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-s10181" xml:space="preserve">
              <pb o="159" file="0165" n="165" rhead="OPTICAE LIBER V."/>
            18 d 11 & 32 p 1 angulus ab axe & latere e g cõprehẽſus, eſt acutus] & à pũcto a ducatur ęquidiſtãs li-
              <lb/>
            n e æ r t:</s>
            <s xml:id="echoid-s10182" xml:space="preserve"> quę ſit a s:</s>
            <s xml:id="echoid-s10183" xml:space="preserve"> & ducatur à pũcto e linea, cõmunis ſuperficiei reflexionis a e h & ſuperficiei, con
              <lb/>
            tingẽti pyra midẽ in linea g e:</s>
            <s xml:id="echoid-s10184" xml:space="preserve"> quæ ſit e o.</s>
            <s xml:id="echoid-s10185" xml:space="preserve"> Cadet quidẽ orthogonaliter ſuper a s:</s>
            <s xml:id="echoid-s10186" xml:space="preserve"> cũ ſit orthogonalis
              <lb/>
            ſuper e t:</s>
            <s xml:id="echoid-s10187" xml:space="preserve"> [quia enim e o tãgit peripheriã circuli in e:</s>
            <s xml:id="echoid-s10188" xml:space="preserve"> erit
              <lb/>
              <figure xlink:label="fig-0165-01" xlink:href="fig-0165-01a" number="91">
                <variables xml:id="echoid-variables81" xml:space="preserve">g m n b f q k l
                  <gap/>
                  <gap/>
                e p o h r a</variables>
              </figure>
            ք 18 p 3 perpẽdicularis ipſi r e t, cui a s parallela eſt ք fabri
              <lb/>
            cationẽ.</s>
            <s xml:id="echoid-s10189" xml:space="preserve"> Quare ք 29 p 1 e o perpendicularis eſt ipſi a s] &
              <lb/>
            ducatur linea b q:</s>
            <s xml:id="echoid-s10190" xml:space="preserve"> quæ ꝓ ducta, neceſſariò cõcurret cũ li-
              <lb/>
            nea a l:</s>
            <s xml:id="echoid-s10191" xml:space="preserve"> [ք lẽma Procli ad 29 p 1] ſit pũctũ cõcurſus l:</s>
            <s xml:id="echoid-s10192" xml:space="preserve"> &
              <lb/>
            ducatur à pũcto q linea, cõmunis ſuperficiei cõtingenti,
              <lb/>
            [ſpeculũ in latere conico e g] & ſuperficiei a b l:</s>
            <s xml:id="echoid-s10193" xml:space="preserve"> quæ ſit
              <lb/>
            o p:</s>
            <s xml:id="echoid-s10194" xml:space="preserve"> & ducãtur l s, p o.</s>
            <s xml:id="echoid-s10195" xml:space="preserve"> Palàm, quoniam ſuperficies a l s eſt
              <lb/>
            ęquidiſtãs ſuperficiei g e k:</s>
            <s xml:id="echoid-s10196" xml:space="preserve"> [Nã quia e t ſemidiameter cir
              <lb/>
            culi, eſt perpẽdicularis axi ք 18.</s>
            <s xml:id="echoid-s10197" xml:space="preserve"> 3 d 11:</s>
            <s xml:id="echoid-s10198" xml:space="preserve"> & angulus g q k re-
              <lb/>
            ctus ք fabricationẽ:</s>
            <s xml:id="echoid-s10199" xml:space="preserve"> ergo ք 32 p 1 angulus g k q eſt acutus,
              <lb/>
            & reliquus t k q obtuſus.</s>
            <s xml:id="echoid-s10200" xml:space="preserve"> Quare e t, f k ultra axẽ cõtinua-
              <lb/>
            tæ efficient angulos duob.</s>
            <s xml:id="echoid-s10201" xml:space="preserve"> rectis minores ք 13 p 1, & ք 11
              <lb/>
            ax.</s>
            <s xml:id="echoid-s10202" xml:space="preserve"> cõcurrent:</s>
            <s xml:id="echoid-s10203" xml:space="preserve"> His uerò parallelæ a l, a s cõcurrũt in pun-
              <lb/>
            cto a:</s>
            <s xml:id="echoid-s10204" xml:space="preserve"> ſuntq́;</s>
            <s xml:id="echoid-s10205" xml:space="preserve"> binę in diuerſis planis.</s>
            <s xml:id="echoid-s10206" xml:space="preserve"> Ergo ք 15 p 11 ipſarũ
              <lb/>
            plana ſunt parallela] & lineæ q e, p o ſunt in ſuքficie con
              <lb/>
            tingẽte:</s>
            <s xml:id="echoid-s10207" xml:space="preserve"> quę ſuքficies ſecat illas ſuperficies ęquidiſtãtes,
              <lb/>
            ſuper duas lineas q e, p o:</s>
            <s xml:id="echoid-s10208" xml:space="preserve"> Igitur [ք 16 p 11] q e æquidiſtat
              <lb/>
            p o.</s>
            <s xml:id="echoid-s10209" xml:space="preserve"> Ducatur aũt linea h e, donec cõcurrat cũ h s in pũcto
              <lb/>
            s [cõcurret aũt ք lẽma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s10210" xml:space="preserve">] Palã [ք 1 p 11] q đ
              <lb/>
            linea e s eſt in ſuperficie h e g:</s>
            <s xml:id="echoid-s10211" xml:space="preserve"> & in eadẽ eſt linea b l:</s>
            <s xml:id="echoid-s10212" xml:space="preserve"> [ք 2
              <lb/>
            p 11] & hęc ſuperficies ſecat prædictas ſuperficies æquidi
              <lb/>
            ſtãtes, in duabus lineis e q, l s.</s>
            <s xml:id="echoid-s10213" xml:space="preserve"> Igitur [ք 16 p 11] e q eſt æ-
              <lb/>
            diſtãs l s:</s>
            <s xml:id="echoid-s10214" xml:space="preserve"> erit igitur [
              <gap/>
            ք 30 p 1] p o ęquidiſtãs l s.</s>
            <s xml:id="echoid-s10215" xml:space="preserve"> Quare [ք
              <lb/>
            2 p 6] a o ad o s, ſicut a p ad p l:</s>
            <s xml:id="echoid-s10216" xml:space="preserve"> ſed palã [per 12 n 4] quod
              <lb/>
            angulus h e r æ qualis eſt angulo r e a:</s>
            <s xml:id="echoid-s10217" xml:space="preserve"> erit angulus e s a ę-
              <lb/>
            qualis angulo e a s:</s>
            <s xml:id="echoid-s10218" xml:space="preserve"> [Nã cũ r t ſit parallela ipſi a s per fabri
              <lb/>
            cationẽ:</s>
            <s xml:id="echoid-s10219" xml:space="preserve"> æquabitur tũ angulus h e r exterior, angulo e s a interiori & oppoſito, tũ e a s alterno r e a
              <lb/>
            per 29 p 1.</s>
            <s xml:id="echoid-s10220" xml:space="preserve"> Quare ք 1 ax.</s>
            <s xml:id="echoid-s10221" xml:space="preserve"> angulus e s a æquabitur angulo e a s] & e o eſt perpẽdicularis ſuper a s:</s>
            <s xml:id="echoid-s10222" xml:space="preserve"> [ut
              <lb/>
            oſtẽſum eſt] erit ergo [per 26 p 1] a o æqualis o s:</s>
            <s xml:id="echoid-s10223" xml:space="preserve"> erit ergo a p æqualis p l [demõſtratũ enim eſt, ut a o
              <lb/>
            ad o s, ſic a p ad p l] & q p perpẽdicularis eſt ſuper a l.</s>
            <s xml:id="echoid-s10224" xml:space="preserve"> cũ ſit perpẽdicularis ſuper f k.</s>
            <s xml:id="echoid-s10225" xml:space="preserve"> [Quia enim f k
              <lb/>
            քpẽdicularis eſt plano tãgẽti, ut patuit, in quo eſt q p:</s>
            <s xml:id="echoid-s10226" xml:space="preserve"> cũ ſit illius, & plani a b l cõmunis ſectio:</s>
            <s xml:id="echoid-s10227" xml:space="preserve"> ergo
              <lb/>
            ք 3 d 11 f k eſt քpẽdicularis ipſi p q, & ք 29 p 1 ipſi a l parallelę.</s>
            <s xml:id="echoid-s10228" xml:space="preserve">] Igitur [ք 4 p 1] q l ęqualis a q:</s>
            <s xml:id="echoid-s10229" xml:space="preserve"> & angul9
              <unsure/>
              <lb/>
            q l a æqualis angulo l a q.</s>
            <s xml:id="echoid-s10230" xml:space="preserve"> Erit ergo angulus b q f æqualis angulo a q f:</s>
            <s xml:id="echoid-s10231" xml:space="preserve"> [Quia enim q f parallela eſt
              <lb/>
            ipſi a l:</s>
            <s xml:id="echoid-s10232" xml:space="preserve"> ęquabitur exterior angulus b q finteriori & oppoſito q l a:</s>
            <s xml:id="echoid-s10233" xml:space="preserve"> & q a l alterno a q f ք 29 p 1.</s>
            <s xml:id="echoid-s10234" xml:space="preserve"> Qua-
              <lb/>
            re b q f æquabitur a q f.</s>
            <s xml:id="echoid-s10235" xml:space="preserve">] Igitur a reflectetur ad b à puncto q [per 12 n 4.</s>
            <s xml:id="echoid-s10236" xml:space="preserve">] Quod eſt propoſitum.</s>
            <s xml:id="echoid-s10237" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div371" type="section" level="0" n="0">
          <head xml:id="echoid-head358" xml:space="preserve" style="it">55. Viſu & uiſibili in plano per uerticem ſpeculi conici
            <lb/>
          conuexi ducto, baſi́ par allelo, poſitis: punctũ reflexio-
            <lb/>
          nis inuenire. 36 p 7.</head>
          <figure number="92">
            <variables xml:id="echoid-variables82" xml:space="preserve">g m q n t e b r a</variables>
          </figure>
          <p>
            <s xml:id="echoid-s10238" xml:space="preserve">SI uerò cẽtrũ uiſus & punctũ uiſum fuerint in ſuperfi
              <lb/>
            cie m g n:</s>
            <s xml:id="echoid-s10239" xml:space="preserve"> ſit unũ in puncto m, aliud in pũcto n:</s>
            <s xml:id="echoid-s10240" xml:space="preserve"> & du
              <lb/>
            cãtur lineæ m g, n g, m n:</s>
            <s xml:id="echoid-s10241" xml:space="preserve"> & diuidatur angulus m g n
              <lb/>
            per æqualia, per lineã q g [per 9 p 1.</s>
            <s xml:id="echoid-s10242" xml:space="preserve">] Palã [per 12 n 4] qđ
              <lb/>
            n à puncto g reflectitur ad m.</s>
            <s xml:id="echoid-s10243" xml:space="preserve"> Palã etiã, quòd linea q g &
              <lb/>
            axis pyramidis ſunt in ſuperficie, ſecãte pyramidẽ ſuper
              <lb/>
            lineã longitudinis:</s>
            <s xml:id="echoid-s10244" xml:space="preserve"> [ſunt enim axis & latus in uno plano,
              <lb/>
            ut è 18 d 11 intelligitur, & in eodẽ plano eſt recta linea q g
              <lb/>
            per 2 p 11] à pũcto q ducatur orthogonalis ſuper hãc lineã
              <lb/>
            lõgitudinis g e:</s>
            <s xml:id="echoid-s10245" xml:space="preserve"> quę ſit q e:</s>
            <s xml:id="echoid-s10246" xml:space="preserve"> & ſuper pũctũ e fiat ſuperficies
              <lb/>
            æquidiſtãs baſi:</s>
            <s xml:id="echoid-s10247" xml:space="preserve"> [ut dictũ eſt 52 n] quæ ſecabit pyramidẽ
              <lb/>
            ſuper circulũ [per 4 th 1 coni.</s>
            <s xml:id="echoid-s10248" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s10249" xml:space="preserve">] linea cõmunis ſuք-
              <lb/>
            ficiei q e g, & huic circulo ſit e t.</s>
            <s xml:id="echoid-s10250" xml:space="preserve"> Palã, quoniã cadet ſuper
              <lb/>
            axem & ſuper cẽtrũ circuli.</s>
            <s xml:id="echoid-s10251" xml:space="preserve"> [Quia enim conus ſectus eſt
              <lb/>
            duplici plano:</s>
            <s xml:id="echoid-s10252" xml:space="preserve"> uno per axem, altero ad baſim parallelo:</s>
            <s xml:id="echoid-s10253" xml:space="preserve"> &
              <lb/>
            illius quidẽ & coni cõmunis ſectio eſt triágulũ, per 3 th 1
              <lb/>
            coni.</s>
            <s xml:id="echoid-s10254" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s10255" xml:space="preserve"> huius uerò circulus per 4 th eiuſdẽ:</s>
            <s xml:id="echoid-s10256" xml:space="preserve"> ergo per
              <lb/>
            cõſectariũ 4 th comunis ſectio circuli & trianguli eſt dia
              <lb/>
            meter circuli, cuius cẽtrum eſt in axe.</s>
            <s xml:id="echoid-s10257" xml:space="preserve">] Deinde à pũcto m
              <lb/>
            ducatur ęquidiſtãs lineę e g:</s>
            <s xml:id="echoid-s10258" xml:space="preserve"> quę quidẽ in ſuperficie illius
              <lb/>
            circuli cadat in pũctũ b:</s>
            <s xml:id="echoid-s10259" xml:space="preserve"> [cadet aũt, ꝗa eſt interplana pa-
              <lb/>
            rallela.</s>
            <s xml:id="echoid-s10260" xml:space="preserve">] Similiter à pũcto n ducatur æquidiſtans g e:</s>
            <s xml:id="echoid-s10261" xml:space="preserve"> quæ
              <lb/>
            cadat in pũctũ a:</s>
            <s xml:id="echoid-s10262" xml:space="preserve"> & ducatur a b:</s>
            <s xml:id="echoid-s10263" xml:space="preserve"> & e t ſecet eá in punctor.</s>
            <s xml:id="echoid-s10264" xml:space="preserve">
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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