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-div391" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s10850" xml:space="preserve">
              <pb o="166" file="0172" n="172" rhead="ALHAZEN"/>
            quantum cunq;</s>
            <s xml:id="echoid-s10851" xml:space="preserve"> productæ, poteſt eſſe locus imaginum.</s>
            <s xml:id="echoid-s10852" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s10853" xml:space="preserve"> ſit a g diameter circuli a m g:</s>
            <s xml:id="echoid-s10854" xml:space="preserve"> cu-
              <lb/>
            ius d centrum.</s>
            <s xml:id="echoid-s10855" xml:space="preserve"> Sumatur in hac diametro punctum z:</s>
            <s xml:id="echoid-s10856" xml:space="preserve"> e cẽtrum uiſus.</s>
            <s xml:id="echoid-s10857" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a" number="108">
                <variables xml:id="echoid-variables98" xml:space="preserve">z t a l m e d b p g</variables>
              </figure>
            Dico, quòd z poteſt eſſe locus imaginis.</s>
            <s xml:id="echoid-s10858" xml:space="preserve"> Ducatur linea e t z per t pun
              <lb/>
            ctum circuli:</s>
            <s xml:id="echoid-s10859" xml:space="preserve"> & ducatur linea d t:</s>
            <s xml:id="echoid-s10860" xml:space="preserve"> erit angulus e t d acutus:</s>
            <s xml:id="echoid-s10861" xml:space="preserve"> [ut demõ-
              <lb/>
            ſtratum eſt 60 n.</s>
            <s xml:id="echoid-s10862" xml:space="preserve">] Fiat aũt ei ęqualis [ք 23 p 1] ꝗ ſit d t l.</s>
            <s xml:id="echoid-s10863" xml:space="preserve"> Palã [per 12 n
              <lb/>
            4] quòd l reflectetur ad e à puncto t:</s>
            <s xml:id="echoid-s10864" xml:space="preserve"> & eius imago erit z [ք 6 n.</s>
            <s xml:id="echoid-s10865" xml:space="preserve">] Simi
              <lb/>
            liter ſumpto l puncto:</s>
            <s xml:id="echoid-s10866" xml:space="preserve"> patebit quod eſt locus imaginis.</s>
            <s xml:id="echoid-s10867" xml:space="preserve"> Ducatur.</s>
            <s xml:id="echoid-s10868" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10869" xml:space="preserve"> li-
              <lb/>
            neale uſq;</s>
            <s xml:id="echoid-s10870" xml:space="preserve"> in b punctũ circuli:</s>
            <s xml:id="echoid-s10871" xml:space="preserve"> & ducatur linea b d:</s>
            <s xml:id="echoid-s10872" xml:space="preserve"> erit [ut prius] an-
              <lb/>
            gulus e b d acutus.</s>
            <s xml:id="echoid-s10873" xml:space="preserve"> Fiat ei ęqualis:</s>
            <s xml:id="echoid-s10874" xml:space="preserve"> qui ſit d b p:</s>
            <s xml:id="echoid-s10875" xml:space="preserve"> reflectetur quidẽ pun-
              <lb/>
            ctum p ad e à puncto b:</s>
            <s xml:id="echoid-s10876" xml:space="preserve"> [per 12 n 4] & locus imaginis eius erit l:</s>
            <s xml:id="echoid-s10877" xml:space="preserve"> [per
              <lb/>
            6 n.</s>
            <s xml:id="echoid-s10878" xml:space="preserve">] Et ita ſumpto quocunq;</s>
            <s xml:id="echoid-s10879" xml:space="preserve"> alio puncto:</s>
            <s xml:id="echoid-s10880" xml:space="preserve"> erit eadem probatio.</s>
            <s xml:id="echoid-s10881" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div393" type="section" level="0" n="0">
          <head xml:id="echoid-head372" xml:space="preserve" style="it">69. Si uiſu et uiſibili in eadẽ diametro circuli (ꝗ eſt cõmunis ſectio
            <lb/>
          ſuperficierũ, reflexionis & ſpeculi ſphærici caui) ſitis: imago uidea
            <lb/>
          tur in ipſo uiſu: ab uno ſemicirculi, uel à quolibet alteri{us} definiti
            <lb/>
          circuli puncto poteſt ad uiſum reflexio fieri. 23 p 8.</head>
          <p>
            <s xml:id="echoid-s10882" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s10883" xml:space="preserve"> punctorum, quæ cõprehẽduntur in his ſpeculis:</s>
            <s xml:id="echoid-s10884" xml:space="preserve"> quo-
              <lb/>
            rundã imagines quaturor loca ſortiuntur:</s>
            <s xml:id="echoid-s10885" xml:space="preserve"> quorundã tria:</s>
            <s xml:id="echoid-s10886" xml:space="preserve"> quo-
              <lb/>
            rundã duo:</s>
            <s xml:id="echoid-s10887" xml:space="preserve"> quorundã unum.</s>
            <s xml:id="echoid-s10888" xml:space="preserve"> Punctũ, cuius imago in quatuor
              <lb/>
            ceciderit loca:</s>
            <s xml:id="echoid-s10889" xml:space="preserve"> â quatuor pũctis determinatis reflectitur, nõ ab alijs,
              <lb/>
            uel plurib.</s>
            <s xml:id="echoid-s10890" xml:space="preserve"> Punctum, cuius imago tria ſibi uſurpat loca:</s>
            <s xml:id="echoid-s10891" xml:space="preserve"> à tribus pun-
              <lb/>
            ctis ſpeculi reflectitur, nõ à plurib.</s>
            <s xml:id="echoid-s10892" xml:space="preserve"> cuius duo:</s>
            <s xml:id="echoid-s10893" xml:space="preserve"> à duobus.</s>
            <s xml:id="echoid-s10894" xml:space="preserve"> Puncti aũt, cuius imago in unicum caditlo-
              <lb/>
            cũ:</s>
            <s xml:id="echoid-s10895" xml:space="preserve"> poterit eſſe:</s>
            <s xml:id="echoid-s10896" xml:space="preserve"> quòd ab uno tãtùm puncto ſit reflexio:</s>
            <s xml:id="echoid-s10897" xml:space="preserve"> & poterit eſſe:</s>
            <s xml:id="echoid-s10898" xml:space="preserve"> quòd à quolibet circuli deter-
              <lb/>
            minati puncto, non ab alio.</s>
            <s xml:id="echoid-s10899" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s10900" xml:space="preserve"> ſit e cẽtrum uiſus:</s>
            <s xml:id="echoid-s10901" xml:space="preserve"> h ſit punctũ uiſum in eadẽ diametro:</s>
            <s xml:id="echoid-s10902" xml:space="preserve"> d ſit
              <lb/>
            centrum circuli.</s>
            <s xml:id="echoid-s10903" xml:space="preserve"> Ducatur diameter z e h a:</s>
            <s xml:id="echoid-s10904" xml:space="preserve"> aut e d eſt æqualis d h:</s>
            <s xml:id="echoid-s10905" xml:space="preserve"> aut nõ.</s>
            <s xml:id="echoid-s10906" xml:space="preserve"> Sit æqualis:</s>
            <s xml:id="echoid-s10907" xml:space="preserve"> & ſuper e h du
              <lb/>
            catur à puncto d perpendiculariter diameter g d b:</s>
            <s xml:id="echoid-s10908" xml:space="preserve"> & ducãtur lineę h g, g e, h b, b e.</s>
            <s xml:id="echoid-s10909" xml:space="preserve"> Palã [per 4 p 1]
              <lb/>
            quòd triangulũ h g d æquale triãgulo e d g, & ęqua-
              <lb/>
              <figure xlink:label="fig-0172-02" xlink:href="fig-0172-02a" number="109">
                <variables xml:id="echoid-variables99" xml:space="preserve">g c z e d h a b</variables>
              </figure>
            le triãgulo h b d, & triãgulo e b d.</s>
            <s xml:id="echoid-s10910" xml:space="preserve"> Palàm, quòd, cum
              <lb/>
            angulus h g e diuiſus ſit per ęqualia, h à puncto g re
              <lb/>
            flectetur ad e:</s>
            <s xml:id="echoid-s10911" xml:space="preserve"> [per 12 n 4] & locus imaginis eius eſt
              <lb/>
            e [per 6 n.</s>
            <s xml:id="echoid-s10912" xml:space="preserve">] Similiter h à puncto b reflectetur ad e:</s>
            <s xml:id="echoid-s10913" xml:space="preserve">
              <lb/>
            & locus imaginis eius e.</s>
            <s xml:id="echoid-s10914" xml:space="preserve"> Si igitur diametro z e h a
              <lb/>
            immota, moueatur ſemicirculus a g z per ſphæram
              <lb/>
            ſpeculi aut ſolũ triangulũ h g e:</s>
            <s xml:id="echoid-s10915" xml:space="preserve"> deſcribet quidẽ pun
              <lb/>
            ctum g motu ſuo circulũ:</s>
            <s xml:id="echoid-s10916" xml:space="preserve"> & à quolibet puncto circu
              <lb/>
            li reflectetur h ad e:</s>
            <s xml:id="echoid-s10917" xml:space="preserve"> & locus imaginis eius ſemper
              <lb/>
            erit punctũ e.</s>
            <s xml:id="echoid-s10918" xml:space="preserve"> Et ita patet propoſitum.</s>
            <s xml:id="echoid-s10919" xml:space="preserve"> Quòd aũt ab
              <lb/>
            alio puncto, ꝗ̃ aliquo illius circuli, nõ poſsit fieri re-
              <lb/>
            flexio puncti h ad e:</s>
            <s xml:id="echoid-s10920" xml:space="preserve"> palã per hoc.</s>
            <s xml:id="echoid-s10921" xml:space="preserve"> Sumatur punctũ
              <lb/>
            c, & ducatur e c, c h:</s>
            <s xml:id="echoid-s10922" xml:space="preserve"> erit quidẽ [per 7 p 3] e c maior
              <lb/>
            linea e g, & linea h c minor linea h g.</s>
            <s xml:id="echoid-s10923" xml:space="preserve"> Quare non erit
              <lb/>
            proportio e c ad c h, ſicut e d ad d h.</s>
            <s xml:id="echoid-s10924" xml:space="preserve"> [Quia.</s>
            <s xml:id="echoid-s10925" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10926" xml:space="preserve"> ex the
              <lb/>
            ſi punctum h reflectitur ad e à puncto g:</s>
            <s xml:id="echoid-s10927" xml:space="preserve"> erit per 12
              <lb/>
            n 4.</s>
            <s xml:id="echoid-s10928" xml:space="preserve"> 3 p 6 e g ad g h, ſicut e d ad d h.</s>
            <s xml:id="echoid-s10929" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s10930" xml:space="preserve"> cum e c, h c
              <lb/>
            ſint inęquales ipſis e g, g h:</s>
            <s xml:id="echoid-s10931" xml:space="preserve"> nõ erunt proportionales ipſis e d, d h.</s>
            <s xml:id="echoid-s10932" xml:space="preserve">] Igitur [per 3 p 6] linea d c nõ diui-
              <lb/>
            det angulum e c h per æqualia.</s>
            <s xml:id="echoid-s10933" xml:space="preserve"> Quare h à puncto c non põtreflecti ad e.</s>
            <s xml:id="echoid-s10934" xml:space="preserve"> [Idẽ breuius cõcludetur ք
              <lb/>
            4 ꝓ.</s>
            <s xml:id="echoid-s10935" xml:space="preserve"> geometrię Iordani.</s>
            <s xml:id="echoid-s10936" xml:space="preserve"> Quia.</s>
            <s xml:id="echoid-s10937" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10938" xml:space="preserve"> triãguli e c h latera e c, h c ſuntinęqualia:</s>
            <s xml:id="echoid-s10939" xml:space="preserve"> & recta c d ab ipſorũ angu
              <lb/>
            lo eſt in mediũ baſis e h ex theſi:</s>
            <s xml:id="echoid-s10940" xml:space="preserve"> erit ք allegatã 4 ꝓpoſitionẽ angulus e c d minorangulo h c d.</s>
            <s xml:id="echoid-s10941" xml:space="preserve"> Qua
              <lb/>
            re h à puncto c ad uiſum e nõ reflectetur.</s>
            <s xml:id="echoid-s10942" xml:space="preserve"> Quòd aũt e c, h c latera ſint inæqualia, patet:</s>
            <s xml:id="echoid-s10943" xml:space="preserve"> quia per 7 p 3 e
              <lb/>
            c maior eſt e g, id eſt, h g (ęquales.</s>
            <s xml:id="echoid-s10944" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10945" xml:space="preserve"> ſunt è cõcluſo) & h g maior h c:</s>
            <s xml:id="echoid-s10946" xml:space="preserve"> erit e c multò maior h c.</s>
            <s xml:id="echoid-s10947" xml:space="preserve">] Eadem
              <lb/>
            erit probatio, ſi ſumatur c inter g & z.</s>
            <s xml:id="echoid-s10948" xml:space="preserve"> Si uerò e d fuerit maior d h:</s>
            <s xml:id="echoid-s10949" xml:space="preserve"> mutetur figura:</s>
            <s xml:id="echoid-s10950" xml:space="preserve"> & addatur lineæ
              <lb/>
            d h, linea h q, ut productum ex e q in q h, ſit æquale quadrato d q.</s>
            <s xml:id="echoid-s10951" xml:space="preserve"> Erit igitur proportio e q ad d q, ſi-
              <lb/>
            cut d q ad h q, ſicut probat Euclides [17 p 6.</s>
            <s xml:id="echoid-s10952" xml:space="preserve">] Fiat circulus ad quãtitatẽ ſemidiametri q d:</s>
            <s xml:id="echoid-s10953" xml:space="preserve"> cuius q cẽ
              <lb/>
            trum:</s>
            <s xml:id="echoid-s10954" xml:space="preserve"> g, b loca ſectionis duorũ circulorum:</s>
            <s xml:id="echoid-s10955" xml:space="preserve"> & ducãtur lineę e g, e b, q g, q b, d g, d b, h g, h b.</s>
            <s xml:id="echoid-s10956" xml:space="preserve"> Palã ergo,
              <lb/>
            quòd erit proportio e q ad q g, ſicut q g ad q h:</s>
            <s xml:id="echoid-s10957" xml:space="preserve"> [ęquales enim ſunt q g, d q ք 15 d 1:</s>
            <s xml:id="echoid-s10958" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s10959" xml:space="preserve"> e q ad d q & q
              <lb/>
            g eãdẽ habet rationẽ ք 7 p 5:</s>
            <s xml:id="echoid-s10960" xml:space="preserve"> & iã patuit, ut e q ad d q, ſic d q ad h q:</s>
            <s xml:id="echoid-s10961" xml:space="preserve"> ergo per 7 p 5, ute q ad g q, ſic g q
              <lb/>
            ad h q] & angulus g q h cõmunis utriq;</s>
            <s xml:id="echoid-s10962" xml:space="preserve"> triãgulo e q g, h q g.</s>
            <s xml:id="echoid-s10963" xml:space="preserve"> Igitur illa duo triãgula ſunt ſimilia.</s>
            <s xml:id="echoid-s10964" xml:space="preserve"> [ք 6.</s>
            <s xml:id="echoid-s10965" xml:space="preserve">
              <lb/>
            4 p.</s>
            <s xml:id="echoid-s10966" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s10967" xml:space="preserve">] Erit igitur proportio e q ad q g, ſicute e g ad g h.</s>
            <s xml:id="echoid-s10968" xml:space="preserve"> Erit igitur e d ad d h, ſicut e g ad g h.</s>
            <s xml:id="echoid-s10969" xml:space="preserve"> [oſten
              <lb/>
            ſũ.</s>
            <s xml:id="echoid-s10970" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10971" xml:space="preserve"> eſt, ut tota e q ad totã d q, ſic ablata d q ad ablatã h q:</s>
            <s xml:id="echoid-s10972" xml:space="preserve"> ergo ք 19 p 5, uttota e q ad totã d q, id eſt q
              <lb/>
            g.</s>
            <s xml:id="echoid-s10973" xml:space="preserve"> ſic reliqua e d ad reliquã d h:</s>
            <s xml:id="echoid-s10974" xml:space="preserve"> ſed ut e q ad q g, ſic e g ad g h:</s>
            <s xml:id="echoid-s10975" xml:space="preserve"> ergo ք 11 p 5 ut e d ad d h, ſice g ad g h.</s>
            <s xml:id="echoid-s10976" xml:space="preserve">]
              <lb/>
            Quare [ք 3 p 6] linea d g diuidet angulũ e g h ք ęqualia.</s>
            <s xml:id="echoid-s10977" xml:space="preserve"> Vnde punctũ h à pũcto g reflectetur ad e:</s>
            <s xml:id="echoid-s10978" xml:space="preserve"> [ք
              <lb/>
            12 n 4] & locus imaginis eius pũctũ e [ք 6 n.</s>
            <s xml:id="echoid-s10979" xml:space="preserve">] Similiter h à pũcto b reflectetur ad e:</s>
            <s xml:id="echoid-s10980" xml:space="preserve"> & locus imaginis
              <lb/>
            eſt pũctũ e.</s>
            <s xml:id="echoid-s10981" xml:space="preserve"> Si ergo moueatur triãgulũ e g h, pũctis e, h immotis:</s>
            <s xml:id="echoid-s10982" xml:space="preserve"> pũctũ g deſcribet in ſphęra circulũ,
              <lb/>
            à cuius quolibet pũcto reflectetur h ad e:</s>
            <s xml:id="echoid-s10983" xml:space="preserve"> & ſemք erit locus imaginis e.</s>
            <s xml:id="echoid-s10984" xml:space="preserve"> Et qđ ab alio pũcto, ꝗ̃ aliquo
              <lb/>
            illius circuli, nõ poſsit h reflecti ad e:</s>
            <s xml:id="echoid-s10985" xml:space="preserve"> palã, ut prius.</s>
            <s xml:id="echoid-s10986" xml:space="preserve"> Si.</s>
            <s xml:id="echoid-s10987" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10988" xml:space="preserve"> ſumatur c inter g & a:</s>
            <s xml:id="echoid-s10989" xml:space="preserve"> erit e c maior e g, & h c
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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