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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          <pb o="115" file="0121" n="121" rhead="OPTICAE LIBER IIII."/>
        </div>
        <div xml:id="echoid-div237" type="section" level="0" n="0">
          <head xml:id="echoid-head270" xml:space="preserve" style="it">23. Superficies reflexionis quatuor habet puncta: uiſibilis: reflexionis: uiſ{us}: & terminũ per-
            <lb/>
          pendicularis ductæ à puncto reflexionis ſuper planum in eodem puncto ſpeculum tangens. Ita
            <lb/>
          perpendicularis hæc cõmunis eſt omnib{us} reflexionis ſuperficieb{us}. 27 p 5.6 p 6.24 p 7.3 p 8.3 p 9.</head>
          <p>
            <s xml:id="echoid-s6766" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s6767" xml:space="preserve"> ſi opponatur ſpeculum uiſui:</s>
            <s xml:id="echoid-s6768" xml:space="preserve"> & intelligatur à cẽtro uiſus ad ſuperficiem ſpeculi py-
              <lb/>
            ramis & baſis illius pyramidis:</s>
            <s xml:id="echoid-s6769" xml:space="preserve"> & ſumatur punctum:</s>
            <s xml:id="echoid-s6770" xml:space="preserve"> & intelligatur linea pyramidis à centro
              <lb/>
            uiſus ad illud punctum:</s>
            <s xml:id="echoid-s6771" xml:space="preserve"> cum à puncto illo infinitæ poſsint produci lineæ:</s>
            <s xml:id="echoid-s6772" xml:space="preserve"> ſi aliqua earũ cum
              <lb/>
            latere pyramidis eundem habeat ſitum, & æqualem cum perpendiculari teneat angulum, & ita ac-
              <lb/>
            cidat quolibet puncto ſpeculi ſumpto:</s>
            <s xml:id="echoid-s6773" xml:space="preserve"> planũ, quòd à quolibet puncto ſpeculi poteſt fieri reflexio.</s>
            <s xml:id="echoid-s6774" xml:space="preserve">
              <lb/>
            Dico igitur, quòd inter lineas à puncto ſumpto productas, eſt linea, quæ eundẽ habet ſitum cum la-
              <lb/>
            tere pyramidis, & æqualem tenet angulum cum perpẽdiculari ſuper illud punctum:</s>
            <s xml:id="echoid-s6775" xml:space="preserve"> & illa linea eſt
              <lb/>
            latus pyramidis intellectæ à puncto illo ſuperficiei rei occurrẽtis:</s>
            <s xml:id="echoid-s6776" xml:space="preserve"> & quod ſuper terminum illius li-
              <lb/>
            neæ ceciderit, cum per eam ad punctũ ſumptum uenerit:</s>
            <s xml:id="echoid-s6777" xml:space="preserve"> reflectetur ad uiſum, per latus pyramidis
              <lb/>
            iam dictũ.</s>
            <s xml:id="echoid-s6778" xml:space="preserve"> Et huius pyramidis latus cum linea à puncto illo producta erit in eadẽ ſuperficie, ortho-
              <lb/>
            gonali ſuper ſuperficiẽ tãgentẽ ſpeculũ in illo pũcto.</s>
            <s xml:id="echoid-s6779" xml:space="preserve"> Et hoc dico, cũ lateris pyramidis ſuper punctũ
              <lb/>
            ſumptũ fuerit declinatio.</s>
            <s xml:id="echoid-s6780" xml:space="preserve"> Si enim orthogonaliter cadat ſuper ſuperficiẽ tangentẽ ſpeculũ in pũcto
              <lb/>
            ſumpto, latus pyramidis productum à cẽtro uiſus reflectetur in ſe, & redibit in uiſum ad originem
              <lb/>
            ſui motus [per 11 n.</s>
            <s xml:id="echoid-s6781" xml:space="preserve">] In ſpeculo plano planũ eſt:</s>
            <s xml:id="echoid-s6782" xml:space="preserve"> quod diximus.</s>
            <s xml:id="echoid-s6783" xml:space="preserve"> Quo
              <lb/>
              <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a" number="24">
                <variables xml:id="echoid-variables14" xml:space="preserve">e d f a c b</variables>
              </figure>
            niã in quodcunq;</s>
            <s xml:id="echoid-s6784" xml:space="preserve"> punctũ ſuperficiei planæ ceciderit radius:</s>
            <s xml:id="echoid-s6785" xml:space="preserve"> à pũcto
              <lb/>
            illo poteſt erigi linea orthogonalis ſuper ſuperficiẽ illã:</s>
            <s xml:id="echoid-s6786" xml:space="preserve"> & à cẽtro ui
              <lb/>
            ſus poteſt intelligi linea perpendiculariter cadẽs in ſuperficiẽ planã
              <lb/>
            prædictæ continuam, aut in eandẽ:</s>
            <s xml:id="echoid-s6787" xml:space="preserve"> & [per 35 d 1] hæ duæ perpendi-
              <lb/>
            culares erũt in eadẽ ſuperficie:</s>
            <s xml:id="echoid-s6788" xml:space="preserve"> quoniã ſunt æquidiſtãtes [per 6 p 11]
              <lb/>
            & linea à termino unius uſq;</s>
            <s xml:id="echoid-s6789" xml:space="preserve"> ad terminũ alterius protracta in ſuper-
              <lb/>
            ficie plana tenebit angulũ cum utraq;</s>
            <s xml:id="echoid-s6790" xml:space="preserve">: & erit in eadẽ ſuperficie cum
              <lb/>
            utraq;</s>
            <s xml:id="echoid-s6791" xml:space="preserve"> [per 2 p 11] & radius, qui à linea illa eleuatur:</s>
            <s xml:id="echoid-s6792" xml:space="preserve"> tenebit acutum
              <lb/>
            angulum cũ perpendiculari ſpeculi, & ſimiliter cum perpendiculari
              <lb/>
            uiſus [angulus enim d c e acutus eſt:</s>
            <s xml:id="echoid-s6793" xml:space="preserve"> quia pars recti d c a:</s>
            <s xml:id="echoid-s6794" xml:space="preserve"> & huic æ-
              <lb/>
            quatur a e c per 29 p 1:</s>
            <s xml:id="echoid-s6795" xml:space="preserve"> quia a e, d c ſunt parallelæ.</s>
            <s xml:id="echoid-s6796" xml:space="preserve">] Et ſi intelligatur
              <lb/>
            in partem alteram produci linea ſuperficiei planæ, tranſiens ortho-
              <lb/>
            gonaliter ſuper terminos perpendicularium:</s>
            <s xml:id="echoid-s6797" xml:space="preserve"> tenebit ex parte alte-
              <lb/>
            ra cum perpendiculari ſpeculi angulum rectum [per 29 p 1:</s>
            <s xml:id="echoid-s6798" xml:space="preserve">] unde
              <lb/>
            ex illo recto poterit abſcindi angulus acutus, æqualis angulo acu-
              <lb/>
            to, quem cum eadem perpendiculari tenet radius.</s>
            <s xml:id="echoid-s6799" xml:space="preserve"> Et hi duo anguli
              <lb/>
            ſunt in eadem ſuperficie.</s>
            <s xml:id="echoid-s6800" xml:space="preserve"> Quare radius exiens & reflexus in eadem
              <lb/>
            ſunt ſuperficie, & in ſuperficie perpendicularium dictarum.</s>
            <s xml:id="echoid-s6801" xml:space="preserve"> Inſpe-
              <lb/>
            cto autem alio puncto, idem ſitus accidet radiorum cum perpendi-
              <lb/>
            cularibus:</s>
            <s xml:id="echoid-s6802" xml:space="preserve"> quarum una à centro uiſus:</s>
            <s xml:id="echoid-s6803" xml:space="preserve"> alia à puncto uiſo.</s>
            <s xml:id="echoid-s6804" xml:space="preserve"> In omni ergo ſuperficie reflexionis accidit
              <lb/>
            quatuor punctorũ concurſus, quæ ſunt:</s>
            <s xml:id="echoid-s6805" xml:space="preserve"> centrũ uiſus:</s>
            <s xml:id="echoid-s6806" xml:space="preserve"> & punctũ apprehenſum:</s>
            <s xml:id="echoid-s6807" xml:space="preserve"> & terminus perpen-
              <lb/>
            dicularis à cẽtro uiſus ductæ:</s>
            <s xml:id="echoid-s6808" xml:space="preserve"> & punctũ reflexionis.</s>
            <s xml:id="echoid-s6809" xml:space="preserve"> Et oẽs reflexionis ſuքficies ſecãt ſe in քpẽdicu-
              <lb/>
            lari, à pũcto reflexionis intellecta:</s>
            <s xml:id="echoid-s6810" xml:space="preserve"> & eſt ipſa cõmunis omnib.</s>
            <s xml:id="echoid-s6811" xml:space="preserve"> ſuperficieb.</s>
            <s xml:id="echoid-s6812" xml:space="preserve"> reflexionis.</s>
            <s xml:id="echoid-s6813" xml:space="preserve"> Et cũ idẽ ac-
              <lb/>
            cidat, quolibet pũcto ſuքficiei planæ inſpecto:</s>
            <s xml:id="echoid-s6814" xml:space="preserve"> erit ex omnib.</s>
            <s xml:id="echoid-s6815" xml:space="preserve"> pũctis ſimilis reflexio & eodẽ modo.</s>
            <s xml:id="echoid-s6816" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div239" type="section" level="0" n="0">
          <head xml:id="echoid-head271" xml:space="preserve" style="it">24. Si uiſ{us} ſit extra ſuperficiem ſpeculi ſphærici conuexi, uelipſi continuam: communis ſe-
            <lb/>
          ctio baſis pyramidis opticæ & ſuperficiei ſpeculi, erit peripheria
            <lb/>
          minimi in ſphæra circuli. 3 p 6.</head>
          <p>
            <s xml:id="echoid-s6817" xml:space="preserve">IN ſpeculis autem ſphęricis palàm erit, quod diximus:</s>
            <s xml:id="echoid-s6818" xml:space="preserve"> oppoſito
              <lb/>
              <figure xlink:label="fig-0121-02" xlink:href="fig-0121-02a" number="25">
                <variables xml:id="echoid-variables15" xml:space="preserve">a s b c</variables>
              </figure>
            uiſui ſpeculo ſphærico:</s>
            <s xml:id="echoid-s6819" xml:space="preserve"> (& eſt oppoſitio, ut uiſus nõ ſit in ſuper-
              <lb/>
            ficie illius ſpeculi:</s>
            <s xml:id="echoid-s6820" xml:space="preserve"> aut in ſuperficie ei continua) & inſpecto hoc
              <lb/>
            ſpeculo:</s>
            <s xml:id="echoid-s6821" xml:space="preserve"> pars eius à uiſu comprehenſa, erit pars ſphæræ circulo mi-
              <lb/>
            nore incluſa, quem efficit motu ſuo radius, tangẽs ſuperficiẽ ſphæ-
              <lb/>
            ræ, ſi per gyrum moueatur contingendo ſphæram, donec redeat ad
              <lb/>
            punctum primum, à quo ſumpſit motus principium:</s>
            <s xml:id="echoid-s6822" xml:space="preserve"> quia ſi intelli-
              <lb/>
            gantur ſuperficies ſe ſecantes ſuper diametrum ſphæræ, à polo cir-
              <lb/>
            culi prædicti intellectam:</s>
            <s xml:id="echoid-s6823" xml:space="preserve"> quilibet arcuum ſuperficiei ſphęræ, & his
              <lb/>
            ſuperficiebus communium, à polo circuli ad ipſum circulum intel-
              <lb/>
            lectorum, erit minor quarta circuli magni.</s>
            <s xml:id="echoid-s6824" xml:space="preserve"> Quoniam linea à centro
              <lb/>
            ſphæræ ad terminum radij, ſphæram contingentis protracta (quæ
              <lb/>
            eſt ad circulum prædictum) tenet cum radio angulum rectum ra-
              <lb/>
            tione contingentiæ [per 18 p 3.</s>
            <s xml:id="echoid-s6825" xml:space="preserve">] Tenet ergo angulum acutum cum
              <lb/>
            ſemidiametro à polo circuli producta [per 17 p 1] & hunc angulum
              <lb/>
            reſpicit arcus interiacens
              <gap/>
            polum circuli & circulum [Quare per 33
              <lb/>
            p 6 peripheria c s minor eſt quadrãte peripheriæ maximi in ſphæ-
              <lb/>
            ra circuli.</s>
            <s xml:id="echoid-s6826" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s6827" xml:space="preserve"> cum per 16 th.</s>
            <s xml:id="echoid-s6828" xml:space="preserve"> 1 ſphęr.</s>
            <s xml:id="echoid-s6829" xml:space="preserve"> Theodoſij peripheria maximi
              <lb/>
            </s>
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