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-div1150" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s28949" xml:space="preserve">
              <pb o="148" file="0450" n="450" rhead="VITELLONIS OPTICAE"/>
            metro:</s>
            <s xml:id="echoid-s28950" xml:space="preserve"> eſt autem linea g d diameter baſis pyramidis uiſionis:</s>
            <s xml:id="echoid-s28951" xml:space="preserve"> minus ergo hemiſphærio uidetur.</s>
            <s xml:id="echoid-s28952" xml:space="preserve">
              <lb/>
            Quod eſt prop oſitum.</s>
            <s xml:id="echoid-s28953" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1152" type="section" level="0" n="0">
          <head xml:id="echoid-head915" xml:space="preserve" style="it">67. Viſu ſphæræ illuminatæ conuexæ approximante, minus ſuperficiei ſphæræ uidetur: appa-
            <lb/>
          ret autem quaſi magis uideatur. Euclides 24 th. opt.</head>
          <p>
            <s xml:id="echoid-s28954" xml:space="preserve">Eſto, ut in præ miſſa, ſphæra, cuius centrum a:</s>
            <s xml:id="echoid-s28955" xml:space="preserve"> ſit quoq;</s>
            <s xml:id="echoid-s28956" xml:space="preserve"> centrum uiſus b:</s>
            <s xml:id="echoid-s28957" xml:space="preserve"> & ducatur linea a b:</s>
            <s xml:id="echoid-s28958" xml:space="preserve"> & cir
              <lb/>
            ca diam etrum a b deſcribatur circulus g b d:</s>
            <s xml:id="echoid-s28959" xml:space="preserve"> & ducatur à pũcto
              <lb/>
              <figure xlink:label="fig-0450-01" xlink:href="fig-0450-01a" number="495">
                <variables xml:id="echoid-variables475" xml:space="preserve">e a z g k l d c b</variables>
              </figure>
            a linea e a z perpendiculariter ſuper lineam a b per 11 p 1.</s>
            <s xml:id="echoid-s28960" xml:space="preserve"> Et quia
              <lb/>
            lineæ a b & e z ſunt in una ſuperficie per 2 p 11:</s>
            <s xml:id="echoid-s28961" xml:space="preserve"> intelligatur hæc ſu
              <lb/>
            perficies plana ſecare ſphæram:</s>
            <s xml:id="echoid-s28962" xml:space="preserve"> ipſa autem per 69 th.</s>
            <s xml:id="echoid-s28963" xml:space="preserve"> 1 huius ſe-
              <lb/>
            cabit ſphæram ſecundum circulum, qui ſit g e z d:</s>
            <s xml:id="echoid-s28964" xml:space="preserve"> eruntq́;</s>
            <s xml:id="echoid-s28965" xml:space="preserve"> puncta
              <lb/>
            ſectionis duorum propoſitorum circulorum, quę g & d:</s>
            <s xml:id="echoid-s28966" xml:space="preserve"> & ducan
              <lb/>
            tur lineæ g a, d a, b g, b d:</s>
            <s xml:id="echoid-s28967" xml:space="preserve"> & patet per modum proximæ præcedẽ,
              <lb/>
            tis, quoniam lineę b g & b d contingunt ſphæram, & uidetur ab
              <lb/>
            oculo exiſtente in puncto b pars ſphæræ g d.</s>
            <s xml:id="echoid-s28968" xml:space="preserve"> Sit ergo, ut appro-
              <lb/>
            pinquet oculus ſphęrę, & fiat in pũcto c:</s>
            <s xml:id="echoid-s28969" xml:space="preserve"> ducaturq́;</s>
            <s xml:id="echoid-s28970" xml:space="preserve"> c a, circa quã,
              <lb/>
            ut diametrum, deſcribatur circulus a k c l:</s>
            <s xml:id="echoid-s28971" xml:space="preserve"> ducanturq́;</s>
            <s xml:id="echoid-s28972" xml:space="preserve"> lineæ c k,
              <lb/>
            c l, a k, a l:</s>
            <s xml:id="echoid-s28973" xml:space="preserve"> ergo ք pręmiſſam uidebitur ab oculo exiſtẽte in pũcto
              <lb/>
            c, pars ſphærę, quę eſt k l, quæ minor eſt parte ſphærę g d uiſæ ab
              <lb/>
            oculo exiſtente in puncto b:</s>
            <s xml:id="echoid-s28974" xml:space="preserve"> quoniã arcus cadẽs inter puncta cõ
              <lb/>
            tingentię linearum c k & c l, quę per 64 huius contingunt ſphę.</s>
            <s xml:id="echoid-s28975" xml:space="preserve">
              <lb/>
            ram, minor eſt arcu g d, qui cadit inter puncta contingentiæ li-
              <lb/>
            nearũ b g & b d:</s>
            <s xml:id="echoid-s28976" xml:space="preserve"> quod patet per 60 th.</s>
            <s xml:id="echoid-s28977" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s28978" xml:space="preserve"> Palàm ergo quo-
              <lb/>
            niam appropinquante oculo ipſi ſphęrę, minus ſuperficiei ſphę
              <lb/>
            ricę uidetur.</s>
            <s xml:id="echoid-s28979" xml:space="preserve"> Quia uerò, ut patet per 60 th.</s>
            <s xml:id="echoid-s28980" xml:space="preserve"> 1 huius, lineę g b & c k concurrunt, ſi producantur uerſus
              <lb/>
            punctum g:</s>
            <s xml:id="echoid-s28981" xml:space="preserve"> palàm per 16 p 1, quoniam angulus k c a maior eſt angulo g b a:</s>
            <s xml:id="echoid-s28982" xml:space="preserve"> ſimiliter angulus a clma
              <lb/>
            ior eſt angulo a b d:</s>
            <s xml:id="echoid-s28983" xml:space="preserve"> totus ergo angulus k c l eſt maior toto angulo g b d.</s>
            <s xml:id="echoid-s28984" xml:space="preserve"> Pars ergo ſphęrę, in qua eſt
              <lb/>
            arcus k l, ſub maiori angulo uidebitur, quã pars ſphęrę, in qua eſt arcus g d.</s>
            <s xml:id="echoid-s28985" xml:space="preserve"> Apparet ergo per 20 hu-
              <lb/>
            ius maior uiſui pars ſphęrę, quę eſt k l, quàm pars eius, quæ eſt g d.</s>
            <s xml:id="echoid-s28986" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s28987" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1154" type="section" level="0" n="0">
          <head xml:id="echoid-head916" xml:space="preserve" style="it">68. Diametro ſphæræ illuminatæ conuexæ, lineæ connectentic entra amborum oculorumæ-
            <lb/>
          quali exiſtente: hemiſphærium eſt, quod ambobus uiſibus uidetur.
            <lb/>
          Euclides 26 th. opt.</head>
          <figure number="496">
            <variables xml:id="echoid-variables476" xml:space="preserve">g a b e z d</variables>
          </figure>
          <p>
            <s xml:id="echoid-s28988" xml:space="preserve">Sphæræ datę ſit centrum a:</s>
            <s xml:id="echoid-s28989" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s28990" xml:space="preserve"> circulus eius maior, cuius diame
              <lb/>
            ter ſti b g:</s>
            <s xml:id="echoid-s28991" xml:space="preserve"> quę ex hypotheſi ſit ęqualis diſtantiæ oculorum, hoc eſt
              <lb/>
            lineę connectenti centra uiſuum amborum, qui ſint e & d.</s>
            <s xml:id="echoid-s28992" xml:space="preserve"> Ducan-
              <lb/>
            tur quoq;</s>
            <s xml:id="echoid-s28993" xml:space="preserve"> à punctis b & g perpendiculares b d & g e, quę fiant ęqua-
              <lb/>
            les per 3 p 1:</s>
            <s xml:id="echoid-s28994" xml:space="preserve"> & copuletur linea d e:</s>
            <s xml:id="echoid-s28995" xml:space="preserve"> quæ per 33 p 1 & ex hypotheſi erit
              <lb/>
            æqualis & ęquidιſtans lineæ g b.</s>
            <s xml:id="echoid-s28996" xml:space="preserve"> Ducatur quo que perpendicularis
              <lb/>
            à puncto a centro ſphęrę ſuper lineam g b per 11 p 1:</s>
            <s xml:id="echoid-s28997" xml:space="preserve"> quę producta ad
              <lb/>
            lineam d e ſecet ipſam in puncto z.</s>
            <s xml:id="echoid-s28998" xml:space="preserve"> Palàm ergo per 29 p 1, quoniam
              <lb/>
            linea a z eſt per pendicularis ſuper lineam e d, & per 28 p 1 erit linea
              <lb/>
            a z ęquidiſtãs lineę g e:</s>
            <s xml:id="echoid-s28999" xml:space="preserve"> ergo per 33 p 1 patet, quòd linea e d diuiditur
              <lb/>
            per æqualia in puncto z, quia, ut patet ex hypotheſi, oculi ſunt in
              <lb/>
            punctis d & e:</s>
            <s xml:id="echoid-s29000" xml:space="preserve"> dico, quòd hemiſphęrium eſt quod uidetur.</s>
            <s xml:id="echoid-s29001" xml:space="preserve"> Manen-
              <lb/>
            te enim fixa linea a z, circumuoluatur parallelogrãmum a b z d, do-
              <lb/>
            nec redeat ad locum, unde incœpit:</s>
            <s xml:id="echoid-s29002" xml:space="preserve"> linea ergo a b mota deſcribet cir
              <lb/>
            culum ęqualem circulo g b, cuius ipſa eſt ſemidiam eter:</s>
            <s xml:id="echoid-s29003" xml:space="preserve"> eſt autẽ cir-
              <lb/>
            culus magnus ſphęrę datæ circulus g d:</s>
            <s xml:id="echoid-s29004" xml:space="preserve"> ergo per motũ lineę a b de-
              <lb/>
            ſcribitur circulus magnus:</s>
            <s xml:id="echoid-s29005" xml:space="preserve"> hic autem ſphęram diuidit in duo ęqua-
              <lb/>
            lia.</s>
            <s xml:id="echoid-s29006" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s29007" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1155" type="section" level="0" n="0">
          <figure number="497">
            <variables xml:id="echoid-variables477" xml:space="preserve">u e f c a h d b g</variables>
          </figure>
          <head xml:id="echoid-head917" xml:space="preserve" style="it">69. Linea connectens centra amborum oculorum, ſimaior diametro ſphæræ illuminatæ con-
            <lb/>
          uexæ fuerit: plus hemiſphærio eſt, quod ambo-
            <lb/>
          bus uiſibus uidetur. Euclides 27 th. opt.</head>
          <p>
            <s xml:id="echoid-s29008" xml:space="preserve">Sit ſphæra data, cuius centrum a:</s>
            <s xml:id="echoid-s29009" xml:space="preserve"> & eius circu
              <lb/>
            lus magnus ſit e c d i:</s>
            <s xml:id="echoid-s29010" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s29011" xml:space="preserve"> centra amborum o-
              <lb/>
            culorum b & g:</s>
            <s xml:id="echoid-s29012" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s29013" xml:space="preserve"> linea b g producta maior dia
              <lb/>
            metro datę ſphęræ & eius circuli magni.</s>
            <s xml:id="echoid-s29014" xml:space="preserve"> Dico,
              <lb/>
            quòd ambobus uiſibus maius hemiſphęrio ui-
              <lb/>
            debitur.</s>
            <s xml:id="echoid-s29015" xml:space="preserve"> Ducantur enim à centris oculorum li-
              <lb/>
            neæ b e & g d contingentes circulum e d ci per
              <lb/>
            17 p 3:</s>
            <s xml:id="echoid-s29016" xml:space="preserve"> contingantq́;</s>
            <s xml:id="echoid-s29017" xml:space="preserve"> in punctis e & d:</s>
            <s xml:id="echoid-s29018" xml:space="preserve"> & ducatur
              <lb/>
            à puncto a diameter ſphęrę ęquidiſtãs lineę b g
              <lb/>
            </s>
          </p>
        </div>
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