Ibn-al-Haitam, al-Hasan Ibn-al-Hasan; Witelo; Risner, Friedrich, Opticae thesavrvs Alhazeni Arabis libri septem, nunc primùm editi. Eivsdem liber De Crepvscvlis & Nubium ascensionibus. Item Vitellonis Thuvringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentarijs, a Federico Risnero, 1572

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        <div xml:id="echoid-div654" type="section" level="0" n="0">
          <head xml:id="echoid-head574" xml:space="preserve">THEOREMATA
            <gap/>
          </head>
          <head xml:id="echoid-head575" xml:space="preserve" style="it">1. Omnes lineæ æquidiſt antes in eadem ſuperficie plana neceſſariò conſiſtunt.
            <lb/>
          E' 35 definit. 1 element.</head>
          <p>
            <s xml:id="echoid-s20464" xml:space="preserve">Sint duæ lineæ æquidiſtantes, quæ a b & c d utcunque diſpoſitæ:</s>
            <s xml:id="echoid-s20465" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a" number="257">
                <variables xml:id="echoid-variables243" xml:space="preserve">a c b d</variables>
              </figure>
            dico quòd ipſæ ſunt in eadem ſuperficie plana:</s>
            <s xml:id="echoid-s20466" xml:space="preserve"> copulentur enim per
              <lb/>
            lineam b d.</s>
            <s xml:id="echoid-s20467" xml:space="preserve"> Quoniam ergo lineæ a b & b d angulariter coniungun-
              <lb/>
            tur:</s>
            <s xml:id="echoid-s20468" xml:space="preserve"> palàm quoniam ipſæ ſunt in eadem ſuperficie per 2 p 11.</s>
            <s xml:id="echoid-s20469" xml:space="preserve"> Simi-
              <lb/>
            liter, quia lineę e d & b d angulariter coniunguntur, eruntipſæ in ea-
              <lb/>
            dem ſuperficie:</s>
            <s xml:id="echoid-s20470" xml:space="preserve"> Sed linea b d eſt in una tantum ſuperficie plana, quo-
              <lb/>
            niam ipſius partem eſſe in ſublimi, partem in plano, eſt impoſsibile ք
              <lb/>
            1 p 11.</s>
            <s xml:id="echoid-s20471" xml:space="preserve"> Palàm ergo, quoniam lineę a b & c d neceſſariò conſiſtunt in ea-
              <lb/>
            dem plana ſuperficie contenta inter eas & inter lineas, extremitates
              <lb/>
            illarum linearum copulantes:</s>
            <s xml:id="echoid-s20472" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s20473" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div656" type="section" level="0" n="0">
          <head xml:id="echoid-head576" xml:space="preserve" style="it">2. Lineam à puncto unius linearum æquidiſtantium in eadem
            <lb/>
          ſuperficie protr actam, cum alter a indefinitæ quantitatis concurre
            <lb/>
          re eſt neceſſe. Lemma Procli ad 29 p relement.</head>
          <p>
            <s xml:id="echoid-s20474" xml:space="preserve">Sint duæ lineæ æquidiſtantes, quæ a b & c d:</s>
            <s xml:id="echoid-s20475" xml:space="preserve"> quarum unam, ſcilicet a b, ſecet linea b e in puncte
              <lb/>
            b.</s>
            <s xml:id="echoid-s20476" xml:space="preserve"> Dico, quòd linea b e ſecabit etiam lineam c d.</s>
            <s xml:id="echoid-s20477" xml:space="preserve"> Quia enim linea c d
              <lb/>
              <figure xlink:label="fig-0307-02" xlink:href="fig-0307-02a" number="258">
                <variables xml:id="echoid-variables244" xml:space="preserve">c a b d e</variables>
              </figure>
            indefinitæ quantitatis eſſe ſupponitur, protrahatur uerſus ipſam li-
              <lb/>
            nea b e:</s>
            <s xml:id="echoid-s20478" xml:space="preserve"> quę ſi concurrit cum c d, habetur propoſitum.</s>
            <s xml:id="echoid-s20479" xml:space="preserve"> Sinon concur-
              <lb/>
            rat:</s>
            <s xml:id="echoid-s20480" xml:space="preserve"> palàm per definitionem æquidiſtantium linearum, quoniam linea
              <lb/>
            b e eſt æquidiſtans lineæ c d:</s>
            <s xml:id="echoid-s20481" xml:space="preserve"> & quia lineæ a b & b e ambę ſunt æquidi-
              <lb/>
            ſtãtes lineę c d:</s>
            <s xml:id="echoid-s20482" xml:space="preserve"> erit per 30 p 1 linea e b ęquidiſtans lineę a b:</s>
            <s xml:id="echoid-s20483" xml:space="preserve"> ſed palã ex
              <lb/>
            hypotheſi, quoniam concurrunt, ut in puncto b:</s>
            <s xml:id="echoid-s20484" xml:space="preserve"> non ergo ęquidiſtat li
              <lb/>
            nea b e lineę c d:</s>
            <s xml:id="echoid-s20485" xml:space="preserve"> ergo neceſſariò cõcurrit linea b e cum linea c d:</s>
            <s xml:id="echoid-s20486" xml:space="preserve"> quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s20487" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div658" type="section" level="0" n="0">
          <head xml:id="echoid-head577" xml:space="preserve" style="it">3. Datis tribus lineis, cuilibet tertiæ ſecundum proportionẽ alia-
            <lb/>
          rum duarum proportionalem inuenire. E' 12 p 6 element.</head>
          <p>
            <s xml:id="echoid-s20488" xml:space="preserve">Sint datæ tres lineæ, quę ſint a b, c d, e f, quarum uni ut a b, ſecun-
              <lb/>
            dum proportionem aliarum duarum, quę ſunt c d & e f, quarta propor
              <lb/>
            tionalis debeat inueniri.</s>
            <s xml:id="echoid-s20489" xml:space="preserve"> Duæ itaque lineæ æquales duabus lineis,
              <lb/>
            quæ ſunt c d & e f, ab una linea continua abſcin dantur, quę ſit a e f per
              <lb/>
            3 p 1, & illi lineę a e fangulariter tertia data ſcilicet a b coniungatur in puncto a:</s>
            <s xml:id="echoid-s20490" xml:space="preserve"> & à puncto commu
              <lb/>
            ni diſtinguẽte duas lineas reſectas, (quod ſit punctum e) ducatur li-
              <lb/>
              <figure xlink:label="fig-0307-03" xlink:href="fig-0307-03a" number="259">
                <variables xml:id="echoid-variables245" xml:space="preserve">a b c d e f</variables>
              </figure>
            nea e b a d extremitatem tertię datarum, quę eſt a b:</s>
            <s xml:id="echoid-s20491" xml:space="preserve"> & à puncto f du-
              <lb/>
            catur linea ęquidiſtans lineę e b per 31 p 1, quę ſit f g.</s>
            <s xml:id="echoid-s20492" xml:space="preserve"> Deinde protraha
              <lb/>
            tur linea a b in cõtinuum & directum, quouſque ſecet lineã f g:</s>
            <s xml:id="echoid-s20493" xml:space="preserve"> ſeca-
              <lb/>
              <figure xlink:label="fig-0307-04" xlink:href="fig-0307-04a" number="260">
                <variables xml:id="echoid-variables246" xml:space="preserve">a e b f g</variables>
              </figure>
            bit aũt per pręmiſſam:</s>
            <s xml:id="echoid-s20494" xml:space="preserve"> ſit itaq;</s>
            <s xml:id="echoid-s20495" xml:space="preserve"> punctus cõcurſus g.</s>
            <s xml:id="echoid-s20496" xml:space="preserve"> Dico, quod per 2
              <lb/>
            p 6 eadem eſt proportio lineę a b ad lineam b g, quę eſt lineę a e datę
              <lb/>
            ad lineam e f datam.</s>
            <s xml:id="echoid-s20497" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s20498" xml:space="preserve"> de qualibet aliarum reſpectu re
              <lb/>
            liquarum duarum demonſtrari poteſt:</s>
            <s xml:id="echoid-s20499" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20500" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div660" type="section" level="0" n="0">
          <figure number="261">
            <variables xml:id="echoid-variables247" xml:space="preserve">a b c d g c d g f</variables>
          </figure>
          <head xml:id="echoid-head578" xml:space="preserve" style="it">4. Cum duabus lineis inæqualibus notæ proportionis, æqualiũ
            <lb/>
          linearum facta fuerit ad
            <lb/>
          ditio: maioris adminorẽ minuitur proportio. Ex 8 p 5 element.</head>
          <p>
            <s xml:id="echoid-s20501" xml:space="preserve">Sint duæ lineæ a b & c d
              <lb/>
            inæquales, notæ propor-
              <lb/>
            tionis:</s>
            <s xml:id="echoid-s20502" xml:space="preserve"> ſitq́ue linea a b ma-
              <lb/>
            ior quàm linea c d:</s>
            <s xml:id="echoid-s20503" xml:space="preserve"> adda-
              <lb/>
            tur quoq;</s>
            <s xml:id="echoid-s20504" xml:space="preserve"> linea b e ipſi a b,
              <lb/>
            & linea d f ipſi c d:</s>
            <s xml:id="echoid-s20505" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s20506" xml:space="preserve"> lineę b e & d f ęquales.</s>
            <s xml:id="echoid-s20507" xml:space="preserve"> Dico, quòd minor eſt proportio lineę a e ad lineam
              <lb/>
            c f, quàm lineę a b ad lineam c d.</s>
            <s xml:id="echoid-s20508" xml:space="preserve"> Quoniam enim datę ſunt tres lineę, quę ſunt a b & c d & b e:</s>
            <s xml:id="echoid-s20509" xml:space="preserve"> inue-
              <lb/>
            niatur per pręcedẽtem linea proportionalis lineę b e, ſecundum proportionem linearum a b & c d,
              <lb/>
            quę ſit d g.</s>
            <s xml:id="echoid-s20510" xml:space="preserve"> Quia ergo linea a b eſt maior quàm linea c d, patet, quia linea b e eſt maior quã linea d g:</s>
            <s xml:id="echoid-s20511" xml:space="preserve">
              <lb/>
            ergo & linea d f eſt maior quã linea d g.</s>
            <s xml:id="echoid-s20512" xml:space="preserve"> Abſcindatur ergo per 3 p 1 è linea d f ęqualis ipſi d g.</s>
            <s xml:id="echoid-s20513" xml:space="preserve"> Quia er-
              <lb/>
            go eſt proportio lineę a b ad lineam c d, ſicut lineę b e ad lineam d g:</s>
            <s xml:id="echoid-s20514" xml:space="preserve"> erit per 15 p 5 proportio totius
              <lb/>
            lineę a e ad totalem lineã c g, ſicut lineę a b ad lineam c d:</s>
            <s xml:id="echoid-s20515" xml:space="preserve"> ſed per 8 p 5 minor eſt proportio lineæ a e
              <lb/>
            </s>
          </p>
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