Biancani, Giuseppe, Aristotelis loca mathematica, 1615

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="s.002105">
                <pb pagenum="121" xlink:href="009/01/121.jpg"/>
              probare P, eſſe polum prædicti ambitus, ſic. </s>
              <s id="s.002106">Primò enim ſciendum in præ­
                <lb/>
              miſſa conſtructione eſſe, vt F, ad G K, & B, ad K P, ita D, ad P M. nam ſi non
                <lb/>
              ſit eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiuſdem D,
                <lb/>
              ad aliam maiorem, vel minorem ipſa P M. ſit ad minorem P R. nihil enim
                <lb/>
              refert ſiue dixeris habere eandem rationem ad minorem, ſiue ad maiorem,
                <lb/>
              ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. ſed li­
                <lb/>
              neæ F, B, D, erant proportionales
                <expan abbr="componẽdo">componendo</expan>
              hoc modo, vt F B, ad D, ita
                <lb/>
              D, ad B: quare ſimiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. ſi igi­
                <lb/>
              tur à punctis G, & K, figuræ nu. </s>
              <s id="s.002107">164.
                <expan abbr="iungãtur">iungantur</expan>
              lineæ ad R, quæ ſint G R, K R,
                <lb/>
              erit vt G R, ad K R, ita G P, ad P R. quia orta
                <expan abbr="sũt">sunt</expan>
              duo
                <expan abbr="triãgula">triangula</expan>
              G P R, K P R,
                <lb/>
              quæ habent eundem angulum ad P. & latera proportionalia circa dictum
                <lb/>
              angulum. </s>
              <s id="s.002108">eſt etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in
                <lb/>
              minori, ex conſtructione, quare per 6. 6. erunt illa duo triangula æquian­
                <lb/>
              gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia;
                <lb/>
              quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita
                <lb/>
              B, ad D. & ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R,
                <lb/>
              ad R G, intra eandem circunferentiam, & in eodem plano: quod eſſe im­
                <lb/>
              poſſibile ſupra oſtendimus, hoc autem impoſſibile, ſequitur ſi neges eſſe vt
                <lb/>
              F, ad G K; & B, ad K P, ita D, ad P M.</s>
            </p>
            <p type="main">
              <s id="s.002109">
                <arrow.to.target n="marg168"/>
              </s>
            </p>
            <p type="margin">
              <s id="s.002110">
                <margin.target id="marg168"/>
              167</s>
            </p>
            <p type="main">
              <s id="s.002111">Ibidem
                <emph type="italics"/>
              (Quoniăm igitur quæ D,
                <expan abbr="neq;">neque</expan>
              ad minorem ea, quæ P M,
                <expan abbr="neq;">neque</expan>
              ad maiorem
                <lb/>
              (ſimiliter enim demonſtrabimus) palam eſt, quod ad ipſam
                <expan abbr="vtiq;">vtique</expan>
              erit, in qua P M,
                <lb/>
              quare erit, quod quæ M P, ad P K, quæ P G, ad M P. </s>
              <s id="s.002112">Si igitur eo in quo P, polo
                <lb/>
              vtens, diſtantia autem ea, in qua P M, circulus deſcribatur, omnes angulos attin­
                <lb/>
              get, quos reflexæ faciunt, quæ à K, G. ſi autem non, ſimiliter oſtendentur eandem
                <lb/>
              babere rationem, quæ alibi, quam in ſemicirculo conſtituuntur; quod quidem erat
                <lb/>
              impoſſibile)
                <emph.end type="italics"/>
              quoniam igitur, inquit, linea D,
                <expan abbr="neq;">neque</expan>
              ad minorem,
                <expan abbr="neq;">neque</expan>
              ad ma­
                <lb/>
              iorem quam P M, habet eam rationem, quæ eſt ipſius F, ad G K, aut ipſius
                <lb/>
              B, ad K P. ſimiliter enim demonſtratur abſurdum ſequi. </s>
              <s id="s.002113">palàm eſt, quoniam
                <lb/>
              erit D, ad P M, vt prædictæ ad prædictas: quare componendo, & permu­
                <lb/>
              tando, erunt tandem vt G P, ad P M, ita P M, ad P K, & ita G M, ad M K,
                <lb/>
              aſſumpſimus enim in conſtructione eſſe G M, ad M K, ita F B, ad D, & D, ad
                <lb/>
              B. quare cum ſit vt G M, ad M K, ita F B, ad D. & G P, ad P M. & P M, ad
                <lb/>
              K P; erunt per 11. 5. vt G M, ad M K. ita G P, ad P M. & P M, ad P K. ſi quis
                <lb/>
              igitur vtens puncto P, tanquam polo, & interuallo P M, circulum deſcribat,
                <lb/>
              omnes angulos reflexionis attinget, quos faciunt lineæ productæ à K, & re­
                <lb/>
              flexæ ab M, ad G. harum enim infinitam multitudinem debemus imaginari
                <lb/>
              à K, ad infinita puncta M, produci in ambitu illo conſtituta,
                <expan abbr="reſlectiq́">reflectique</expan>
              ; ad G.
                <lb/>
              ſi enim non attingat omnes illos angulos, ſequitur, vt ſupra, in eodem ſemi­
                <lb/>
              circulo
                <expan abbr="cõſtitui">conſtitui</expan>
              poſſe duas alias rectas proportionales prioribus G M, M K,
                <lb/>
              quod eſt impoſſibile. </s>
              <s id="s.002114">Porrò ſub angulo G M K, linearum G M, M K, Iris
                <lb/>
              apparet: quare apparebit etiam ſub alijs omnibus, quæ à punctis G K, duci
                <lb/>
              poſſunt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum
                <lb/>
              non deſinant in eundem
                <expan abbr="ſemicirculũ">ſemicirculum</expan>
              , ſed in ambitum Iridis M N, in quo M,
                <lb/>
              punctum imaginamur circumduci. </s>
              <s id="s.002115">Ex quibus pater P, eſſe polum Iridis, ex
                <lb/>
              quo per puncta M, vbi ſit reflexio, deſcribitur arcus attingens omnes Iridis
                <lb/>
              reflexiones.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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