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-div1912" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s51768" xml:space="preserve">
              <pb o="469" file="0771" n="771" rhead="LIBER DECIMVS."/>
            cundum præmiſſa nota.</s>
            <s xml:id="echoid-s51769" xml:space="preserve"> Copuletur itaq;</s>
            <s xml:id="echoid-s51770" xml:space="preserve"> à puncto p ad punctũ m linea in ſuperficie circuli altitudi-
              <lb/>
            nis, quæ ſit p m:</s>
            <s xml:id="echoid-s51771" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s51772" xml:space="preserve"> neceſſariò, ut quæ eſt ꝓportio lineæ c d ad h k, uel lineæ b c ad k p, eadẽ ſit ꝓ-
              <lb/>
            portio lineæ a b ad lineã p m.</s>
            <s xml:id="echoid-s51773" xml:space="preserve"> Quòd ſi dicatur hoc nõ eſſe poſsibile:</s>
            <s xml:id="echoid-s51774" xml:space="preserve"> quę eſt ergo proportio lineæ c d
              <lb/>
            ad h k, uel b c ad k p:</s>
            <s xml:id="echoid-s51775" xml:space="preserve"> eadẽ erit lineæ a b ad aliquã aliã lineam maiorẽ uel minorẽ linea p m, per 3 th.</s>
            <s xml:id="echoid-s51776" xml:space="preserve"> 1
              <lb/>
            huius.</s>
            <s xml:id="echoid-s51777" xml:space="preserve"> Sit ergo nũc illa proportio lineæ a b ad quandã minorem linea m p, quæ ſit p r.</s>
            <s xml:id="echoid-s51778" xml:space="preserve"> Quæ eſt ergo
              <lb/>
            proportio lineæ c d ad lineã h k, uel b c ad lineã k p, eadẽ eſt lineæ a b ad lineã p r:</s>
            <s xml:id="echoid-s51779" xml:space="preserve"> quæ autẽ eſt pro-
              <lb/>
            portio lineæ c d ad lineã h k, eadẽ eſt lineæ b c ad lineã k p:</s>
            <s xml:id="echoid-s51780" xml:space="preserve"> ergo per 16 p 5 quæ eſt proportio lineæ
              <lb/>
            c d ad b c, eadẽ eſt h k ad k p:</s>
            <s xml:id="echoid-s51781" xml:space="preserve"> & quæ eſt proportio lineæ b c ad k p, eadẽ eſt lineæ a b ad lineã p r:</s>
            <s xml:id="echoid-s51782" xml:space="preserve"> ergo
              <lb/>
            itẽ per 16 p 5 quæ eſt proportio lineæ b c ad a b, eadẽ eſt lineæ k p ad p r:</s>
            <s xml:id="echoid-s51783" xml:space="preserve"> & ſic lineæ c d, b c, a b pro-
              <lb/>
            portionales erunt lineis h k, k p, p r:</s>
            <s xml:id="echoid-s51784" xml:space="preserve"> ſed quæ eſt proportio lineæ a b ad b c, eadẽ eſt lineæ b d ad a b:</s>
            <s xml:id="echoid-s51785" xml:space="preserve">
              <lb/>
            ergo & in ipſarũ comproportionalibus ſic erit, quòd ſicut ſe habet linea r p ad p k, ſic coniunctim ſe
              <lb/>
            habebit tota p h ad lineã p r.</s>
            <s xml:id="echoid-s51786" xml:space="preserve"> Ducãtur ergo lineæ h r & k r:</s>
            <s xml:id="echoid-s51787" xml:space="preserve">fientq́;</s>
            <s xml:id="echoid-s51788" xml:space="preserve"> duo triãguli, qui h r p & k r p, quo-
              <lb/>
            rum cõmunis eſt angulus r p h, & latera dictũ angulũ continẽtia reſpectu diuerſorũ trigonorũ ſunt
              <lb/>
            proportionalia:</s>
            <s xml:id="echoid-s51789" xml:space="preserve"> quæ enim eſt ꝓportio lineæ p r lateris maioris trianguli ad lineã p k latus minoris
              <lb/>
            trianguli:</s>
            <s xml:id="echoid-s51790" xml:space="preserve"> eadẽ ꝓportio lineę h p lateris maioris trigoni ad lineã p r latus trigoni p r k minoris:</s>
            <s xml:id="echoid-s51791" xml:space="preserve"> ergo
              <lb/>
            per 6 p 6 illi trianguli ſunt æ quianguli:</s>
            <s xml:id="echoid-s51792" xml:space="preserve"> ergo per 4 p 6 latera ipſorũ æ quos angulos reſpiciẽtia ſunt
              <lb/>
            proportionalia.</s>
            <s xml:id="echoid-s51793" xml:space="preserve"> Eſt ergo ꝓportio lineæ h p ad lineã p r, & lineæ p r ad lineã p k, ſicut lineæ h r ad li-
              <lb/>
            neã k r:</s>
            <s xml:id="echoid-s51794" xml:space="preserve">ſed quam proportionẽ habet linea h p ad lineã p r, hanc habet linea b d ad lineã a b:</s>
            <s xml:id="echoid-s51795" xml:space="preserve"> & quam
              <lb/>
            habet linea b d ad a b, hãc habet linea a b ad b c:</s>
            <s xml:id="echoid-s51796" xml:space="preserve"> & quam habet a b ad b c, hãc habet linea h m ad k m
              <lb/>
            ex hypotheſi:</s>
            <s xml:id="echoid-s51797" xml:space="preserve"> per 11 ergo p 5 patet quòd quam proportionẽ habet linea h r ad lineã k r, hãc habet li-
              <lb/>
            nea h m ad lineã k m:</s>
            <s xml:id="echoid-s51798" xml:space="preserve"> hoc aũt eſt impoſsibile, & cõtra 56 th.</s>
            <s xml:id="echoid-s51799" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s51800" xml:space="preserve"> quoniã in ſemicirculo quocunq;</s>
            <s xml:id="echoid-s51801" xml:space="preserve">
              <lb/>
            duab.</s>
            <s xml:id="echoid-s51802" xml:space="preserve"> lineis ductis ad quẽcũq;</s>
            <s xml:id="echoid-s51803" xml:space="preserve"> pũctũ քipherię, ſcilicet una à termino diametri & alia à cẽtro, ut ſunt
              <lb/>
            in ꝓpoſito lineę h m & k m, duas alias lineas ab eiſdẽ pũctis ad ali udpũctũ circũſerentiæ quodcũq;</s>
            <s xml:id="echoid-s51804" xml:space="preserve">
              <lb/>
            duabus prioribus ꝓportionales ducere eſt impoſsibile.</s>
            <s xml:id="echoid-s51805" xml:space="preserve"> Eſt ergo impoſsibile lineã a b ad aliá mino-
              <lb/>
            rem lineá quam linea p m, eandẽ habere ꝓportionẽ quam linea b d ad lineã h p, uel quam linea c d
              <lb/>
            ad h k, uel quã linea b c ad k p.</s>
            <s xml:id="echoid-s51806" xml:space="preserve"> Sed neq;</s>
            <s xml:id="echoid-s51807" xml:space="preserve"> poteſt linea a b habere illã proportionẽ ad aliquá lineá ma-
              <lb/>
            iorẽ linea p m:</s>
            <s xml:id="echoid-s51808" xml:space="preserve"> quoniã eadẽ eſt ratio, & eodẽ modo deducitur ad impoſsibile.</s>
            <s xml:id="echoid-s51809" xml:space="preserve"> Ergo quę eſt ꝓportio
              <lb/>
            c d ad lineã h k, uel lineæ b c ad k p:</s>
            <s xml:id="echoid-s51810" xml:space="preserve"> eadę erit lineæ a b ad p m:</s>
            <s xml:id="echoid-s51811" xml:space="preserve"> & ſequetur repetita priori demõſtra-
              <lb/>
            tione, quæ ducebat ad impoſsibile, ſcilicet, ut quæ eſt ꝓportio lineæ h p ad p m, & lineæ m p ad p k,
              <lb/>
            eadẽ ſit lineæ h m ad k m.</s>
            <s xml:id="echoid-s51812" xml:space="preserve"> Ductis itaq;</s>
            <s xml:id="echoid-s51813" xml:space="preserve"> pluribus ſemicirculis altitudinis circa centrũ k ſub horizõte,
              <lb/>
            proportionales lineæ prędictis lineis h m & k m ducãtur ſecundũ modũ 56 th.</s>
            <s xml:id="echoid-s51814" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s51815" xml:space="preserve"> Si ergo linea
              <lb/>
            m p ſit perpẽdiculariter inſiſtẽs diametro h g:</s>
            <s xml:id="echoid-s51816" xml:space="preserve"> tũc poſito cẽtro p ſecundũ ſemidiametrũ p m deſcri-
              <lb/>
            batur circulus:</s>
            <s xml:id="echoid-s51817" xml:space="preserve"> quòd ſi linea p m nõ ſit perpẽdicularis ſuper diametrũ h g:</s>
            <s xml:id="echoid-s51818" xml:space="preserve"> polo itaq;</s>
            <s xml:id="echoid-s51819" xml:space="preserve"> exiſtẽte pũcto
              <lb/>
            p per 65 th.</s>
            <s xml:id="echoid-s51820" xml:space="preserve"> 1 huius (quoniã ille punctus æqualiter diſtabit ab omnibus in illis ſemicirculis ſignatis
              <lb/>
            pũctis, ſimilibus pũcto m) ducatur circulus ſecundũ diſtantiã lineæ p m:</s>
            <s xml:id="echoid-s51821" xml:space="preserve"> qui attinget omnia dicta
              <lb/>
            pũcta ſemicirculorũ altitudinis, in quæ cadũt prædictæ proportionales lineæ, ſiue anguli reflexio-
              <lb/>
            num iridẽ cauſſantes.</s>
            <s xml:id="echoid-s51822" xml:space="preserve"> Si enim dicatur quòd nõ attingat:</s>
            <s xml:id="echoid-s51823" xml:space="preserve"> accidet ſecundũ pręmiſſa contrariũ 56 th.</s>
            <s xml:id="echoid-s51824" xml:space="preserve"> 1
              <lb/>
            huius, quod eſt impoſsibile.</s>
            <s xml:id="echoid-s51825" xml:space="preserve"> Poteſt etiá ſic fieri, ut ſemicirculus h m g ſit medietas horizõtis, & facta
              <lb/>
            diuiſione in pũcto m, intelligatur circũduci idẽ ſemicirculus:</s>
            <s xml:id="echoid-s51826" xml:space="preserve"> nihil enim refert ſemicirculos diuer-
              <lb/>
            ſos deſcribere uel unũ circũducere:</s>
            <s xml:id="echoid-s51827" xml:space="preserve"> punctusq́;</s>
            <s xml:id="echoid-s51828" xml:space="preserve"> m circumductus deſcribet circulũ iridis, qui eſt n m,
              <lb/>
            circa centrũ uel polũ p ſecundũ diſtantiã lineæ p m:</s>
            <s xml:id="echoid-s51829" xml:space="preserve"> eruntq́;</s>
            <s xml:id="echoid-s51830" xml:space="preserve"> anguli à termino diametri, ſcilicet pũ-
              <lb/>
            cto h & à centro k ductarum linearũ ad circulũ n m, omnes æquales in qualibet ſuperficie reflexio-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s51831" xml:space="preserve"> quia triangulus h m k in tota circum ductione ſimiles ſibi triangulos cauſſat in qualibet ſuper-
              <lb/>
            ficie reflexionis:</s>
            <s xml:id="echoid-s51832" xml:space="preserve"> & ſimiliter triangulus h m p motu ſuo deſcribet ſimiles triangulos:</s>
            <s xml:id="echoid-s51833" xml:space="preserve"> & triangulus k
              <lb/>
            m p ſimiliter ſimiles triangulos deſcribet.</s>
            <s xml:id="echoid-s51834" xml:space="preserve"> Si itaq;</s>
            <s xml:id="echoid-s51835" xml:space="preserve"> linea m p non ſit perpẽdicularis ſuper diametrũ h
              <lb/>
            g:</s>
            <s xml:id="echoid-s51836" xml:space="preserve"> ducatur ergo perpẽdicularis à pũcto m per 12 p 1 ſuper diametrũ h g:</s>
            <s xml:id="echoid-s51837" xml:space="preserve"> cadetq́;</s>
            <s xml:id="echoid-s51838" xml:space="preserve"> illa perpendicularis
              <lb/>
            per 29 th.</s>
            <s xml:id="echoid-s51839" xml:space="preserve"> 1 huius inter pũcta k & p, uel inter pũcta p & g:</s>
            <s xml:id="echoid-s51840" xml:space="preserve"> quoniã linea m p cũ diametro h g ex aliqua
              <lb/>
            ſui parte angulũ acutũ continet, ut patet ex pręmiſsis:</s>
            <s xml:id="echoid-s51841" xml:space="preserve"> & ſimiliter linea m k;</s>
            <s xml:id="echoid-s51842" xml:space="preserve"> quia iris nõ apparet niſi
              <lb/>
            ultra mediũ diametri horizontis, ut prius patuit:</s>
            <s xml:id="echoid-s51843" xml:space="preserve"> cadat ergo illa perpẽdicularis in punctũ o.</s>
            <s xml:id="echoid-s51844" xml:space="preserve"> Simili-
              <lb/>
            ter quoq;</s>
            <s xml:id="echoid-s51845" xml:space="preserve"> ad idem punctũ diametri neceſſariò cadent ab omnibus aliorũ ſemicirculorum angulis
              <lb/>
            lineæ perpẽdiculares:</s>
            <s xml:id="echoid-s51846" xml:space="preserve"> uel angulus k o m motu ſuo in omnibus ſuքficiebus reflexionũ æquales an-
              <lb/>
            gulos cauſſabit.</s>
            <s xml:id="echoid-s51847" xml:space="preserve"> Punctũ ergo o eſt centrũ circuli reflexionis factę ad uiſum.</s>
            <s xml:id="echoid-s51848" xml:space="preserve"> Cũ ergo centrũ iridis ſit
              <lb/>
            in horizontis diametro:</s>
            <s xml:id="echoid-s51849" xml:space="preserve"> medietas eius erit ſupra horizontẽ, quæ eſt n m, & medietas ſub horizõte:</s>
            <s xml:id="echoid-s51850" xml:space="preserve">
              <lb/>
            quoniã tũc cõmunis ſectio ſuքficierũ horizontis & iridis eſt diameter iridis.</s>
            <s xml:id="echoid-s51851" xml:space="preserve"> Idẽq́;</s>
            <s xml:id="echoid-s51852" xml:space="preserve"> accideret ſi linea
              <lb/>
            m p eſſet քpẽdicularis ſuք diametrũ.</s>
            <s xml:id="echoid-s51853" xml:space="preserve"> Et hic eſt modus, quo Ariſtoteles ꝓpoſitũ cõcluſit.</s>
            <s xml:id="echoid-s51854" xml:space="preserve"> Sed tamen
              <lb/>
            nõ eſt nobis uiſa fore neceſſaria notitia linearũ, quia ſine illa idem & eodẽ modo declarari poteſt.</s>
            <s xml:id="echoid-s51855" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1914" type="section" level="0" n="0">
          <head xml:id="echoid-head1403" xml:space="preserve" style="it">75. In aliquo circulo altitudinis ſuper horizontem exiſtente centro corporis luminoſi, ſecun-
            <lb/>
          dum eius eleuationem centrum circuli iridis ſub horizonte deprimitur: & portio iridis minor
            <lb/>
          ſemicirculo uidetur.</head>
          <p>
            <s xml:id="echoid-s51856" xml:space="preserve">Eſto ſecundum diſpoſitionem proximæ, ſcilicet ut ſit horizon circulus h m g:</s>
            <s xml:id="echoid-s51857" xml:space="preserve"> cuius diameter ſit
              <lb/>
            linea m h.</s>
            <s xml:id="echoid-s51858" xml:space="preserve"> & centrum k:</s>
            <s xml:id="echoid-s51859" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s51860" xml:space="preserve"> circulus altitudinis tranſiens per zenith capitis & per centrum corpo-
              <lb/>
            ris luminoſi:</s>
            <s xml:id="echoid-s51861" xml:space="preserve"> qui eſt l m n h:</s>
            <s xml:id="echoid-s51862" xml:space="preserve"> & ſit centrum ſolis eleuatum ſupra horizontem in circulo altitudinis in
              <lb/>
            puncto n.</s>
            <s xml:id="echoid-s51863" xml:space="preserve"> Et quoniam per 64 th.</s>
            <s xml:id="echoid-s51864" xml:space="preserve"> huius centrum corporis luminoſi, & cẽtrum oculi, & centrũ baſis
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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