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-div1794" type="section" level="0" n="0">
          <pb o="405" file="0707" n="707" rhead="LIBER DECIMVS."/>
        </div>
        <div xml:id="echoid-div1795" type="section" level="0" n="0">
          <head xml:id="echoid-head1328" xml:space="preserve">THEOREMATA</head>
          <head xml:id="echoid-head1329" xml:space="preserve" style="it">1. In omni ſuperficie refractionis neceſſariò ſunt punctum, cuius forma refringitur: & pun-
            <lb/>
          ctum refractionis: & centrum ipſius uiſus: & perpendicularis ducta à puncto refractionis ſu-
            <lb/>
          per ſuperficiem, à qua fit refractio. Ex quo patet quòd unius refractionis unica tantùm eſt
            <lb/>
          ſuperficies.</head>
          <p>
            <s xml:id="echoid-s47131" xml:space="preserve">Sit ſuperficies ſecundi diaphani denſioris uel rarioris primo diaphano, in qua ſit linea a b c:</s>
            <s xml:id="echoid-s47132" xml:space="preserve"> & ſit
              <lb/>
            punctũ, cuius forma refringitur, punctum d:</s>
            <s xml:id="echoid-s47133" xml:space="preserve"> ſitq́ue centrum uiſus e:</s>
            <s xml:id="echoid-s47134" xml:space="preserve"> fiatq́ue refractio in puncto ſu-
              <lb/>
            perficiei ſecundi diaphani, quod eſt b:</s>
            <s xml:id="echoid-s47135" xml:space="preserve"> & à puncto b ſuper ſuperficiem a b c ducatur perpendicula-
              <lb/>
            ris b f.</s>
            <s xml:id="echoid-s47136" xml:space="preserve"> Dico quòd puncta d, e, b, & linea b f ſunt ſemper in eadem ſuperficie refractionis.</s>
            <s xml:id="echoid-s47137" xml:space="preserve"> Quoniam
              <lb/>
            eni m, ut patet per definitionem præmiſſam in principijs libri huius, & per 46 th.</s>
            <s xml:id="echoid-s47138" xml:space="preserve"> 2 huius linea radia
              <lb/>
            lis incidens (quæ eſt d b) & refracta (quæ eſt b e) ſunt in eadem ſuperficie refractionis:</s>
            <s xml:id="echoid-s47139" xml:space="preserve"> punctum er
              <lb/>
            go d, cuius forma incidit & refringitur, & punctum refractionis, ſcilicet pũctum, à quo fit refra ctio,
              <lb/>
            (quod eſt b) & centrum uiſus (quod eſt e) ſunt in eadem ſuperficie per 1 p 11:</s>
            <s xml:id="echoid-s47140" xml:space="preserve"> ſed & per 2 p 11 linea
              <lb/>
            b f, quæ eſt perpendicularis ſuper ſuperficiem a b c,
              <lb/>
            eſt in eadem ſuperficie cum linea b c:</s>
            <s xml:id="echoid-s47141" xml:space="preserve"> ergo & cum
              <lb/>
              <figure xlink:label="fig-0707-01" xlink:href="fig-0707-01a" number="841">
                <variables xml:id="echoid-variables818" xml:space="preserve">e g f a b c d</variables>
              </figure>
            lineis d b & b e:</s>
            <s xml:id="echoid-s47142" xml:space="preserve"> quoniam linea b f eſt perpendi-
              <lb/>
            cularis ſuper lineam a b c, & cum illa in eadem ſu-
              <lb/>
            perficie.</s>
            <s xml:id="echoid-s47143" xml:space="preserve"> Similiter protracta linea d b ultra punctum
              <lb/>
            b ad punctum g, eſt in eadem ſuperficie.</s>
            <s xml:id="echoid-s47144" xml:space="preserve"> Puncta ita-
              <lb/>
            que d, b, e & linea b f ſunt in eadem ſuperficie per 1
              <lb/>
            & 2 p 11.</s>
            <s xml:id="echoid-s47145" xml:space="preserve"> Omnis enim refractio aut fit ad ipſam per-
              <lb/>
            pendicularem b f, aut ab ipſa:</s>
            <s xml:id="echoid-s47146" xml:space="preserve"> & ſemper in eadem
              <lb/>
            ſuperficie, in qua fiebat incidentia formę refringen
              <lb/>
            dæ.</s>
            <s xml:id="echoid-s47147" xml:space="preserve"> Quoniam enim omnis refractio fit ad omnem
              <lb/>
            differentiam poſitionis (quia qua ratione fit ad u-
              <lb/>
            nam partem, eadem ratiõe fit ad quamlibet aliam)
              <lb/>
            determinatio ergo refractionis ad certam differen
              <lb/>
            tiam poſitionis fit tantùm per uiſum:</s>
            <s xml:id="echoid-s47148" xml:space="preserve"> quia in qua-
              <lb/>
            cunque ſuperficie centrum uiſus fuerit, in illa tan-
              <lb/>
            tùm percipitur fieri refractio.</s>
            <s xml:id="echoid-s47149" xml:space="preserve"> Patet ergo propoſi-
              <lb/>
            tum.</s>
            <s xml:id="echoid-s47150" xml:space="preserve"> Et ex hoc patet, cum iſta puncta refractionis omnia ſcilicet d, e, b & linea b f ſuperficiem refra-
              <lb/>
            ctionis conſtituant, quòd horum aliquo deficiente non eſt ſuperficies refractionis:</s>
            <s xml:id="echoid-s47151" xml:space="preserve"> & quòd unius
              <lb/>
            refractionis unica tantùm eſt ſuperficies refractionis:</s>
            <s xml:id="echoid-s47152" xml:space="preserve"> quoniam hæc omnia puncta in unica tantùm
              <lb/>
            ſuperficie ſimili concurrere eſt poſsibile, & non in pluribus.</s>
            <s xml:id="echoid-s47153" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s>
            <s xml:id="echoid-s47154" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1797" type="section" level="0" n="0">
          <head xml:id="echoid-head1330" xml:space="preserve" style="it">2. Neceſſe eſt omnem ſuperficiem refractionis ſuper ſuperficiem corporis, à qua fit refractio
            <lb/>
          (ſiue illa ſuperficies ſit plana conuexa uel concaua) erectam eſſe. Alhazen 9 n 7.</head>
          <p>
            <s xml:id="echoid-s47155" xml:space="preserve">Hoc, quod hic proponitur, patet per præmiſſam.</s>
            <s xml:id="echoid-s47156" xml:space="preserve"> Quoniam enim in omni ſuperficie refractionis
              <lb/>
            neceſſariò ſunt:</s>
            <s xml:id="echoid-s47157" xml:space="preserve"> punctum, cuius forma refringitur:</s>
            <s xml:id="echoid-s47158" xml:space="preserve"> & punctum ſuperficiei corporis, à quo fit refra-
              <lb/>
            ctio:</s>
            <s xml:id="echoid-s47159" xml:space="preserve"> & centrum uiſus & perpendicularis ducta à puncto refractionis ſuper ſuperficiem corporis il
              <lb/>
            lius, à qua fit refractio:</s>
            <s xml:id="echoid-s47160" xml:space="preserve"> ergo per 18 p 11 patet quòd omnis ſuperficies refractionis eſt perpendicula-
              <lb/>
            ris ſuper ſuperficiem corporis, à qua fit refractio.</s>
            <s xml:id="echoid-s47161" xml:space="preserve"> Si enim illa ſuperficies fuerit plana:</s>
            <s xml:id="echoid-s47162" xml:space="preserve"> tunc euiden-
              <lb/>
            ter patet propoſitum per 18 p 11, ut præ miſſum eſt.</s>
            <s xml:id="echoid-s47163" xml:space="preserve"> Si uerò fuerit illa ſuperficies conuexa uel con-
              <lb/>
            caua ſphærica:</s>
            <s xml:id="echoid-s47164" xml:space="preserve"> tunc patet per 72 th.</s>
            <s xml:id="echoid-s47165" xml:space="preserve"> 1 huius quoniam perpendicularis ducta à puncto refractionis ſu
              <lb/>
            per ipſam ſuperficiem corporis, à qua fit refractio, ſemper tranſit centrũ illius corporis:</s>
            <s xml:id="echoid-s47166" xml:space="preserve"> & eſt perpẽ-
              <lb/>
            dicularis ſuper ſuperficiem illud corpus in puncto refractionis contingentem:</s>
            <s xml:id="echoid-s47167" xml:space="preserve"> ergo itẽ per 18 p 11
              <lb/>
            ſuperficies refractionis eſt erecta ſuper illã ſuperficiẽ contingentẽ:</s>
            <s xml:id="echoid-s47168" xml:space="preserve"> ergo & ſuper ipſam corporis ſu-
              <lb/>
            perficiem.</s>
            <s xml:id="echoid-s47169" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s47170" xml:space="preserve"> demonſtrandum, ſiue figura corporis, à qua fit refractio, fuerit columna
              <lb/>
            ris ſiue pyramidalis ſiue alterius figuræ cuiuſcunq;</s>
            <s xml:id="echoid-s47171" xml:space="preserve">: ſemper enim ſuperficies refractionis erit erecta
              <lb/>
            ſuper ſuperficiem corporis, à qua fit refractio.</s>
            <s xml:id="echoid-s47172" xml:space="preserve"> Et ſi accidat, ut illa ſuperficies corporis, à qua fit refra
              <lb/>
            ctio, fuerit æquidiſtans horizonti:</s>
            <s xml:id="echoid-s47173" xml:space="preserve"> tunc perpendicularis ducta à puncto refractionis ſuper ſuperfi-
              <lb/>
            ciem corporis, à qua fit reſractio, eſt ctiam perpendicularis ſuper ſuperficiem horizontis per 23 th.</s>
            <s xml:id="echoid-s47174" xml:space="preserve"> 1
              <lb/>
            huius:</s>
            <s xml:id="echoid-s47175" xml:space="preserve"> ergo & per 18 p 11 ſuperficies refractionis eſt perpendicularis, & erecta ſuper ſuperficiem ho-
              <lb/>
            rizontis.</s>
            <s xml:id="echoid-s47176" xml:space="preserve"> Sed & hoc patet per declarationem, quæ fit in inſtrumento, quod in 1 th.</s>
            <s xml:id="echoid-s47177" xml:space="preserve"> 2 huius præmiſi-
              <lb/>
            mus.</s>
            <s xml:id="echoid-s47178" xml:space="preserve"> Quoniam enim linea radialis incidens & refracta ab aliqua ſuperficie unius corporis diapha-
              <lb/>
            ni ad aliud corpus diaphanum, ut patet per 46 th.</s>
            <s xml:id="echoid-s47179" xml:space="preserve"> 2 huius, ſemper ſunt in una plana ſuperficie, quæ
              <lb/>
            eſt medius circulus illorũ triũ circulorũ ſignatorũ in interiori parte oræ inſtrumenti, æquidiſtans
              <lb/>
            ſuperficiei interioris laminæ inſtrumẽti:</s>
            <s xml:id="echoid-s47180" xml:space="preserve"> ſed illa ſuperficies laminæ æ quidiſtat ſuքficiei dorſi inſtru
              <lb/>
            mẽti, cui extrinſecus ſuքponitur ſuperficies regulæ cubitalis tenentis inſtrumentũ.</s>
            <s xml:id="echoid-s47181" xml:space="preserve"> Suքficies itaq;</s>
            <s xml:id="echoid-s47182" xml:space="preserve">
              <lb/>
            medij circuli ęquidiſtat ſuքficiei regulę lógę quadrãgulę ſuքpoſitę dorſo laminę ք 24.</s>
            <s xml:id="echoid-s47183" xml:space="preserve"> th.</s>
            <s xml:id="echoid-s47184" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s47185" xml:space="preserve"> ſed
              <lb/>
            illa ſuքficies քpẽdicularis eſt ſuք ſuքficies laterũ lógitudinis regulę erectas ſuք oras inſtrumẽti.</s>
            <s xml:id="echoid-s47186" xml:space="preserve"> Su
              <lb/>
            perficies itaq;</s>
            <s xml:id="echoid-s47187" xml:space="preserve"> medij circuli eſt ք cõuerſam 14 p 11 քpendicularis ſuper ſuքficies lõgitudinis regulæ
              <lb/>
            </s>
          </p>
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