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-div357" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s9766" xml:space="preserve">
              <pb o="155" file="0161" n="161" rhead="OPTICAE LIBER V."/>
            ſtans baſi [per 5 th Sereni de ſectione cylindri.</s>
            <s xml:id="echoid-s9767" xml:space="preserve">] Et iam patuit [29 n] quòd ab alio pũcto illius cir-
              <lb/>
            culi non poteſt fieri ad a reflexio.</s>
            <s xml:id="echoid-s9768" xml:space="preserve"> Et ſi ab alio
              <lb/>
              <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a" number="87">
                <variables xml:id="echoid-variables77" xml:space="preserve">a q k b f l n g c e l d h</variables>
              </figure>
            puncto ſpeculi fiat reflexio perpẽdicularis du
              <lb/>
            cta à puncto illo, cadet orthogonaliter ſuper
              <lb/>
            axẽ.</s>
            <s xml:id="echoid-s9769" xml:space="preserve"> [Nã cũ per 34 n 4 perpẽdicularis illa in-
              <lb/>
            tus cõtinuata fiat diameter circuli baſibus pa
              <lb/>
            ralleli:</s>
            <s xml:id="echoid-s9770" xml:space="preserve"> erit per 21 d 11.</s>
            <s xml:id="echoid-s9771" xml:space="preserve"> 29 p 1 ad axem perpendi
              <lb/>
            cularis] & ſecabit lineã a b in puncto aliquo.</s>
            <s xml:id="echoid-s9772" xml:space="preserve">
              <lb/>
            À
              <unsure/>
            pũcto illo ducatur linea ad axem in ſuper-
              <lb/>
            ficie, æquidiſtante baſi colũnæ:</s>
            <s xml:id="echoid-s9773" xml:space="preserve"> erit quidẽ or-
              <lb/>
            thogonalis ſuper axem [per 21 d 11.</s>
            <s xml:id="echoid-s9774" xml:space="preserve"> 29 p 1.</s>
            <s xml:id="echoid-s9775" xml:space="preserve">] Et
              <lb/>
            ita duæ perpẽdiculares efficient cũ axe trian-
              <lb/>
            gulum, cuius duo anguli ſunt recti:</s>
            <s xml:id="echoid-s9776" xml:space="preserve"> quod eſt
              <lb/>
            impoſsibile [& contra 32 p 1.</s>
            <s xml:id="echoid-s9777" xml:space="preserve">] Palàm ergo, quòd in hoc ſitu non reflectetur b ad a, niſi à puncto g.</s>
            <s xml:id="echoid-s9778" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div359" type="section" level="0" n="0">
          <head xml:id="echoid-head350" xml:space="preserve" style="it">47. Si communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei conuexi fuerit elli-
            <lb/>
          pſis: ab uno puncto unum uiſibilis punctum ad unum uiſum reflectetur. 28 p 7.</head>
          <p>
            <s xml:id="echoid-s9779" xml:space="preserve">SIuerò ſuperficies a b g ſecet ſpeculũ ſectione columnari:</s>
            <s xml:id="echoid-s9780" xml:space="preserve"> dico, quòd à ſolo pũcto g fit reflexio.</s>
            <s xml:id="echoid-s9781" xml:space="preserve">
              <lb/>
            Ducatur à puncto a ſuperficies æquidiſtans baſi columnæ:</s>
            <s xml:id="echoid-s9782" xml:space="preserve"> [ductis nimirũ duabus perpen di-
              <lb/>
            cularibus ſuper axem ſe interſecãtibus:</s>
            <s xml:id="echoid-s9783" xml:space="preserve"> una quidẽ à puncto a per 12 p 1:</s>
            <s xml:id="echoid-s9784" xml:space="preserve"> altera uerò ab axis pun
              <lb/>
            cto, in quod illa cadit per 11 p 1.</s>
            <s xml:id="echoid-s9785" xml:space="preserve"> Sic enim axis, qui per 21 d 11 eſt perpendicularis baſi:</s>
            <s xml:id="echoid-s9786" xml:space="preserve"> erit per 4 p 11
              <lb/>
            perpendicularis plano ductarũ perpen diculariũ.</s>
            <s xml:id="echoid-s9787" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9788" xml:space="preserve"> per 14 p 11 baſis & hoc planũ erũt parallela]
              <lb/>
            quæ ſit e z i:</s>
            <s xml:id="echoid-s9789" xml:space="preserve"> & à puncto g ſimiliter ſuperficies æquidiſtans baſi ſpeculi:</s>
            <s xml:id="echoid-s9790" xml:space="preserve"> in qua ducatur ab axe linea
              <lb/>
            ad pũctũ g:</s>
            <s xml:id="echoid-s9791" xml:space="preserve"> quæ ſit t g:</s>
            <s xml:id="echoid-s9792" xml:space="preserve"> erit quidẽ perpẽdicularis ſuper ſuperficiẽ, cõtingẽtẽ ſpeculũ in pũcto g [per
              <lb/>
            34 n 4:</s>
            <s xml:id="echoid-s9793" xml:space="preserve"> quia eſt diameter circuli baſibus cylindri paralleli] & cõcurrat cũ a b in puncto k [cõcurret
              <lb/>
            aũt:</s>
            <s xml:id="echoid-s9794" xml:space="preserve"> quia diuidit angulũ a g b] & ducatur à puncto g linea lõgitudinis ſpeculi:</s>
            <s xml:id="echoid-s9795" xml:space="preserve"> [educto nẽpe plano
              <lb/>
            per axem & per rectã, cũ ipſo à puncto g utlibet cõcurrentẽ:</s>
            <s xml:id="echoid-s9796" xml:space="preserve"> erit enim huius plani & cylindraceæ ſu
              <lb/>
            perficiei cõmunis ſectio latus cylindri per 21 d 11] quæ ſit g z:</s>
            <s xml:id="echoid-s9797" xml:space="preserve"> & ſit axis t q:</s>
            <s xml:id="echoid-s9798" xml:space="preserve"> & à puncto b perpẽdicu
              <lb/>
            laris ducatur ad ſuperficiẽ e z i:</s>
            <s xml:id="echoid-s9799" xml:space="preserve"> quę ſit b h:</s>
            <s xml:id="echoid-s9800" xml:space="preserve"> & ducãtur lineę a z, h z:</s>
            <s xml:id="echoid-s9801" xml:space="preserve"> & ducatur à pũcto z in ſuperficie
              <lb/>
            illa ad axem linea, quæ ſit z q:</s>
            <s xml:id="echoid-s9802" xml:space="preserve"> erit quidẽ perpẽdicularis ſuper axem [per 3 d 11] cũ axis ſit perpẽdi-
              <lb/>
            cularis ſuper hãc ſuperficiẽ [per 21 d 11] & erit perpẽdicularis ſuper ſuperficiẽ, cõtingentẽ ſpeculũ
              <lb/>
            in puncto z [ut paulò antè oſtẽſum eſt] & cõcurrat cũ linea a k in pũcto l.</s>
            <s xml:id="echoid-s9803" xml:space="preserve"> [cõcurret uerò, quia ſe-
              <lb/>
            cat angulũ a z h.</s>
            <s xml:id="echoid-s9804" xml:space="preserve">] Dico, quòd forma puncti h reflectetur ad a, à puncto z.</s>
            <s xml:id="echoid-s9805" xml:space="preserve"> Ducatur à pũcto a æ quidi-
              <lb/>
            ſtãs lineę k g:</s>
            <s xml:id="echoid-s9806" xml:space="preserve"> quę ſit a m:</s>
            <s xml:id="echoid-s9807" xml:space="preserve"> quę quidẽ cõcurret cũ b g.</s>
            <s xml:id="echoid-s9808" xml:space="preserve"> [per lẽma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s9809" xml:space="preserve">] Sit cõcurſus in pun
              <lb/>
            cto m.</s>
            <s xml:id="echoid-s9810" xml:space="preserve"> Palàm [per 6 p 11] quòd g z eſt æquidiſtãs lineæ b h:</s>
            <s xml:id="echoid-s9811" xml:space="preserve"> cũ utraq;</s>
            <s xml:id="echoid-s9812" xml:space="preserve"> ſit orthogonalis ſuper ſuperfi
              <lb/>
            ciẽ æquidiſtantẽ baſibus colũnæ.</s>
            <s xml:id="echoid-s9813" xml:space="preserve"> Quare [per 7 p 11] linea b g m eſt in ſuperficie harũ linearũ.</s>
            <s xml:id="echoid-s9814" xml:space="preserve"> Igitur
              <lb/>
            tria pũcta m, z, h ſunt in hac
              <lb/>
              <figure xlink:label="fig-0161-02" xlink:href="fig-0161-02a" number="88">
                <variables xml:id="echoid-variables78" xml:space="preserve">a ſ f K b h d z g e s n q o t m i p</variables>
              </figure>
            ſuքficie.</s>
            <s xml:id="echoid-s9815" xml:space="preserve"> Sed iterũ a m eſt æ-
              <lb/>
            quidiſtans k g [per fabrica-
              <lb/>
            tionẽ] & l z æquidiſtãs k g:</s>
            <s xml:id="echoid-s9816" xml:space="preserve">
              <lb/>
            quoniã g z æquidiſtãs t q &
              <lb/>
            inter ſuperficies æquidiſtan
              <lb/>
            tes.</s>
            <s xml:id="echoid-s9817" xml:space="preserve"> [nã per 21 d 11 latus z g &
              <lb/>
            axis q t paralleli & æquales,
              <lb/>
            circulis oppoſitis & paral-
              <lb/>
            lelis terminantur, in quibus
              <lb/>
            ſemidiametritg, q z ſunt pa
              <lb/>
            rallelę per 33 p 1:</s>
            <s xml:id="echoid-s9818" xml:space="preserve"> & t g conti-
              <lb/>
            nuata eſt in k.</s>
            <s xml:id="echoid-s9819" xml:space="preserve">] Igitur l z æ-
              <lb/>
            quidiſtãs a m [ք 30 p 1:</s>
            <s xml:id="echoid-s9820" xml:space="preserve"> ſunt
              <lb/>
            enim m a, z l eidẽ t g k paral-
              <lb/>
            lelæ.</s>
            <s xml:id="echoid-s9821" xml:space="preserve">] Quare ſunt in eadem
              <lb/>
            ſuperficie [per 35 d 1] & in ea eſt linea a h [per 7 p 11:</s>
            <s xml:id="echoid-s9822" xml:space="preserve"> quia cõnectit m a, z l parallelas.</s>
            <s xml:id="echoid-s9823" xml:space="preserve">] Igitur in hac
              <lb/>
            ſuperficie ſunt tria puncta, m, z, h:</s>
            <s xml:id="echoid-s9824" xml:space="preserve"> & iã patuit, quòd ſint in ſuperficie b m h:</s>
            <s xml:id="echoid-s9825" xml:space="preserve"> igitur ſunt in linea cõmu
              <lb/>
            ni his duabus ſuperficiebus.</s>
            <s xml:id="echoid-s9826" xml:space="preserve"> Igitur [per 3 p 11] h z m eſt linea recta.</s>
            <s xml:id="echoid-s9827" xml:space="preserve"> Palàm igitur, cum g ſit punctum
              <lb/>
            reflexionis:</s>
            <s xml:id="echoid-s9828" xml:space="preserve"> erit [per 12 n 4] angulus a g k æqualis angulo k g b:</s>
            <s xml:id="echoid-s9829" xml:space="preserve"> & ita [per 29 p 1.</s>
            <s xml:id="echoid-s9830" xml:space="preserve">1 ax.</s>
            <s xml:id="echoid-s9831" xml:space="preserve">] ęqualis an-
              <lb/>
            gulo a m g:</s>
            <s xml:id="echoid-s9832" xml:space="preserve"> ſed [per 29 p 1] eſt æqualis m a g:</s>
            <s xml:id="echoid-s9833" xml:space="preserve"> quia coalternus.</s>
            <s xml:id="echoid-s9834" xml:space="preserve"> Igitur [per 6 p 1] a g, m g ſunt æ qua
              <lb/>
            les.</s>
            <s xml:id="echoid-s9835" xml:space="preserve"> Sed quoniam g z eſt orthogonalis ſuper quãlibet lineã ſuperficiei z a h:</s>
            <s xml:id="echoid-s9836" xml:space="preserve"> [per 3 d 11] erit quadra
              <lb/>
            tũ m g æquale quadratis m z, g z [per 47 p 1] erit igitur a z æqualis m z [Nam propter eandẽ cauſ-
              <lb/>
            ſam quadratum a g æquatur quadratis a z, g z:</s>
            <s xml:id="echoid-s9837" xml:space="preserve"> at quadrata a g, m g æquãtur:</s>
            <s xml:id="echoid-s9838" xml:space="preserve"> quia ipſorum latera a g,
              <lb/>
            m g æquãtur:</s>
            <s xml:id="echoid-s9839" xml:space="preserve"> communi igitur quadrato g z ablato, reliquum quadratũ a z ęquabitur quadrato m z:</s>
            <s xml:id="echoid-s9840" xml:space="preserve">
              <lb/>
            quare ipſorũ latera m z, a z ęquabuntur.</s>
            <s xml:id="echoid-s9841" xml:space="preserve">] Quare [per 5 p 1] angulus a m z eſt æqualis angulo m a z:</s>
            <s xml:id="echoid-s9842" xml:space="preserve">
              <lb/>
            ſed [per 29 p 1] angulus a m z eſt æqualis angulo l z h:</s>
            <s xml:id="echoid-s9843" xml:space="preserve"> & angulus z a m eſt æqualis l z a:</s>
            <s xml:id="echoid-s9844" xml:space="preserve"> quia coal-
              <lb/>
            ternus.</s>
            <s xml:id="echoid-s9845" xml:space="preserve"> Igitur angulus a z l eſt æqualis angulo l z h.</s>
            <s xml:id="echoid-s9846" xml:space="preserve"> Quare forma puncti h accedẽs ad punctũ z, re-
              <lb/>
            </s>
          </p>
        </div>
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    </echo>
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