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-s23509" xml:space="preserve">
              <pb o="54" file="0356" n="356" rhead="VITELLONIS OPTICAE"/>
            in d g:</s>
            <s xml:id="echoid-s23510" xml:space="preserve"> erit ergo per 16 p 6 proportio a n primi ad g d ſecundum, ſicut b g tertij ad d h quartũ:</s>
            <s xml:id="echoid-s23511" xml:space="preserve"> ſed pro
              <lb/>
            batum eſt in præcedentibus, quòd proportio lineæ a n ad lineam d g eſt, ſicut diameter b g ad lineã
              <lb/>
            e q.</s>
            <s xml:id="echoid-s23512" xml:space="preserve"> Igitur per 9 p 5 linea d h eſt æqualis lineę e q.</s>
            <s xml:id="echoid-s23513" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s23514" xml:space="preserve"> Si uerò linea a g ſit minor ꝗ̃
              <lb/>
            linea a b, ſecabit linea d a circulum in arcu a b.</s>
            <s xml:id="echoid-s23515" xml:space="preserve"> Sit ergo ut ſecet ipſum in puncto h:</s>
            <s xml:id="echoid-s23516" xml:space="preserve"> & ducatur linea
              <lb/>
              <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a" number="388">
                <variables xml:id="echoid-variables372" xml:space="preserve">d q n g a e b h</variables>
              </figure>
            g h & linea g n, æquidiſtans lineę b a.</s>
            <s xml:id="echoid-s23517" xml:space="preserve"> Palã ergo ք 29 p 1, quoniã an-
              <lb/>
            gulus n g d eſt ęqualis angulo a b g:</s>
            <s xml:id="echoid-s23518" xml:space="preserve"> ſed angulus a b g eſt ęqualis an-
              <lb/>
            gulo a h g per 27 p 3:</s>
            <s xml:id="echoid-s23519" xml:space="preserve"> quoniã ambo cadunt in arcũ g a, & ſunt ſuք cir
              <lb/>
            cumferentiã circuli:</s>
            <s xml:id="echoid-s23520" xml:space="preserve"> ergo angulus n g d eſt æqualis angulo a h g:</s>
            <s xml:id="echoid-s23521" xml:space="preserve"> &
              <lb/>
            angulus n d g cõmunis eſt ambobus trigonis, ſcilicet n d g & d h g:</s>
            <s xml:id="echoid-s23522" xml:space="preserve">
              <lb/>
            eſt ergo tertius d n g ęqualis tertio, ſcilicet d g h ք 32 p 1.</s>
            <s xml:id="echoid-s23523" xml:space="preserve"> Ergo per 4
              <lb/>
            p 6 erit proportio lineę h d ad lineam d g, ſicut lineæ d g ad lineam
              <lb/>
            d n:</s>
            <s xml:id="echoid-s23524" xml:space="preserve"> ergo per 17 p 6 illud, quod fit ex ductu h d in d n eſt ęquale qua-
              <lb/>
            drato lineę g d:</s>
            <s xml:id="echoid-s23525" xml:space="preserve"> ſed illud, qđ fit ex ductu b d in d g ք 36 p 3, eſt ęquale
              <lb/>
            ei quod fit ex ductu h d in d a:</s>
            <s xml:id="echoid-s23526" xml:space="preserve"> illud aũt, quod fit ex ductu h d in d a,
              <lb/>
            eſt ք 1 p 2 ęquale ei, qđ fit ex ductu lineę h d in d n, & lineę h d in n a:</s>
            <s xml:id="echoid-s23527" xml:space="preserve">
              <lb/>
            illud uerò quod fit ex ductu lineę b d in d g, per 3 p 2 ualet illud, qđ
              <lb/>
            fit ex ductu lineæ b g in g d & quadratũ g d.</s>
            <s xml:id="echoid-s23528" xml:space="preserve"> Ablatis ergo ęqualibus
              <lb/>
            hinc inde, erit illud, quod fit ex ductu h d in n a ęquale ei, quod fit ex
              <lb/>
            ductu b g in g d:</s>
            <s xml:id="echoid-s23529" xml:space="preserve"> erit ergo, ut prius, ꝓportio lineæ a n ad lineam d g,
              <lb/>
            ſicut lineę b g ad lineã h d.</s>
            <s xml:id="echoid-s23530" xml:space="preserve"> Sed iã oſtenſum eſt ſuprà, quòd eſt ꝓpor
              <lb/>
            tio lineę a n ad lineã d g, ſicut lineę b g ad lineã e q.</s>
            <s xml:id="echoid-s23531" xml:space="preserve"> Igitur linea e q
              <lb/>
            eſt æqualis lineę h d per 9 p 5.</s>
            <s xml:id="echoid-s23532" xml:space="preserve"> Quod eſt propoſitum:</s>
            <s xml:id="echoid-s23533" xml:space="preserve"> quoniã à pun-
              <lb/>
            cto a dato ducta eſt linea ſecãs circulũ, cuius pars à pũcto ſectionis
              <lb/>
            uſque ad concurſum cum diametro producta, æqualis eſt datæ li-
              <lb/>
            neæ.</s>
            <s xml:id="echoid-s23534" xml:space="preserve"> Patet ergo quod proponebatur.</s>
            <s xml:id="echoid-s23535" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div866" type="section" level="0" n="0">
          <head xml:id="echoid-head705" xml:space="preserve" style="it">131. Inter duas rectas ſe ſecantes ex unaparte à puncto dato hyperbolẽ, illas lineas nõ cõtingẽ
            <lb/>
          tem ducere, ex alia parte cõmunis puncti illarũ linearũ hyperbolẽ priori oppoſit ã deſignare. Ex
            <lb/>
          quo patet, quòd cũ fuerint duæ ſectiones oppoſitæ inter duas lineas, et producatur linea minima
            <lb/>
          ab una ſectione ad aliã: erit pars illi{us} lineæ interiacens unã ſectionũ, & reliquãlineam æqualis
            <lb/>
          ſuæ partialiam ſectionem, & reliquam lineam interiacenti. 4. 8 th. 2 conicorum Apollonij.</head>
          <p>
            <s xml:id="echoid-s23536" xml:space="preserve">Quod hic proponitur, demonſtratum eſt ab Apollonio in libro ſuo de conicis elementis:</s>
            <s xml:id="echoid-s23537" xml:space="preserve"> dicun-
              <lb/>
              <figure xlink:label="fig-0356-02" xlink:href="fig-0356-02a" number="389">
                <variables xml:id="echoid-variables373" xml:space="preserve">n u l c u x g t c m f q t k p h p z</variables>
              </figure>
            tur aũt ſectiones am
              <lb/>
            blygonię ſiue hyper
              <lb/>
            bolæ oppoſitæ, qñ
              <lb/>
            gibboſitas unius i-
              <lb/>
            pſarũ ſequitur gib-
              <lb/>
            boſitatẽ alterius, ita
              <lb/>
            utillæ gibboſitates
              <lb/>
            ſe reſpiciãt, & amba
              <lb/>
            rum diametri ſintin
              <lb/>
            una linea recta.</s>
            <s xml:id="echoid-s23538" xml:space="preserve"> Ver
              <lb/>
            bi gratia:</s>
            <s xml:id="echoid-s23539" xml:space="preserve"> ſit, ut duæ
              <lb/>
            lineæ h l & z n ſecẽt
              <lb/>
            ſe in puncto x, & ex una parte ipſarum, ſcilicet ſub angulo h x z, uel ſub angulo h x n à dato puncto,
              <lb/>
            qui ſit t, ducatur ſectio amblygonia, quæ ſit t p, & ex altera parte ſub angulo n x l, uel ſub angulo z x
              <lb/>
            l ducatur ſectio illi opp oſita, quæ ſit c u, ita, quòd diametri quarumlibet oppoſitarum ambarum ſe-
              <lb/>
            ctionum illarũ ſint in una linea, quæ t c, à uertice unius ad uerticẽ alterius producta:</s>
            <s xml:id="echoid-s23540" xml:space="preserve"> quę neceſſariò
              <lb/>
            eſt minima omnium linearum inter illas duas ſectiones productarum.</s>
            <s xml:id="echoid-s23541" xml:space="preserve"> Et ex ijs declarauit Apollo-
              <lb/>
            nius illud, quod corollatiuè proponitur, ſcilicet, quòd ſi linea t c ſecet lineam h l in puncto f, & lineã
              <lb/>
            z n in puncto q, quòd linea t q erit ęqualis lineæ c f:</s>
            <s xml:id="echoid-s23542" xml:space="preserve"> & ſi linea t c pertrãſeat punctum x, erit linea t x
              <lb/>
            ęqualis lineę x c:</s>
            <s xml:id="echoid-s23543" xml:space="preserve"> & nos utimur hoc illo, ut per Apollonium demonſtrato, & propter conformitatẽ
              <lb/>
            portionis ſectionum reſpectu linearum ſe interſecantium.</s>
            <s xml:id="echoid-s23544" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s23545" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div868" type="section" level="0" n="0">
          <head xml:id="echoid-head706" xml:space="preserve" style="it">132. In uertice alteri{us} conicarum ſectionum poſito pede circini immobili, ſecundum quan-
            <lb/>
          titatem lineæ breuißimæ inter illas ſectiones ductæ, deſcriptus circulus ſectionem reliquam con-
            <lb/>
          tinget: ſecundum uerò maiorem, in duobus tantùm punctis reliquam ſecabit.</head>
          <p>
            <s xml:id="echoid-s23546" xml:space="preserve">Quod hic proponitur, facile eſt, & ſola indiget declaratione.</s>
            <s xml:id="echoid-s23547" xml:space="preserve"> Sint ut enim in præcedenti propoſi
              <lb/>
            tione duæ ſectiones conicæ oppoſitæ adinuicẽ, quę ſint t p & c u, inter quas linea minima uertices,
              <lb/>
            ſcilicet ambarum ſectionum continuans, ſit linea t c:</s>
            <s xml:id="echoid-s23548" xml:space="preserve"> & poſito in altero punctorum tuel c pede cir-
              <lb/>
            cini, utpote in puncto t, deſcribatur circulus ſecundum quantitatem diametri t c.</s>
            <s xml:id="echoid-s23549" xml:space="preserve"> Hic ergo cir-
              <lb/>
            culus, quia ſectionem c u non attingit niſi in puncto c, & omnes alię lineæ ducibiles interipſas ſe-
              <lb/>
            ctiones, ſunt maiores quã linea t c:</s>
            <s xml:id="echoid-s23550" xml:space="preserve"> ſunt ergo maiores ſemidiametro circuli:</s>
            <s xml:id="echoid-s23551" xml:space="preserve"> ſecabuntur ergo oẽs
              <lb/>
            per circulũ, nec attinget circulus alicubi ſectionem niſi in puncto c.</s>
            <s xml:id="echoid-s23552" xml:space="preserve"> Patet ergo primũ propoſitorũ.</s>
            <s xml:id="echoid-s23553" xml:space="preserve">
              <lb/>
            Qđ ſi linea t c ſemidiameter circuli ſit maior ꝗ̃ linearũ minima, inter oppoſitas ſectiões ꝓductarũ,
              <lb/>
            </s>
          </p>
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