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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          <p>
            <s xml:id="echoid-s9952" xml:space="preserve">
              <pb o="157" file="0163" n="163" rhead="OPTICAE LIBER V."/>
            culum [per 4 th.</s>
            <s xml:id="echoid-s9953" xml:space="preserve"> 1 conicorũ Apollonij] ꝗ ſit p g:</s>
            <s xml:id="echoid-s9954" xml:space="preserve"> & ducãtur lineę a g, b g, a b:</s>
            <s xml:id="echoid-s9955" xml:space="preserve"> & à pũcto g ducatur ad
              <lb/>
            cẽtrũ circuli linea:</s>
            <s xml:id="echoid-s9956" xml:space="preserve"> q̃ ſit g t:</s>
            <s xml:id="echoid-s9957" xml:space="preserve"> & uertex pyramidis ſit e:</s>
            <s xml:id="echoid-s9958" xml:space="preserve"> à quo ducatur axis:</s>
            <s xml:id="echoid-s9959" xml:space="preserve"> ꝗ erit e t.</s>
            <s xml:id="echoid-s9960" xml:space="preserve"> [per 3 d 1 coni.</s>
            <s xml:id="echoid-s9961" xml:space="preserve"> A-
              <lb/>
            pol.</s>
            <s xml:id="echoid-s9962" xml:space="preserve">] Et ducatur [per 12 p 11] perpẽdicularis ſuper ſuperficiẽ, cõtingentẽ fpeculũ in pũcto g:</s>
            <s xml:id="echoid-s9963" xml:space="preserve"> q̃ ſit h
              <lb/>
            g:</s>
            <s xml:id="echoid-s9964" xml:space="preserve"> q̃ cũ diuidat angulũ a g b per æqualia, [per 13 n 4] cadet ſuper a b:</s>
            <s xml:id="echoid-s9965" xml:space="preserve"> pũctũ caſus ſit z.</s>
            <s xml:id="echoid-s9966" xml:space="preserve"> Et à uertice py
              <lb/>
            ramidis ducatur linea lõgitudinis ſpeculi ad punctũ g:</s>
            <s xml:id="echoid-s9967" xml:space="preserve"> [educto nẽpe plano per axem, & perrectã à
              <lb/>
            puncto g, cũ ipſo utlibet cõcurrentẽ:</s>
            <s xml:id="echoid-s9968" xml:space="preserve"> cõmunis enim fectio huius plani & conicæ ſuperficiei erit la-
              <lb/>
            tus coni, ք 18 d 11, uel 3 th.</s>
            <s xml:id="echoid-s9969" xml:space="preserve"> 1 coni.</s>
            <s xml:id="echoid-s9970" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s9971" xml:space="preserve">] quę ſit e g:</s>
            <s xml:id="echoid-s9972" xml:space="preserve"> cui lineæ ducatur æquidiſtãs à pũcto a:</s>
            <s xml:id="echoid-s9973" xml:space="preserve"> [per 31 p 1]
              <lb/>
            quę neceſſariò ſecabit ſuperficiẽ circuli g p:</s>
            <s xml:id="echoid-s9974" xml:space="preserve"> [ſi enim circulũ cũ diametro infinitè extẽſum cogites:</s>
            <s xml:id="echoid-s9975" xml:space="preserve">
              <lb/>
            diameter ſecãs e g conilatus, ſecabit etiã rectã lateri parallelã, per lẽma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s9976" xml:space="preserve"> Quare eadẽ
              <lb/>
            parallela circulũ ipſum quoq;</s>
            <s xml:id="echoid-s9977" xml:space="preserve"> ſecabit] ſecet in pũcto n:</s>
            <s xml:id="echoid-s9978" xml:space="preserve"> & ſit n a.</s>
            <s xml:id="echoid-s9979" xml:space="preserve"> Similiter à pũcto b ducatur æqui-
              <lb/>
            diſtãs eidẽ e g, ſcilicet b m:</s>
            <s xml:id="echoid-s9980" xml:space="preserve"> quę ſecet ſuperficiẽ p g in pũcto m.</s>
            <s xml:id="echoid-s9981" xml:space="preserve"> Et à pũcto n ducatur ę ꝗ diſtãs ipſi g t:</s>
            <s xml:id="echoid-s9982" xml:space="preserve">
              <lb/>
            quę ſit n f:</s>
            <s xml:id="echoid-s9983" xml:space="preserve"> & ducãtur lineæ n g, m g, n m.</s>
            <s xml:id="echoid-s9984" xml:space="preserve"> Palàm, quòd t g ſecabit m n:</s>
            <s xml:id="echoid-s9985" xml:space="preserve"> [per lẽma Procli ad 29 p 1] ſe-
              <lb/>
            cetin pũcto q.</s>
            <s xml:id="echoid-s9986" xml:space="preserve"> Palàm etiã, quòd m g ſecabit n f:</s>
            <s xml:id="echoid-s9987" xml:space="preserve"> cũ ſecet ei æquidiſtãtẽ:</s>
            <s xml:id="echoid-s9988" xml:space="preserve"> ſit pũctũ ſectionis f.</s>
            <s xml:id="echoid-s9989" xml:space="preserve"> Et à pun
              <lb/>
            cto a ducatur æquidiſtãs h z:</s>
            <s xml:id="echoid-s9990" xml:space="preserve"> quę ſit a l.</s>
            <s xml:id="echoid-s9991" xml:space="preserve"> Palàm [per lẽma Procli ad 29 p 1] quòd b g cõcurret cũ a l:</s>
            <s xml:id="echoid-s9992" xml:space="preserve">
              <lb/>
            ſit cõcurſus l.</s>
            <s xml:id="echoid-s9993" xml:space="preserve"> Deinde ducatur linea cõmunis ſuperficiei,
              <lb/>
              <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a" number="89">
                <variables xml:id="echoid-variables79" xml:space="preserve">f d a e p t m f k h i g z o q n b</variables>
              </figure>
            cõtingẽti ſpeculũ in puncto g, & ſuperficiei circuli p g:</s>
            <s xml:id="echoid-s9994" xml:space="preserve"> q̃
              <lb/>
            ſit g o.</s>
            <s xml:id="echoid-s9995" xml:space="preserve"> Palàm [per 18 p 3] quòd erit orthogonalis ſuper
              <lb/>
            g t:</s>
            <s xml:id="echoid-s9996" xml:space="preserve"> & ſimiliter [ք 29 p 1] ſuper n f.</s>
            <s xml:id="echoid-s9997" xml:space="preserve"> Sumatur etiã linea cõ-
              <lb/>
            munis ſuperficiei, cõtingẽti ſpeculũ, & ſuperficiei reflexi
              <lb/>
            onis:</s>
            <s xml:id="echoid-s9998" xml:space="preserve"> quę ſit g d:</s>
            <s xml:id="echoid-s9999" xml:space="preserve"> q̃ quidẽ cũ ſecet g h, ſecabit a l.</s>
            <s xml:id="echoid-s10000" xml:space="preserve"> [per lẽma
              <lb/>
            Procli ad 29 p 1.</s>
            <s xml:id="echoid-s10001" xml:space="preserve">] Sit punctũ ſectionis d:</s>
            <s xml:id="echoid-s10002" xml:space="preserve"> & erit orthogo-
              <lb/>
            nalis ſuper a l.</s>
            <s xml:id="echoid-s10003" xml:space="preserve"> [Quia enim h g perpẽdicularis eſt plano,
              <lb/>
            tãgẽti ſpeculũ in pũcto reflexionis g, ք fabricationẽ:</s>
            <s xml:id="echoid-s10004" xml:space="preserve"> erit
              <lb/>
            ք 3 d 11 perpẽdicularis rectæ lineæ g d ipſam in puncto g
              <lb/>
            tãgẽti.</s>
            <s xml:id="echoid-s10005" xml:space="preserve"> Et quoniã a l, h z ſunt parallelę, ք fabricationẽ:</s>
            <s xml:id="echoid-s10006" xml:space="preserve"> erit
              <lb/>
            g d perpẽdicularis ipſi a l per 29 p 1.</s>
            <s xml:id="echoid-s10007" xml:space="preserve">] Palàm ex prędictis,
              <lb/>
            quoniã n f eſt æquidiſtãs g t, & a l ęquidiſtãs g h:</s>
            <s xml:id="echoid-s10008" xml:space="preserve"> igitur [ք
              <lb/>
            15 p 11] ſuperficies, in qua ſunt n f, al, eſt ęquidiſtãs ſuper-
              <lb/>
            ficiei g t h:</s>
            <s xml:id="echoid-s10009" xml:space="preserve"> ſed linea e g æquidiſtat b m [ք fabricationẽ]
              <lb/>
            quare ſunt in eadẽ ſuperficie [ք 35 d 1] q̃ ſuperficies ſecat
              <lb/>
            preędictas æquidiſtãtes:</s>
            <s xml:id="echoid-s10010" xml:space="preserve"> unã ſuper lineã e g:</s>
            <s xml:id="echoid-s10011" xml:space="preserve"> aliã ſuper li-
              <lb/>
            neã fl.</s>
            <s xml:id="echoid-s10012" xml:space="preserve"> Quare [ք 16 p 11] fl eſt æquidiſtãs e g:</s>
            <s xml:id="echoid-s10013" xml:space="preserve"> ſed a n æqui
              <lb/>
            diſtat eidẽ.</s>
            <s xml:id="echoid-s10014" xml:space="preserve"> Igitur [ք 30 p 1] fl eſt æquidiſtãs an.</s>
            <s xml:id="echoid-s10015" xml:space="preserve"> Verũ ſu
              <lb/>
            perficies cõtingẽs ſpeculũ in pũcto g, ſecat ſuperficies e-
              <lb/>
            aſdẽ æquidiſtãtes:</s>
            <s xml:id="echoid-s10016" xml:space="preserve"> unã in linea e g:</s>
            <s xml:id="echoid-s10017" xml:space="preserve"> aliã in linea o d.</s>
            <s xml:id="echoid-s10018" xml:space="preserve"> Igitur
              <lb/>
            [ք 16 p 11] o d eſt æquidiſtãs e g.</s>
            <s xml:id="echoid-s10019" xml:space="preserve"> Igitur [ք 30 p 1] eſt æ ꝗ-
              <lb/>
            diſtãs a n & l f.</s>
            <s xml:id="echoid-s10020" xml:space="preserve"> Et à pũcto f ducatur linea æ quidiſtãs l a,
              <lb/>
            ſecãs d o in k, & a n in i:</s>
            <s xml:id="echoid-s10021" xml:space="preserve"> ergo f k æqualis l d, & k i æqualis
              <lb/>
            d a.</s>
            <s xml:id="echoid-s10022" xml:space="preserve"> [ք 34 p 1.</s>
            <s xml:id="echoid-s10023" xml:space="preserve">] Quare erit ꝓ portio a d ad d l, ſicut n o ad
              <lb/>
            o f.</s>
            <s xml:id="echoid-s10024" xml:space="preserve"> [nã ք 7 p 5 eſt, ut a d ad d l, ſic i k ad k f:</s>
            <s xml:id="echoid-s10025" xml:space="preserve"> ſed ք 2 p 6, ut
              <lb/>
            i k ad k f, ſic n o ad o f:</s>
            <s xml:id="echoid-s10026" xml:space="preserve"> ergo ք 11 p 5, ut a d ad d l, ſic n o ad o
              <lb/>
            f.</s>
            <s xml:id="echoid-s10027" xml:space="preserve">] Palã etiã, quòd angulus b g z æqualis eſt angulo z g a:</s>
            <s xml:id="echoid-s10028" xml:space="preserve"> [recta enim linea g z bifariã ſecat angulũ a
              <lb/>
            g b, ut patuit] & etiã angulo g l a:</s>
            <s xml:id="echoid-s10029" xml:space="preserve"> [interiori & oppoſito per 29 p 1] & etiã angulo g a l:</s>
            <s xml:id="echoid-s10030" xml:space="preserve"> [alterno ք 29
              <lb/>
            p 1.</s>
            <s xml:id="echoid-s10031" xml:space="preserve">] Quare [per 1 ax.</s>
            <s xml:id="echoid-s10032" xml:space="preserve">] g a l, g l a ſunt æquales:</s>
            <s xml:id="echoid-s10033" xml:space="preserve"> & [ք 6 p 1] g a, g l æquales:</s>
            <s xml:id="echoid-s10034" xml:space="preserve"> & g d քpẽdicularis ſuper
              <lb/>
            al:</s>
            <s xml:id="echoid-s10035" xml:space="preserve"> [per cõcluſionẽ] erit [per 26 p 1] a d æqualis d l.</s>
            <s xml:id="echoid-s10036" xml:space="preserve"> Erit igitur n o ęqualis o f:</s>
            <s xml:id="echoid-s10037" xml:space="preserve"> [demõſtratũ enim eſt,
              <lb/>
            ut a d ad d l, ſic n o ad o f:</s>
            <s xml:id="echoid-s10038" xml:space="preserve"> & alternè, ut a d ad n o, ſic d l ad o f:</s>
            <s xml:id="echoid-s10039" xml:space="preserve"> ſed a d æquatur ipſi d l:</s>
            <s xml:id="echoid-s10040" xml:space="preserve"> ergo ք 14 p 5 n o
              <lb/>
            æquabitur ipſi o f] & g o perpẽdicularis ſuper n f:</s>
            <s xml:id="echoid-s10041" xml:space="preserve"> [parallelæ enim ſunt n f, g t ք fabricationẽ, & g o
              <lb/>
            perpẽdicularis eſt ipſi gt per 18 p 3:</s>
            <s xml:id="echoid-s10042" xml:space="preserve"> ergo per 29 p 1 g o eſt perpendicularis ipſi n f:</s>
            <s xml:id="echoid-s10043" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s10044" xml:space="preserve"> angulus ad
              <lb/>
            o uterq;</s>
            <s xml:id="echoid-s10045" xml:space="preserve"> rectus eſt] erit [per 4 p 1] angulus o f g ęqualis angulo o n g.</s>
            <s xml:id="echoid-s10046" xml:space="preserve"> Erit igitur angulus n g q ęqua
              <lb/>
            lis angulo m g q.</s>
            <s xml:id="echoid-s10047" xml:space="preserve"> [Nã cũ t q, f n ductę ſint parallelę:</s>
            <s xml:id="echoid-s10048" xml:space="preserve"> æquabitur ք 29 p 1 angulus m g q angulo n f g:</s>
            <s xml:id="echoid-s10049" xml:space="preserve"> ք
              <lb/>
            æqualis cõcluſus eſt ipſi f n g:</s>
            <s xml:id="echoid-s10050" xml:space="preserve"> æquali angulo n g q alterno per 29 p 1.</s>
            <s xml:id="echoid-s10051" xml:space="preserve"> Quare anguli m g q, n g q inter
              <lb/>
            ſe ęquãtur.</s>
            <s xml:id="echoid-s10052" xml:space="preserve">] Igitur [per 12 n 4] à puncto circuli p g, quod eſt g, poteſt punctum m reflecti ad n, nõ
              <lb/>
            impediente pyramide.</s>
            <s xml:id="echoid-s10053" xml:space="preserve"> [Hęc concluſio uidetur repugnare 41 n 4 & 50 n, quibus demonſtratum eſt
              <lb/>
            communem ſectionem ſuperficierum reflexionis & ſpeculi conici cõuexi non eſfe circulum.</s>
            <s xml:id="echoid-s10054" xml:space="preserve"> Qua-
              <lb/>
            re punctum g circuli p g, à quo hic reflexio fieri concluditur, intelligendum eſt punctum circuli,
              <lb/>
            qui eſt communis ſectio ſphæræuel cylindri, quos mens intra conum fingit ac concipit.</s>
            <s xml:id="echoid-s10055" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div367" type="section" level="0" n="0">
          <head xml:id="echoid-head356" xml:space="preserve" style="it">53. Si communis ſectio ſuperficierum, reflexionis, & ſpeculi conici cõuexifuerit latus conicũ:
            <lb/>
          ab uno puncto unum uiſibilis punctum ad unum uiſum reflectetur. 33 p 7.</head>
          <p>
            <s xml:id="echoid-s10056" xml:space="preserve">DIco igitur, quòd punctũ b à ſolo g reflectitur ad a.</s>
            <s xml:id="echoid-s10057" xml:space="preserve"> Si enim dicatur, quòd ab alio pũcto poteſt
              <lb/>
            reflecti:</s>
            <s xml:id="echoid-s10058" xml:space="preserve"> illud aut erit in linea lõgitudinis:</s>
            <s xml:id="echoid-s10059" xml:space="preserve"> quę eſt e g:</s>
            <s xml:id="echoid-s10060" xml:space="preserve"> aut nõ.</s>
            <s xml:id="echoid-s10061" xml:space="preserve"> Sit in ea:</s>
            <s xml:id="echoid-s10062" xml:space="preserve"> & ſit x:</s>
            <s xml:id="echoid-s10063" xml:space="preserve"> & ab eo ducatur
              <lb/>
            perpẽdicularis ſuper ſuperficiẽ, cõtingẽtẽ ſpeculũ in pũcto illo:</s>
            <s xml:id="echoid-s10064" xml:space="preserve"> [per 12 p 11] q̃ quidẽ perpẽdi-
              <lb/>
            cularis, erit [ք 6 p 11] ęquidiſtãs z g:</s>
            <s xml:id="echoid-s10065" xml:space="preserve"> & ita [per 30 p 1] æquidiſtãs a l.</s>
            <s xml:id="echoid-s10066" xml:space="preserve"> Igitur a l eſt in ſuperficie reflexio-
              <lb/>
            nis huius perpẽdicularis:</s>
            <s xml:id="echoid-s10067" xml:space="preserve"> [per 35 d 1] & eſt ſimiliter in ſuperficie reflexionis perpẽdicularis z g:</s>
            <s xml:id="echoid-s10068" xml:space="preserve"> [ք
              <lb/>
            35 d 1:</s>
            <s xml:id="echoid-s10069" xml:space="preserve"> parallela enim ducta eſt a l ipſi z g] igitur illæ duæ ſuperficies reflexiõis ſecãt ſe ſuper lineam
              <lb/>
            </s>
          </p>
        </div>
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