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

Page concordance

< >
Scan Original
121 115
122 116
123 117
124 118
125 119
126 120
127 121
128 122
129 123
130 124
131 125
132 126
133 127
134 128
135 129
136 130
137 131
138 132
139 133
140 134
141 135
142 136
143 137
144 138
145 139
146 140
147 141
148 142
149 143
150 144
< >
page |< < (115) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div236" type="section" level="0" n="0">
          <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>
          </p>
        </div>
      </text>
    </echo>
Impressum