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-div1182" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s29424" xml:space="preserve">
              <pb o="156" file="0458" n="458" rhead="VITELLONIS OPTICAE"/>
            fint a b & a g:</s>
            <s xml:id="echoid-s29425" xml:space="preserve"> & ducatur linea b g:</s>
            <s xml:id="echoid-s29426" xml:space="preserve"> & producatur uſq;</s>
            <s xml:id="echoid-s29427" xml:space="preserve"> ad punctum l:</s>
            <s xml:id="echoid-s29428" xml:space="preserve"> & à puncto t, quod ſitinferius
              <lb/>
            puncto a uertice coni, ducatur linea æquidiſtãs lineę
              <lb/>
              <figure xlink:label="fig-0458-01" xlink:href="fig-0458-01a" number="507">
                <variables xml:id="echoid-variables487" xml:space="preserve">a t k g h b p l i</variables>
              </figure>
            a b per 31 p 1, quæ producta uerſus lineam b l, ſecetil-
              <lb/>
            lam in pũcto p:</s>
            <s xml:id="echoid-s29429" xml:space="preserve"> & ſit aliquis pũctus eius inſerior pun
              <lb/>
            cto t pũctus k:</s>
            <s xml:id="echoid-s29430" xml:space="preserve"> & ſit illa linea t k p.</s>
            <s xml:id="echoid-s29431" xml:space="preserve"> Dico, quòd oculo
              <lb/>
            poſito ſuper pũctum t, qui eſt eleuatior pũcto k:</s>
            <s xml:id="echoid-s29432" xml:space="preserve"> pars
              <lb/>
            ſuperficiei conicę uiſa, maior quidem erit, minor aũt
              <lb/>
            uidebitur, quàm uideatur oculo exiſtẽte in pũcto k.</s>
            <s xml:id="echoid-s29433" xml:space="preserve">
              <lb/>
            Ducátur enim lineæ a k & a t:</s>
            <s xml:id="echoid-s29434" xml:space="preserve"> & producatur linea a t,
              <lb/>
            donec cõcurrat cum linea b l:</s>
            <s xml:id="echoid-s29435" xml:space="preserve"> cõcurrent aũt per con-
              <lb/>
            uerſam 2 p 6.</s>
            <s xml:id="echoid-s29436" xml:space="preserve"> Quoniã enim linea t p eſt minor quàm
              <lb/>
            linea a b, ut patet ex præmiſsis, & illæ lineæ æquidi-
              <lb/>
            ſtant, patet quòd lineæ a t & b l cõcurrẽt:</s>
            <s xml:id="echoid-s29437" xml:space="preserve"> ſit ergo pun
              <lb/>
            ctus cócurſus i:</s>
            <s xml:id="echoid-s29438" xml:space="preserve"> & ſimiliter lineæ a k & b l concurrẽt:</s>
            <s xml:id="echoid-s29439" xml:space="preserve">
              <lb/>
            ſitq́;</s>
            <s xml:id="echoid-s29440" xml:space="preserve"> pũctus concurſus l.</s>
            <s xml:id="echoid-s29441" xml:space="preserve"> Palàm itaq;</s>
            <s xml:id="echoid-s29442" xml:space="preserve"> quia magis ui-
              <lb/>
            debitur de cono ſuper punctũ i, quàm ſuper pũctum
              <lb/>
            l per 86 huius:</s>
            <s xml:id="echoid-s29443" xml:space="preserve"> ꝓpinquior enim eſt ipſi cono pũctus
              <lb/>
            l, quàm pũctus i.</s>
            <s xml:id="echoid-s29444" xml:space="preserve"> Quod autẽ de ſuperficie conica ui-
              <lb/>
            detur, oculo exiſtente in pũcto i, idem per præceden
              <lb/>
            tem proximam uidetur cẽtro uiſus exiſtẽte per totã
              <lb/>
            lineam i a, utpote in pũcto t:</s>
            <s xml:id="echoid-s29445" xml:space="preserve"> & illud, quod uidetur ui
              <lb/>
            ſu exiſtẽte in pũcto l, uidetur in quolibet pũcto lineę
              <lb/>
            l a exiſtẽte uiſu:</s>
            <s xml:id="echoid-s29446" xml:space="preserve"> ergo & in pũcto k.</s>
            <s xml:id="echoid-s29447" xml:space="preserve"> Sed quod uidetur
              <lb/>
            â pũcto i maius eſt eo, quod uidetur à puncto l, & mi
              <lb/>
            nus eſſe uidetur per 86 huius:</s>
            <s xml:id="echoid-s29448" xml:space="preserve"> ergo illud, quod uide-
              <lb/>
            tur à pũcto t maius eſt illo, quod uidetur à pũcto k, & minus uidetur eſſe.</s>
            <s xml:id="echoid-s29449" xml:space="preserve"> Ethoc eſt quod proponi
              <lb/>
            tur.</s>
            <s xml:id="echoid-s29450" xml:space="preserve"> Ethocidẽ etiam ſuo modo de ambobus uiſibus poteſt demonſtrari.</s>
            <s xml:id="echoid-s29451" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s29452" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1184" type="section" level="0" n="0">
          <head xml:id="echoid-head937" xml:space="preserve" style="it">89. Linea à centro uiſus ad uerticem coni duct a perpendiculari existẽte ſuper axem: ſuper-
            <lb/>
          ficiei conicæ medietas uidetur. Alhazen 36 n 4.</head>
          <p>
            <s xml:id="echoid-s29453" xml:space="preserve">Verbi gratia ſit pyramis a c n:</s>
            <s xml:id="echoid-s29454" xml:space="preserve"> cuius axis a d, & uertex a:</s>
            <s xml:id="echoid-s29455" xml:space="preserve"> palàm ergo per 89 th.</s>
            <s xml:id="echoid-s29456" xml:space="preserve"> 1 huius, quòd pun
              <lb/>
            ctum d eſt centrũ circuli baſis ipſius coni:</s>
            <s xml:id="echoid-s29457" xml:space="preserve"> ſitq̀;</s>
            <s xml:id="echoid-s29458" xml:space="preserve"> centrũ uiſus b:</s>
            <s xml:id="echoid-s29459" xml:space="preserve"> & ducatur linea b a faciens angulum
              <lb/>
            b a d rectũ.</s>
            <s xml:id="echoid-s29460" xml:space="preserve"> Dico, quòd conicæ ſuperficiei a c n medietas uidebitur.</s>
            <s xml:id="echoid-s29461" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0458-02" xlink:href="fig-0458-02a" number="508">
                <variables xml:id="echoid-variables488" xml:space="preserve">b a f j g e k y n d c</variables>
              </figure>
            Secet enim aliqua ſuperficies conum a c n æquidiſtáter baſi c n:</s>
            <s xml:id="echoid-s29462" xml:space="preserve"> hæc
              <lb/>
            ergo per 100 th.</s>
            <s xml:id="echoid-s29463" xml:space="preserve"> 1 huius ſecabit ipſam ſecũdum circulũ, qui ſit f g:</s>
            <s xml:id="echoid-s29464" xml:space="preserve"> &
              <lb/>
            eius cẽtrum, quod ſit pũctum l, erit in aliquo puncto axis a d:</s>
            <s xml:id="echoid-s29465" xml:space="preserve"> ſecetq́:</s>
            <s xml:id="echoid-s29466" xml:space="preserve">
              <lb/>
            ſuperficies plana pyramidẽ per axem a d, & per cẽtrũ uiſus d:</s>
            <s xml:id="echoid-s29467" xml:space="preserve"> illa er-
              <lb/>
            go ſuperficies ſecabit circulum f g:</s>
            <s xml:id="echoid-s29468" xml:space="preserve"> linea quoq;</s>
            <s xml:id="echoid-s29469" xml:space="preserve"> cõmunis huic ſuper-
              <lb/>
            ficiei & circulo f g erit orthogonalis ſuper axem:</s>
            <s xml:id="echoid-s29470" xml:space="preserve"> quoniã axis eſt ere-
              <lb/>
            ctus ſuper ſuperficiẽ circuli, & tráſibit cẽtrũ circuli.</s>
            <s xml:id="echoid-s29471" xml:space="preserve"> Sit quoq;</s>
            <s xml:id="echoid-s29472" xml:space="preserve"> illa li-
              <lb/>
            nea k l:</s>
            <s xml:id="echoid-s29473" xml:space="preserve"> quę erit ք 28 p 1 æquidiſtás lineæ b a, & eſt cũilla in eadẽ ſu-
              <lb/>
            perficie.</s>
            <s xml:id="echoid-s29474" xml:space="preserve"> Ducatur quoq;</s>
            <s xml:id="echoid-s29475" xml:space="preserve"> ք cẽtrum circuli diameter f l g orthogona-
              <lb/>
            lis ſuper lineá k l ք 11 p 1:</s>
            <s xml:id="echoid-s29476" xml:space="preserve"> & à terminis huius diametri protrahantur
              <lb/>
            duę lineæ cótingentes circulũ per 17 p 3, quę ſint f e & g h:</s>
            <s xml:id="echoid-s29477" xml:space="preserve"> & ab eiſdẽ
              <lb/>
            pũctis g & h ducãtur duę lineæ lógitudinis ad uerticẽ coni ք 101 th.</s>
            <s xml:id="echoid-s29478" xml:space="preserve"> 1
              <lb/>
            huius, quę ſint f a & g a:</s>
            <s xml:id="echoid-s29479" xml:space="preserve"> duę ergo ſuperficies planę, in quarũ una ſunt
              <lb/>
            lineę f e & f a, & in quarũ altera ſunt lineæ g h & g a, palàm quoniam
              <lb/>
            cõtingẽt pyramidẽ ſecũdũ lineas lõgitudinis, quę ſunt f a & g a ք 95
              <lb/>
            th.</s>
            <s xml:id="echoid-s29480" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s29481" xml:space="preserve"> Et quoniã linea k l æquidiſtat lineæ b a, & lineis cotingen
              <lb/>
            tibus circulũ, quę ſunt f e & g h, ut patet per 16 p 3, & per 28 p 1:</s>
            <s xml:id="echoid-s29482" xml:space="preserve"> erunt
              <lb/>
            per 9 p 11 lineæ f e & g h æquidiſtãtes lineæ b a:</s>
            <s xml:id="echoid-s29483" xml:space="preserve"> quęlibet ergo ipſarũ
              <lb/>
            eſt in eadẽ ſuperficie cũ illa per 1 th.</s>
            <s xml:id="echoid-s29484" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s29485" xml:space="preserve"> Illę ergo duæ ſuperficies
              <lb/>
            neceſſariò ſecabũt ſe ſuper lineã b a per 19 th.</s>
            <s xml:id="echoid-s29486" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s29487" xml:space="preserve"> utraq;</s>
            <s xml:id="echoid-s29488" xml:space="preserve"> ergo ſuperficierũ pyramidẽ propoſitã
              <lb/>
            in terminis diametri unius ſuorũ circulorũ cótigentiũ trãſit per cẽtrum uiſus.</s>
            <s xml:id="echoid-s29489" xml:space="preserve"> Quod ergo ſuperfi-
              <lb/>
            ciei conicę inter illas ſuperficies cadit, apparet uiſui:</s>
            <s xml:id="echoid-s29490" xml:space="preserve"> eſt aũt hæc medietas pyramidis, quoniá illas li
              <lb/>
            neas contingentes interiacet medietas circuli.</s>
            <s xml:id="echoid-s29491" xml:space="preserve"> In hoc ergo ſitu medietas ſuperficiei conicæ uide-
              <lb/>
            tur.</s>
            <s xml:id="echoid-s29492" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s29493" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1186" type="section" level="0" n="0">
          <head xml:id="echoid-head938" xml:space="preserve" style="it">90. Linea à centro uiſus ad uerticem coni duct a angulũ obtuſum cũ axetenente, nec tamen
            <lb/>
          cum aliqua line arum longitudinis coni unita: uidetur ſnperficiei conicæ pars maior medietate.
            <lb/>
          Alhazen 37 n 4.</head>
          <p>
            <s xml:id="echoid-s29494" xml:space="preserve">Sit pyramis b i m:</s>
            <s xml:id="echoid-s29495" xml:space="preserve"> cuius axis b d:</s>
            <s xml:id="echoid-s29496" xml:space="preserve"> uertex b:</s>
            <s xml:id="echoid-s29497" xml:space="preserve"> palamq́;</s>
            <s xml:id="echoid-s29498" xml:space="preserve"> per 89 th.</s>
            <s xml:id="echoid-s29499" xml:space="preserve"> 1 huius, quòd cẽtrũ circuli baſis eſt
              <lb/>
            punctũ d:</s>
            <s xml:id="echoid-s29500" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s29501" xml:space="preserve"> punctũ a centrũ uiſus:</s>
            <s xml:id="echoid-s29502" xml:space="preserve"> & ducta linea a b, fiat angulus a b d obtuſus, ita tamẽ, ut linea
              <lb/>
            a b nõ fiat una linea cũ aliqua linearũ lõgitudinis coni, ſed ſecet eas utcũq;</s>
            <s xml:id="echoid-s29503" xml:space="preserve"> poſsibile eſt productas
              <lb/>
            oẽs:</s>
            <s xml:id="echoid-s29504" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s29505" xml:space="preserve"> tũc uiſus altior uertice pyramidis:</s>
            <s xml:id="echoid-s29506" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s29507" xml:space="preserve">, ut in pręcedẽte, circulus e h æquidiſtás baſipy-
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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