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="191" file="0197" n="197" rhead="OPTICAE LIBER VI."/>
          <p>
            <s xml:id="echoid-s12988" xml:space="preserve">QVòd aũt forma in his ſpeculis aliquando uideatur maior re uiſa:</s>
            <s xml:id="echoid-s12989" xml:space="preserve"> ſcilicet cum cõprehenditur
              <lb/>
            à tali longitudine, à qua eius certa quantitas nõ poſsit diſcerni:</s>
            <s xml:id="echoid-s12990" xml:space="preserve"> declarabitur.</s>
            <s xml:id="echoid-s12991" xml:space="preserve"> Sit a centrum
              <lb/>
            ſpeculi:</s>
            <s xml:id="echoid-s12992" xml:space="preserve"> & ſuperficies ſumatur reflexionis:</s>
            <s xml:id="echoid-s12993" xml:space="preserve"> quæ ſecabit ſpeculum ſuper circulum:</s>
            <s xml:id="echoid-s12994" xml:space="preserve"> [per 1 th.</s>
            <s xml:id="echoid-s12995" xml:space="preserve"> 1
              <lb/>
            ſphær.</s>
            <s xml:id="echoid-s12996" xml:space="preserve">] ſit circulus ille e d b:</s>
            <s xml:id="echoid-s12997" xml:space="preserve"> e d diameter illius circuli:</s>
            <s xml:id="echoid-s12998" xml:space="preserve"> & producatur diameter e d uſq;</s>
            <s xml:id="echoid-s12999" xml:space="preserve"> ad z, ut multi
              <lb/>
            plicatio e z in z d ſit æqualis quadrato a d:</s>
            <s xml:id="echoid-s13000" xml:space="preserve"> quod planũ eſt, cum ſit poſsibile diametro e d talem addi
              <lb/>
            lineam, ut ductus totalis in partem additam, ſit æqualis quadrato a d:</s>
            <s xml:id="echoid-s13001" xml:space="preserve"> [id uerò quomodo expeditè
              <lb/>
            fiat, oſtenſum eſt 32 n 5] & diuidatur linea z d in partes æquales, in puncto h [per 10 p 1.</s>
            <s xml:id="echoid-s13002" xml:space="preserve">] Erit igi-
              <lb/>
            tur a h medietas e z.</s>
            <s xml:id="echoid-s13003" xml:space="preserve"> [Nam ſi a d, a e per 15 d 1 æquales, addantur æqualibus h d, h z:</s>
            <s xml:id="echoid-s13004" xml:space="preserve"> æquabitur a h
              <lb/>
            ipſis z h & a e.</s>
            <s xml:id="echoid-s13005" xml:space="preserve"> Tota igitur e z dupla eſt ipſius a h.</s>
            <s xml:id="echoid-s13006" xml:space="preserve">] Ductus ergo a h in h d erit æqualis quartæ parti
              <lb/>
            quadrati a d.</s>
            <s xml:id="echoid-s13007" xml:space="preserve"> [Quia enim oblongum comprehenſum ſub e z & z d æquatur quadrato a d per fabri-
              <lb/>
            cationem:</s>
            <s xml:id="echoid-s13008" xml:space="preserve"> ergo quod comprehend:</s>
            <s xml:id="echoid-s13009" xml:space="preserve"> tur ſub a h dimidiata baſi & z d altitudine eadem, æquatur di-
              <lb/>
            midiato quadrato a d per 1 p 6:</s>
            <s xml:id="echoid-s13010" xml:space="preserve"> rurſusq́;</s>
            <s xml:id="echoid-s13011" xml:space="preserve"> oblongũ comprehenſum ſub a h baſi eadem & h d altitudi-
              <lb/>
            ne dimidiata, æquatur dimidiato oblongo ſub a h & z d.</s>
            <s xml:id="echoid-s13012" xml:space="preserve"> Quare æquatur quadranti quadrati a d.</s>
            <s xml:id="echoid-s13013" xml:space="preserve">] Et
              <lb/>
            quoniam ductus a h in h d maior eſt quadrato h d:</s>
            <s xml:id="echoid-s13014" xml:space="preserve"> [quia per 3 p 2 æquatur quadrato h d, & oblongo
              <lb/>
            comprehenſo ſub a d, & d h] ſit ductus a h in h t, æqualis quadrato h d [fiet autem æqualis, ſi ipſis
              <lb/>
            a h & h d tertiam proportionalem per 11 p 6 inueneris:</s>
            <s xml:id="echoid-s13015" xml:space="preserve"> tum enim per 17 p 6 oblongum extremarum
              <lb/>
            æquabitur quadrato mediæ h d.</s>
            <s xml:id="echoid-s13016" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s13017" xml:space="preserve"> ſi de h d detraxeris æqualem inuentæ proportionali, manda-
              <lb/>
            tum executus fueris.</s>
            <s xml:id="echoid-s13018" xml:space="preserve">] Fiat circulus ſecundum quantitatem a h:</s>
            <s xml:id="echoid-s13019" xml:space="preserve"> & à puncto h producatur chorda,
              <lb/>
            æqualis medietati lineæ h d:</s>
            <s xml:id="echoid-s13020" xml:space="preserve"> [per 1 p 4] quæ ſit h q:</s>
            <s xml:id="echoid-s13021" xml:space="preserve"> & producantur lineæ q a, q t:</s>
            <s xml:id="echoid-s13022" xml:space="preserve"> & [per 23 p 1] ſuper
              <lb/>
            punctũ q fiat angulus, æqualis angulo q a h:</s>
            <s xml:id="echoid-s13023" xml:space="preserve"> qui ſit h q n.</s>
            <s xml:id="echoid-s13024" xml:space="preserve"> Cum ergo in his duobus triangulis hi duo
              <lb/>
            anguli ſint æquales, & unus cõmunis, ſcilicet q h a:</s>
            <s xml:id="echoid-s13025" xml:space="preserve"> erit [per 32 p 1] tertius tertio æqualis, ſcilicet a q h
              <lb/>
            angulo h n q:</s>
            <s xml:id="echoid-s13026" xml:space="preserve"> & erũt triangula ſimilia:</s>
            <s xml:id="echoid-s13027" xml:space="preserve"> [per 4 p.</s>
            <s xml:id="echoid-s13028" xml:space="preserve"> 1 d 6] & erit proportio a h ad h q, ſi cut h q ad h n.</s>
            <s xml:id="echoid-s13029" xml:space="preserve"> Igi-
              <lb/>
            tur [per 17 p 6] qđ fit ex ductu a h in h n, æquale eſt quadrato h q:</s>
            <s xml:id="echoid-s13030" xml:space="preserve"> ſed, [per conſectariũ 4 p 2] quadra
              <lb/>
            tum h q eſt quarta pars quadrati h d:</s>
            <s xml:id="echoid-s13031" xml:space="preserve"> cũ h q ſit medietas h d [per fabricationẽ.</s>
            <s xml:id="echoid-s13032" xml:space="preserve">] Igitur multiplicatio
              <lb/>
            a h in h n, ęqualis eſt quartæ parti multiplicationis a h in h t.</s>
            <s xml:id="echoid-s13033" xml:space="preserve"> Quare h n eſt quarta pars h t [per 1 p 6.</s>
            <s xml:id="echoid-s13034" xml:space="preserve">]
              <lb/>
            Igitur n cadit inter h & t:</s>
            <s xml:id="echoid-s13035" xml:space="preserve"> reſtat, ut ductus h t in t n ſint tres quartæ quadrati h t.</s>
            <s xml:id="echoid-s13036" xml:space="preserve"> [Quia enim h n eſt
              <lb/>
            quadrans ipſius h t:</s>
            <s xml:id="echoid-s13037" xml:space="preserve"> reliqua igitur n t eſt dodrans, ſeu tres quartæ h t.</s>
            <s xml:id="echoid-s13038" xml:space="preserve"> Et quoniam rectangula com-
              <lb/>
            prehenſa ſub tota h t & ſegmentis n t & n h æquantur quadrato h t per 2 p 2:</s>
            <s xml:id="echoid-s13039" xml:space="preserve"> rectangulum igitur
              <lb/>
            comprehenſum ſub tota h t & ſegmento t n (quod eſt dodrans totius h t) æquatur dodranti quadra
              <lb/>
            ti h t.</s>
            <s xml:id="echoid-s13040" xml:space="preserve">] Verùm angulus q h a acutus eſt:</s>
            <s xml:id="echoid-s13041" xml:space="preserve"> [ut oſtenſum eſt 60 n 5] & [per 5 p 1] æqualis angulo h q a:</s>
            <s xml:id="echoid-s13042" xml:space="preserve">
              <lb/>
            quia reſpiciunt æqualia latera in triangulo maiori.</s>
            <s xml:id="echoid-s13043" xml:space="preserve"> Igitur angulus q h n æqualis angulo h n q:</s>
            <s xml:id="echoid-s13044" xml:space="preserve"> [æqua
              <lb/>
            lis enim concluſus eſt angulus a q h angulo h n q] & ita [per 6 p 1] h q æqualis q n, & angulus h n q
              <lb/>
            acutus:</s>
            <s xml:id="echoid-s13045" xml:space="preserve"> quare [per 13 p 1] angulus q n t obtuſus.</s>
            <s xml:id="echoid-s13046" xml:space="preserve"> Quadratum igitur t q ſuperat quadratum q n & qua-
              <lb/>
            dratum t n, ductu lineæ t n in h n.</s>
            <s xml:id="echoid-s13047" xml:space="preserve"> Quoniam, ut dicit Euclides [12 p 2] quadratum lateris oppoſiti ob-
              <lb/>
            ruſo ſuperat quadrata duorũ laterũ, quantũ eſt, quod fit ex ductu unius lateris bis in partẽ ei adiun-
              <lb/>
            ctam, procedentẽ uſq;</s>
            <s xml:id="echoid-s13048" xml:space="preserve"> ad locũ caſus perpendicularis à capite alterius lateris ductæ.</s>
            <s xml:id="echoid-s13049" xml:space="preserve"> Nam ſi à pũctò q
              <lb/>
            ducatur perpendicularis ſuper lineam h t:</s>
            <s xml:id="echoid-s13050" xml:space="preserve"> cadet in punctũ me-
              <lb/>
              <figure xlink:label="fig-0197-01" xlink:href="fig-0197-01a" number="157">
                <variables xml:id="echoid-variables147" xml:space="preserve">o z
                  <gap/>
                l h m n q t d a b e</variables>
              </figure>
            dium lineæ h n:</s>
            <s xml:id="echoid-s13051" xml:space="preserve"> [non enim cadit extra puncta h & n:</s>
            <s xml:id="echoid-s13052" xml:space="preserve"> ſecus per
              <lb/>
            16 p 1 angulus acutus maior eſſet recto contra 12 d 1:</s>
            <s xml:id="echoid-s13053" xml:space="preserve"> caditigitur
              <lb/>
            in medium rectæ h n per 26 p 1] & [per 1 p 2] ductus t n in medie-
              <lb/>
            tatem h n bis, æquipollet ductui t n in h n.</s>
            <s xml:id="echoid-s13054" xml:space="preserve"> Igitur quadratum t q
              <lb/>
            ſuperat quadrata q n, t n, ductu tn in n h.</s>
            <s xml:id="echoid-s13055" xml:space="preserve"> Sed [per 3 p 2] ductus
              <lb/>
            t n in h n, cum quadrato tn, æqualis eſt ductui h t in tn.</s>
            <s xml:id="echoid-s13056" xml:space="preserve"> Igitur
              <lb/>
            [ſubducto quadrato tn] ductus h t in t n eſt exceſſus quadrati
              <lb/>
            t q ſupra quadratum h q.</s>
            <s xml:id="echoid-s13057" xml:space="preserve"> [nam quadratum q h æquatur quadra-
              <lb/>
            to q n:</s>
            <s xml:id="echoid-s13058" xml:space="preserve"> quia rectæ q h, q n æquales oſtenſæ ſunt.</s>
            <s xml:id="echoid-s13059" xml:space="preserve">] Amplius:</s>
            <s xml:id="echoid-s13060" xml:space="preserve"> ſit
              <lb/>
            proportio a i ad a h, ſicut q t ad q h:</s>
            <s xml:id="echoid-s13061" xml:space="preserve"> [per 12 p 6] erit [per 22 p 6]
              <lb/>
            quadratum [ai] ad quadratum [a h] ficut quadratum [qt] ad
              <lb/>
            quadratum [qh] & erit [per 17 p 5] proportio exceſſus quadra-
              <lb/>
            ti a i ſupra quadratum a h, ad quadratum a h, ſicut ductus h t in
              <lb/>
            t n, ad quadratum q h.</s>
            <s xml:id="echoid-s13062" xml:space="preserve"> [Nam a i maior eſt a h:</s>
            <s xml:id="echoid-s13063" xml:space="preserve"> quia q t maior eſt
              <lb/>
            q h:</s>
            <s xml:id="echoid-s13064" xml:space="preserve"> cum quadratum q t ſit maius quadrato q h:</s>
            <s xml:id="echoid-s13065" xml:space="preserve"> & oblongũ com
              <lb/>
            prehenſum ſub h t, t n eſt exuperantia quadrati q t ſupra quadra
              <lb/>
            tum q h.</s>
            <s xml:id="echoid-s13066" xml:space="preserve">] Et quoniã quadratum q h quater ſumptũ, efficit quadratum h d:</s>
            <s xml:id="echoid-s13067" xml:space="preserve"> [per conſectarium 4 p 2:</s>
            <s xml:id="echoid-s13068" xml:space="preserve">
              <lb/>
            quia q h dimidia eſt ipſius h d per fabricationẽ] & ductus h t in t n quater ſumptus, efficit triplũ qua-
              <lb/>
            drati h t:</s>
            <s xml:id="echoid-s13069" xml:space="preserve"> [oſtenſum eſt enim rectangulũ cõprehenſum ſub h t & t n, eſſe dodrantẽ quadrati h t:</s>
            <s xml:id="echoid-s13070" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s13071" xml:space="preserve">
              <lb/>
            quater ſumptũ, erit triplũ quadrati h t] erit [per 15 p 5] ductus h t in t n ad quadratũ h q, ſicut triplum
              <lb/>
            quadrati h t ad quadratum h d.</s>
            <s xml:id="echoid-s13072" xml:space="preserve"> Sit autem h o tripla ad h t:</s>
            <s xml:id="echoid-s13073" xml:space="preserve"> erit ductus h o in h t triplus ad quadratum
              <lb/>
            h t [per 1 p 6.</s>
            <s xml:id="echoid-s13074" xml:space="preserve">] Sed quoniã proportio a h ad h d eſt, ſicut h d ad h t:</s>
            <s xml:id="echoid-s13075" xml:space="preserve"> [Nam per theſin rectangulum com
              <lb/>
            prehenſum ſub a h & h t ęquatur quadrato h d:</s>
            <s xml:id="echoid-s13076" xml:space="preserve"> ergo per 17 p 6, ut a h ad h d, ſic h d ad h t] erit [per cõ-
              <lb/>
            ſectaria 20 p 6.</s>
            <s xml:id="echoid-s13077" xml:space="preserve"> 4 p 5] h t ad a h, ſicut quadratũ h t ad quadratum h d.</s>
            <s xml:id="echoid-s13078" xml:space="preserve"> Verùm proportio o h ad a h, ſi-
              <lb/>
            cut ductus o h in h t ad ductum a h in h t [per 1 p 6] & [per 11 p 5] proportio o h ad h a, ſicut propor-
              <lb/>
            tio tripli quadrati h t ad quadratũ h d.</s>
            <s xml:id="echoid-s13079" xml:space="preserve"> Sed hæc erat ꝓportio exceſſus quadrati a i ſupra quadratũ a
              <lb/>
            h ad quadratũ a h.</s>
            <s xml:id="echoid-s13080" xml:space="preserve"> Igitur o h ad a h, ſicut exceſſus quadrati a i ſupra quadratũ a h ad quadratum a h.</s>
            <s xml:id="echoid-s13081" xml:space="preserve">
              <lb/>
            </s>
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