>
            <s xml:id="echoid-s18625" xml:space="preserve">per centrum illius tranſeat, ipſum ſecare. </s>
            <s xml:id="echoid-s18626" xml:space="preserve">Alia ſimilia ſunt apud Euclidem, & </s>
            <s xml:id="echoid-s18627" xml:space="preserve">alios Geometra@,
              <lb/>
            quæ tamen fi benè conſiderentureuidentiſſima ſunt, & </s>
            <s xml:id="echoid-s18628" xml:space="preserve">propterea ſine probatione aſſumuntur.</s>
            <s xml:id="echoid-s18629" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18630" xml:space="preserve">Decimatertia, cum malam illationem admittimus, vt ſi quis ſic inferret, trãſitur a minori ad maius, & </s>
            <s xml:id="echoid-s18631" xml:space="preserve">per
              <lb/>
            omnia media, ergo per æquale; </s>
            <s xml:id="echoid-s18632" xml:space="preserve">vide Clauium ad 16. </s>
            <s xml:id="echoid-s18633" xml:space="preserve">propoſitionem 3. </s>
            <s xml:id="echoid-s18634" xml:space="preserve">Elem. </s>
            <s xml:id="echoid-s18635" xml:space="preserve">in fine illius longiſſimi Scholij.
              <lb/>
            </s>
            <s xml:id="echoid-s18636" xml:space="preserve">ſic fallebat Auerr. </s>
            <s xml:id="echoid-s18637" xml:space="preserve">dum ſic ratiocinabatur, vt eſt 6. </s>
            <s xml:id="echoid-s18638" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s18639" xml:space="preserve">ita 4. </s>
            <s xml:id="echoid-s18640" xml:space="preserve">ad 2. </s>
            <s xml:id="echoid-s18641" xml:space="preserve">ergo permutando,|vt 6. </s>
            <s xml:id="echoid-s18642" xml:space="preserve">ad 2. </s>
            <s xml:id="echoid-s18643" xml:space="preserve">ita 4. </s>
            <s xml:id="echoid-s18644" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s18645" xml:space="preserve">hæc
              <lb/>
            enim non eſt permutata proportio, nec vllus alius modus argumentandi ex ijs, qui ab Euclide comproban-
              <lb/>
            tur, & </s>
            <s xml:id="echoid-s18646" xml:space="preserve">apparet falſitas in numeris. </s>
            <s xml:id="echoid-s18647" xml:space="preserve">nec pariter valet inferre, duæ hæ rectę lineæ non ſunt æquidiſtantes, ergo
              <lb/>
            concurrunt.</s>
            <s xml:id="echoid-s18648" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18649" xml:space="preserve">Decimaquarta, cum aliquid aſſumitur, quod æquè obſcurum eſt, ac ipſa propoſitio; </s>
            <s xml:id="echoid-s18650" xml:space="preserve">debet enim ex notio-
              <lb/>
            ribus deduci; </s>
            <s xml:id="echoid-s18651" xml:space="preserve">ſic Proclus decipitur dum ad probandum axioma 13. </s>
            <s xml:id="echoid-s18652" xml:space="preserve">Euclidis aſſumit illud æquè ignotum, ſi
              <lb/>
            ab vno puncto duæ reææ angulo fac
              <unsure/>
            ientes infinitè producerentur, earum diſtantia excedet omnem finitam
              <lb/>
            magnitudinem. </s>
            <s xml:id="echoid-s18653" xml:space="preserve">vide Clauium in ſcholio 28. </s>
            <s xml:id="echoid-s18654" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s18655" xml:space="preserve">primi Elem. </s>
            <s xml:id="echoid-s18656" xml:space="preserve">pag. </s>
            <s xml:id="echoid-s18657" xml:space="preserve">150.</s>
            <s xml:id="echoid-s18658" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18659" xml:space="preserve">Decimaquinta, oritur ex ignoratione terminorum Geometricorum. </s>
            <s xml:id="echoid-s18660" xml:space="preserve">ſic Tyrones aberrant, dum putant
              <lb/>
            ſe quadrare circulum, ſi conſtruant quadratum cuius quatuor latera ſint æqualia peri pheriæ dati circuli: </s>
            <s xml:id="echoid-s18661" xml:space="preserve">vi-
              <lb/>
            de Geometriam practicam P. </s>
            <s xml:id="echoid-s18662" xml:space="preserve">Clauij pag. </s>
            <s xml:id="echoid-s18663" xml:space="preserve">357.</s>
            <s xml:id="echoid-s18664" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18665" xml:space="preserve">Tandem lectorẽ monitum volo ad vitandas fallacias vtile eſſe legere Io: </s>
            <s xml:id="echoid-s18666" xml:space="preserve">Buteonem de varijs circuli qua-
              <lb/>
            draturis: </s>
            <s xml:id="echoid-s18667" xml:space="preserve">& </s>
            <s xml:id="echoid-s18668" xml:space="preserve">Petrum Nonium de Orontij erratis, & </s>
            <s xml:id="echoid-s18669" xml:space="preserve">Io: </s>
            <s xml:id="echoid-s18670" xml:space="preserve">Regiomontanum de quadratura Nicolai Cuſani. </s>
            <s xml:id="echoid-s18671" xml:space="preserve">hi
              <lb/>
            enim aliorum varios paralogiſmos, & </s>
            <s xml:id="echoid-s18672" xml:space="preserve">fallacias detegunt, vnde nos aliorum damno proſiciamus.</s>
            <s xml:id="echoid-s18673" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div470" type="section" level="1" n="356">
          <head xml:id="echoid-head375" style="it" xml:space="preserve">De reſolutione, & compoſitione.</head>
          <p>
            <s xml:id="echoid-s18674" xml:space="preserve">VT ars quæuis benè tractetur non ſolum neceſſarium eſt, vitia illi contraria, vt vitari|poſſint, cognoſcere,
              <lb/>
            ſed etiam, & </s>
            <s xml:id="echoid-s18675" xml:space="preserve">quidem præcipuum|, præcepta tenere quibus facilè, & </s>
            <s xml:id="echoid-s18676" xml:space="preserve">optimè artis finem conſequamur;
              <lb/>
            </s>
            <s xml:id="echoid-s18677" xml:space="preserve">cum ergo de fallacijs egerimus, quæ arti hui
              <unsure/>
            c contrariæ ſunt, ſequitur vt de reſolutione, & </s>
            <s xml:id="echoid-s18678" xml:space="preserve">compoſitione
              <lb/>
            nonnulla dicamus, per eas enim, & </s>
            <s xml:id="echoid-s18679" xml:space="preserve">optimè & </s>
            <s xml:id="echoid-s18680" xml:space="preserve">facilè Geometricè demonſtrationes inueniri poſſunt.</s>
            <s xml:id="echoid-s18681" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s18682" xml:space="preserve">Reſolutionem hanc Geometricam primus omnium Plato adinuenit, eamque Laodamãtem Theſium do-
              <lb/>
            cuit, cuius auxilio, & </s>
            <s xml:id="echoid-s18683" xml:space="preserve">ipſe, & </s>
            <s xml:id="echoid-s18684" xml:space="preserve">reliqui iuniores Geometrę plurimis ſubtiliter inuentis Geometriam magnope-
              <lb/>
            re amplificarunt. </s>
            <s xml:id="echoid-s18685" xml:space="preserve">de hac reſolutione, vt refert Pappus initio 7. </s>
            <s xml:id="echoid-s18686" xml:space="preserve">ſcripſerat Euclides, Apollonius, & </s>
            <s xml:id="echoid-s18687" xml:space="preserve">Ariſtæus,
              <lb/>
            ſed opera eorum interciderunt. </s>
            <s xml:id="echoid-s18688" xml:space="preserve">Marinus autem Ghetaldus in ſuo Apollonio rediuiuo reſolutionem hanc
              <lb/>
            pariter rediuiuam ſe propediem daturum recipit: </s>
            <s xml:id="echoid-s18689" xml:space="preserve">nos tamen interim de ſtac nonnulla diximus in locis Ma-
              <lb/>
            @hem. </s>
            <s xml:id="echoid-s18690" xml:space="preserve">ad titulum, Lib. </s>
            <s xml:id="echoid-s18691" xml:space="preserve">reſolutor. </s>
            <s xml:id="echoid-s18692" xml:space="preserve">& </s>
            <s xml:id="echoid-s18693" xml:space="preserve">hic etiam nonnulla in medius afferemus. </s>
            <s xml:id="echoid-s18694" xml:space="preserve">quid autem ipſa ſit ex veterum
              <lb/>
            de ea reliquijs primo videamus. </s>
            <s xml:id="echoid-s18695" xml:space="preserve">Euclides igitur in ſcholio propoſ. </s>
            <s xml:id="echoid-s18696" xml:space="preserve">primæ 13. </s>
            <s xml:id="echoid-s18697" xml:space="preserve">Elem. </s>
            <s xml:id="echoid-s18698" xml:space="preserve">iuxta fideliſſimam Go-
              <lb/>
            mand. </s>
            <s xml:id="echoid-s18699" xml:space="preserve">ex Zamberti, & </s>
            <s xml:id="echoid-s18700" xml:space="preserve">Græco interpretationem (quod ſchol. </s>
            <s xml:id="echoid-s18701" xml:space="preserve">cum quatuor reſolutiones, pretioſum antiqui-
              <lb/>
            tatis monumentum nonnulli interpretes perperam omiſerunt) eam ſic definit. </s>
            <s xml:id="echoid-s18702" xml:space="preserve">Reſolntio eſt ſumptio tan-
              <lb/>
            quam conceſſi, per ea quę conſequuntur in aliquod verum conceſſum. </s>
            <s xml:id="echoid-s18703" xml:space="preserve">poſtea addit. </s>
            <s xml:id="echoid-s18704" xml:space="preserve">Compoſitio eſt ſumptio
              <lb/>
            conceſſi per ea quæ conſequuntur in quęſiti concluſionem ſiue depręhenſionem. </s>
            <s xml:id="echoid-s18705" xml:space="preserve">Eaſdem definit Proclus lib.
              <lb/>
            </s>
            <s xml:id="echoid-s18706" xml:space="preserve">6. </s>
            <s xml:id="echoid-s18707" xml:space="preserve">primi pag. </s>
            <s xml:id="echoid-s18708" xml:space="preserve">145. </s>
            <s xml:id="echoid-s18709" xml:space="preserve">& </s>
            <s xml:id="echoid-s18710" xml:space="preserve">Pappus initio lib. </s>
            <s xml:id="echoid-s18711" xml:space="preserve">7. </s>
            <s xml:id="echoid-s18712" xml:space="preserve">quos tu conſule. </s>
            <s xml:id="echoid-s18713" xml:space="preserve">Senſus autem Euclidis eſt hic: </s>
            <s xml:id="echoid-s18714" xml:space="preserve">Reſolutio eſt diſcur-
              <lb/>
            ſus, quo inueſtigamus veritatem Theorematis, aut poſſibilitas Problematis quæſiti hoc modo; </s>
            <s xml:id="echoid-s18715" xml:space="preserve">ſi quæſitũ eſt
              <lb/>
            Theorema accipimus illud tanquam verum, & </s>
            <s xml:id="echoid-s18716" xml:space="preserve">conceſſum. </s>
            <s xml:id="echoid-s18717" xml:space="preserve">ſi vero eſt Problema, aſſumimus illud tanquam
              <lb/>
            factum, ideſt, ſupponimus illud verum eſſe, iſtud vero poſſibile, & </s>
            <s xml:id="echoid-s18718" xml:space="preserve">iam factum; </s>
            <s xml:id="echoid-s18719" xml:space="preserve">ex qua ſuppoſitione ratioci-
              <lb/>
            namur per ea, quæ ex ſuppoſitis veré deducuntur, donec aliquod verum vel falſum oecurrat: </s>
            <s xml:id="echoid-s18720" xml:space="preserve">ſi enim occur-
              <lb/>
            rat aliquod verum, & </s>
            <s xml:id="echoid-s18721" xml:space="preserve">conceſſum, ſignum euidens eſt etiam ſuppoſitum illud, ex quo iſtud ſequitur verum eſ-
              <lb/>
            ſe, ſeu poſſibile eſſe. </s>
            <s xml:id="echoid-s18722" xml:space="preserve">quæ conſequentia nititur hoc logico fundamento, verum non niſi ex vero in bona ma-
              <lb/>
            teria, & </s>
            <s xml:id="echoid-s18723" xml:space="preserve">forma ſequitur.</s>
            <s xml:id="echoid-s18724" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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