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-div816" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22514" xml:space="preserve">
              <pb o="38" file="0340" n="340" rhead="VITELLONIS OPTICAE"/>
            æquidiſtans baſi fieri poterit per 31 p 1, ducta ab uno puncto primæ ſectiõis linea æquidiſtante alicui
              <lb/>
            linearum baſis pyramidis, & à terminis illius alijs lineis æquidiſtantibus reliquis lineis baſis produ
              <lb/>
            ctis.</s>
            <s xml:id="echoid-s22515" xml:space="preserve">) Ex hoc autem accidit impoſsibile, quoniã ſequitur ex hypotheſi angulum extrinſecum pro-
              <lb/>
            pter trigonorum ſimilitudinem æqualem fieri intrinſeco:</s>
            <s xml:id="echoid-s22516" xml:space="preserve"> cum ab uno puncto exeant duæ lineæ æ-
              <lb/>
            quales angulos cõtinentes angulis illis, qui fiunt per lineã aliquã longitudinis & per lineam aliquã
              <lb/>
            peripherię baſis.</s>
            <s xml:id="echoid-s22517" xml:space="preserve"> Patet ergo propoſitum in pyramidibus.</s>
            <s xml:id="echoid-s22518" xml:space="preserve"> Et eodem modo demonſtrandũ eſt in co-
              <lb/>
            lumnis lateratis, & facilius propter æqualitatem linearum per 34 primi.</s>
            <s xml:id="echoid-s22519" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div817" type="section" level="0" n="0">
          <head xml:id="echoid-head674" xml:space="preserve" style="it">100. Omnis ſuperficiei planæ ſecantis pyramidem uel columnam rotundam trans axem æ-
            <lb/>
          quidiſtanter baſi, & curuæ ſuperficiei pyramidis uel columnæ communis ſectio eſt circulus: & ſi
            <lb/>
          illa ſectio eſt circulus, ſuperficies ſecans eſt æquidiſtans baſi. Ex quo patet, quòd omnis plana ſu-
            <lb/>
          perficies æquidiſtanter baſi ſecans pyramidem uel columnam, nouam pyramidem conſtituit uel
            <lb/>
          columnam. 4 theor. 1 Conicorum Apollonij, & 5 the. Cylindricorum Sereni.</head>
          <p>
            <s xml:id="echoid-s22520" xml:space="preserve">Sit pyramis rotũda a b c, cuius uertex a:</s>
            <s xml:id="echoid-s22521" xml:space="preserve"> diameter baſis b c, & centrũ baſis d:</s>
            <s xml:id="echoid-s22522" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s22523" xml:space="preserve"> ipſam ſuperfi
              <lb/>
            cies plana æquidiſtanter baſi:</s>
            <s xml:id="echoid-s22524" xml:space="preserve"> & ſit cõmunis ſectio ſuperficiei illius & ſuperficiei conicæ pyramidis
              <lb/>
            linea e f g.</s>
            <s xml:id="echoid-s22525" xml:space="preserve"> Dico, quòd linea e f g eſt peripheria circuli.</s>
            <s xml:id="echoid-s22526" xml:space="preserve"> Secet enim alia ſuperficies plana pyramidem
              <lb/>
            per uerticem & per axem, qui eſt a d.</s>
            <s xml:id="echoid-s22527" xml:space="preserve"> Cõmunis itaq;</s>
            <s xml:id="echoid-s22528" xml:space="preserve"> illius ſuperficiei & pyramidis ſectio eſt trigonũ
              <lb/>
            (quod ſit a b c) per 90 huius:</s>
            <s xml:id="echoid-s22529" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s22530" xml:space="preserve"> ſuperficies e f g axem a d in puncto h:</s>
            <s xml:id="echoid-s22531" xml:space="preserve"> & trigonum a b c ſecet ſu-
              <lb/>
            perficiem e f g in linea e h f.</s>
            <s xml:id="echoid-s22532" xml:space="preserve"> Erit ergo linea e h æquidiſtans lineæ b d
              <lb/>
              <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a" number="354">
                <variables xml:id="echoid-variables338" xml:space="preserve">a e h f g b d c</variables>
              </figure>
            per 16 p 11:</s>
            <s xml:id="echoid-s22533" xml:space="preserve"> eſt ergo per 29 p 1 & 4 p 6 proportio lineæ b a ad e a, ſi-
              <lb/>
            cut lineæ c a ad lineam a f:</s>
            <s xml:id="echoid-s22534" xml:space="preserve"> ergo per 7 huius erit euerſim proportio
              <lb/>
            lineę b a ad lineam b e, ſicut lineę c a ad lineam c f:</s>
            <s xml:id="echoid-s22535" xml:space="preserve"> ergo per 16 p 5 erit
              <lb/>
            permutatim proportio lineę b a ad lineam c a, ſicut lineę b e ad lineã
              <lb/>
            c f.</s>
            <s xml:id="echoid-s22536" xml:space="preserve"> Sed linea b a eſt ęqualis ipſi c a per 89 huius:</s>
            <s xml:id="echoid-s22537" xml:space="preserve"> ergo erit linea b e æ-
              <lb/>
            qualis lineę c f.</s>
            <s xml:id="echoid-s22538" xml:space="preserve"> Ducantur itaq;</s>
            <s xml:id="echoid-s22539" xml:space="preserve"> lineæ d e, d f.</s>
            <s xml:id="echoid-s22540" xml:space="preserve"> Et quoniã per 89 huius,
              <lb/>
            anguli, quos continent lineę longitudinis pyramidum cum ſemidia
              <lb/>
            metris baſium, ſunt æquales:</s>
            <s xml:id="echoid-s22541" xml:space="preserve"> palàm per 4 p 1, quia linea d e eſt æqua-
              <lb/>
            lis lineæ d f:</s>
            <s xml:id="echoid-s22542" xml:space="preserve"> & angulus e d b eſt æqualis angulo f d c.</s>
            <s xml:id="echoid-s22543" xml:space="preserve"> Quia uerò an-
              <lb/>
            gulus h d b æqualis angulo h d c:</s>
            <s xml:id="echoid-s22544" xml:space="preserve"> quoniã ambo ſunt recti:</s>
            <s xml:id="echoid-s22545" xml:space="preserve"> & angulus
              <lb/>
            e d b æqualis angulo f d c:</s>
            <s xml:id="echoid-s22546" xml:space="preserve"> remanet angulus e d h æqualis angulo f d
              <lb/>
            h:</s>
            <s xml:id="echoid-s22547" xml:space="preserve"> quoniã ſunt reſiduę partes rectorũ ſupér angulos æquales.</s>
            <s xml:id="echoid-s22548" xml:space="preserve"> Palàm
              <lb/>
            ergo per 4 p 1 quoniã linea e h eſt ęquà
              <unsure/>
            lis lineę h f.</s>
            <s xml:id="echoid-s22549" xml:space="preserve"> Similiterq́;</s>
            <s xml:id="echoid-s22550" xml:space="preserve"> ductis
              <lb/>
            lineis h g & d g, & cõpleta, prout in præmiſsis, figuratione, declara-
              <lb/>
            bitur, quoniã linea f h eſt æqualis lineæ g h:</s>
            <s xml:id="echoid-s22551" xml:space="preserve"> ſunt enim trigona æqui-
              <lb/>
            angula, ut patet intendenti.</s>
            <s xml:id="echoid-s22552" xml:space="preserve"> Ergo per 9 p 3 punctum h eſt centrũ cir-
              <lb/>
            culi.</s>
            <s xml:id="echoid-s22553" xml:space="preserve"> Eſt ergo e f g linea circũferentia circuli.</s>
            <s xml:id="echoid-s22554" xml:space="preserve"> Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s22555" xml:space="preserve"> Et
              <lb/>
            ſi ſectio e f g eſt circulus, palàm quoniã ſuperficies plana ſecundum
              <lb/>
            illum circulum ſecans pyramidem, eſt æquidiſtans baſi:</s>
            <s xml:id="echoid-s22556" xml:space="preserve"> erit enim e a
              <lb/>
            f pyramis, cuius axis a h, & centrum baſis h:</s>
            <s xml:id="echoid-s22557" xml:space="preserve"> erit itaq;</s>
            <s xml:id="echoid-s22558" xml:space="preserve"> linea longitudinis, quę eſt e a, æqualis lineę f a
              <lb/>
            per 89 huius:</s>
            <s xml:id="echoid-s22559" xml:space="preserve"> ſed & linea b a æqualis eſt ipſi c a:</s>
            <s xml:id="echoid-s22560" xml:space="preserve"> remanet ergo linea b e æqualis ipſi e f.</s>
            <s xml:id="echoid-s22561" xml:space="preserve"> Erit quoq;</s>
            <s xml:id="echoid-s22562" xml:space="preserve"> li-
              <lb/>
            nea e d æqualis lineę f d per 4 p 1.</s>
            <s xml:id="echoid-s22563" xml:space="preserve"> Et quia trigona e h d & f h d ſunt æqualia inter ſe latera habentia:</s>
            <s xml:id="echoid-s22564" xml:space="preserve">
              <lb/>
            ergo per 8 p 1 angulus e h d eſt æqualis angulo f h d.</s>
            <s xml:id="echoid-s22565" xml:space="preserve"> Ergo per definitionem lineæ ſuper ſuperficiem
              <lb/>
            erectę patet, quod linea d h erecta eſt ſuper ſuperficiem e f g:</s>
            <s xml:id="echoid-s22566" xml:space="preserve"> ſed eadem linea h d eſt erecta ſuper ba-
              <lb/>
            ſim pyramidis, cuius diameter eſt b c.</s>
            <s xml:id="echoid-s22567" xml:space="preserve"> Ergo per 14 p 11 ſuperficies e f g eſt æquidiſtans baſi datę pyra
              <lb/>
            midis.</s>
            <s xml:id="echoid-s22568" xml:space="preserve"> Quod eſt propoſitum:</s>
            <s xml:id="echoid-s22569" xml:space="preserve"> quoniam ſimpliciter ſecundum præmiſſum in pyramidibus modum,
              <lb/>
            in columnis quoq;</s>
            <s xml:id="echoid-s22570" xml:space="preserve"> rotundis poteſt demonſtrari, & propter æquidiſtantiam linearum longitudinis
              <lb/>
            columnę facilitas accedit demõſtrationi.</s>
            <s xml:id="echoid-s22571" xml:space="preserve"> Fiunt enim lineę d f, d g, d e æquales:</s>
            <s xml:id="echoid-s22572" xml:space="preserve"> ergo & lineę h e, h g,
              <lb/>
            h f:</s>
            <s xml:id="echoid-s22573" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s22574" xml:space="preserve"> ſectio e g f circulus per 9 p 3.</s>
            <s xml:id="echoid-s22575" xml:space="preserve"> Et conuerſa ſimpliciter patet per 14 p 11, ut prius.</s>
            <s xml:id="echoid-s22576" xml:space="preserve"> Et hoc pro-
              <lb/>
            ponebatur.</s>
            <s xml:id="echoid-s22577" xml:space="preserve"> Per hæc itaq;</s>
            <s xml:id="echoid-s22578" xml:space="preserve"> patet manifeſtè, quoniam omnis plana ſuperficies ſecans quamcunq;</s>
            <s xml:id="echoid-s22579" xml:space="preserve"> py-
              <lb/>
            ramidem ęquidiſtanter ſuę baſi, nouã conſtituit pyramidem, cuius in pyramide rotunda, baſis eſt
              <lb/>
            circulus, & in laterata pyramide ſuperficies ſimilis baſi illius ſectę pyramidis, ut patet per 99 huius.</s>
            <s xml:id="echoid-s22580" xml:space="preserve">
              <lb/>
            Semper tamen uertex illius pyramidis abſciſſę eſt idem cum uertice prioris, & axis abſciſſę, pars a-
              <lb/>
            xis ipſius prioris datę:</s>
            <s xml:id="echoid-s22581" xml:space="preserve"> baſis quoq;</s>
            <s xml:id="echoid-s22582" xml:space="preserve"> æquidiſtat baſi.</s>
            <s xml:id="echoid-s22583" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s22584" xml:space="preserve"> fit in columnis rotundis uel late
              <lb/>
            ratis:</s>
            <s xml:id="echoid-s22585" xml:space="preserve"> ſuperficies enim ęquidiſtanter baſibus ſecans quamcunq;</s>
            <s xml:id="echoid-s22586" xml:space="preserve"> columnam, nouam efficit columnã
              <lb/>
            rotundam uel lateratam:</s>
            <s xml:id="echoid-s22587" xml:space="preserve"> imò duas, ſcilicet abſciſſam & ipſam reſiduam:</s>
            <s xml:id="echoid-s22588" xml:space="preserve"> quod non accidit in pyrami
              <lb/>
            dibus.</s>
            <s xml:id="echoid-s22589" xml:space="preserve"> Patet ergo totum, quod proponebatur.</s>
            <s xml:id="echoid-s22590" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div819" type="section" level="0" n="0">
          <head xml:id="echoid-head675" xml:space="preserve" style="it">101. In qualibet columna uel pyramide à dato in eius ſuperficie puncto, lineam longitudinis
            <lb/>
          ducere. 7 theo. Cylindricorum Sereni.</head>
          <p>
            <s xml:id="echoid-s22591" xml:space="preserve">Imaginetur enim ſuperficies plana ſecãs pyramidem uel columnã trans illius punctum & trans
              <lb/>
            axem:</s>
            <s xml:id="echoid-s22592" xml:space="preserve"> quod fiet, ſi à puncto dato ducatur linea recta ſuper axẽ:</s>
            <s xml:id="echoid-s22593" xml:space="preserve"> illa ergo linea & axis ſunt in una ſu-
              <lb/>
            perficie per 2 p 11:</s>
            <s xml:id="echoid-s22594" xml:space="preserve"> quę ſuperficies ſecabit pyramidem ſecundum lineam longitudinis per illud pun-
              <lb/>
            ctum tranſeuntem per 90 huius:</s>
            <s xml:id="echoid-s22595" xml:space="preserve"> columnam quoq;</s>
            <s xml:id="echoid-s22596" xml:space="preserve"> per 92 huius.</s>
            <s xml:id="echoid-s22597" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22598" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div820" type="section" level="0" n="0">
          <head xml:id="echoid-head676" xml:space="preserve" style="it">102. À
            <unsure/>
          dato puncto, ſiue in axe, ſiue in ſuperficie curua datæ pyramidis rotundæ uel colũnæ,
            <lb/>
          circulum circumducere.</head>
        </div>
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    </echo>
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