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

List of thumbnails

< >
331
331 (29) 		332
332 (30)
333
333 (31) 		334
334 (32)
335
335 (33) 		336
336 (34)
337
337 (35) 		338
338 (36)
339
339 (37) 		340
340 (38)
< >
page |< < (37) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div812" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22454" xml:space="preserve">
              <pb o="37" file="0339" n="339" rhead="LIBER PRIMVS."/>
            bit per axem a g per 90 huius.</s>
            <s xml:id="echoid-s22455" xml:space="preserve"> Trigonum ergo a b g cum linea d e eſt in eadem ſuperficie.</s>
            <s xml:id="echoid-s22456" xml:space="preserve"> Quia ergo
              <lb/>
            linea e d cum uno latere trigoni b a g, quod eſt a b, continet angulũ rectum, qui eſt d e a:</s>
            <s xml:id="echoid-s22457" xml:space="preserve"> angulus ue-
              <lb/>
            rò e a g eſt acutus:</s>
            <s xml:id="echoid-s22458" xml:space="preserve"> palàm, quia linea d e concurret cum linea a g per 14 huius.</s>
            <s xml:id="echoid-s22459" xml:space="preserve"> Tranſit ergo per axem
              <lb/>
            pyramidis uel columnæ rotundę.</s>
            <s xml:id="echoid-s22460" xml:space="preserve"> Quod eſt propoſitum:</s>
            <s xml:id="echoid-s22461" xml:space="preserve"> quoniã in columna rotunda eodem modo
              <lb/>
            demonſtandũ.</s>
            <s xml:id="echoid-s22462" xml:space="preserve"> In illa enim, quia linea longitudinis a b æquidiſtat axi, & lineę d e & a b & axis ſunt
              <lb/>
            in eadem ſuperficie:</s>
            <s xml:id="echoid-s22463" xml:space="preserve"> patet per 2 huius, quia linea d e concurrẽs cum una linearum æquidiſtantium,
              <lb/>
            ideo cum a b & cum axe neceſſariò concurret.</s>
            <s xml:id="echoid-s22464" xml:space="preserve"> Et hoc proponebatur.</s>
            <s xml:id="echoid-s22465" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div813" type="section" level="0" n="0">
          <head xml:id="echoid-head671" xml:space="preserve" style="it">97. Omnis ſuperficies plana ſuperficiei contingenti pyramidem uel columnam in loco con-
            <lb/>
          tact{us} orthogonaliter inſiſtens, neceſſariò ſecat pyramidem uel columnam per ipſi{us} axem.</head>
          <p>
            <s xml:id="echoid-s22466" xml:space="preserve">Sit pyramis uel columna rotunda, quam contingat ſuperficies plana.</s>
            <s xml:id="echoid-s22467" xml:space="preserve"> Palàm ergo per 95 huius,
              <lb/>
            quoniã continget illam ſecundũ lineã longitudinis.</s>
            <s xml:id="echoid-s22468" xml:space="preserve"> Superficies itaq;</s>
            <s xml:id="echoid-s22469" xml:space="preserve"> huic ſuperficiei orthogonali-
              <lb/>
            ter in loco contactus inſiſtẽs, eſt perpendicularis ſuper ſuperficiẽ curuam pyramidis uel columnę:</s>
            <s xml:id="echoid-s22470" xml:space="preserve">
              <lb/>
            & ipſarũ cõmunis ſectio eſt linea longitudinis, ſuper quã in ſuperficie erecta ducantur perpendicu-
              <lb/>
            lares.</s>
            <s xml:id="echoid-s22471" xml:space="preserve"> Eæ itaq;</s>
            <s xml:id="echoid-s22472" xml:space="preserve"> lineæ per præmiſſam tranſibunt axem pyramidis uel columnæ rotundæ.</s>
            <s xml:id="echoid-s22473" xml:space="preserve"> Er go & ſu-
              <lb/>
            perficies illa axem tranſiens, ſecabit pyramidẽ uel columnã ſecundum axem.</s>
            <s xml:id="echoid-s22474" xml:space="preserve"> Et hoc proponebatur.</s>
            <s xml:id="echoid-s22475" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div814" type="section" level="0" n="0">
          <head xml:id="echoid-head672" xml:space="preserve" style="it">98. Omnis ſuperficiei planæ ſecantis pyramidem rotundam non per uerticẽ, & ſuperficiei co-
            <lb/>
          nicæ pyramidis communem ſectionem figuram triangularem eſſe impoßibile.</head>
          <p>
            <s xml:id="echoid-s22476" xml:space="preserve">Eſto pyramis, cuius uertex a, diameter baſis b c, centrũ baſis d, & axis a d, quã ſecundum axis lon
              <lb/>
            gitudinem ſecet ſuperficies plana ſecundum trigonũ a b c per 90 huius:</s>
            <s xml:id="echoid-s22477" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s22478" xml:space="preserve"> ipſam alia ſuperfi-
              <lb/>
            cies erecta ſuper trigonũ a
              <lb/>
              <figure xlink:label="fig-0339-01" xlink:href="fig-0339-01a" number="353">
                <variables xml:id="echoid-variables337" xml:space="preserve">d f f f g g b h h d c h e e c</variables>
              </figure>
            b c, nõ per uerticem, ſecun-
              <lb/>
            dum ſectionẽ, quæ ſit e f g,
              <lb/>
            cuius ſupremus pũctus ſit
              <lb/>
            f, & ſit linea e g æquidiſtãs
              <lb/>
            alicui diametro baſis pyra-
              <lb/>
            midis, cuius medius pun-
              <lb/>
            ctus ſit h:</s>
            <s xml:id="echoid-s22479" xml:space="preserve"> & ducatur linea f
              <lb/>
            h à ſupremo puncto ſectio-
              <lb/>
            nis ad mediũ ſuæ baſis.</s>
            <s xml:id="echoid-s22480" xml:space="preserve"> Et
              <lb/>
            quia linea e g eſt linea re-
              <lb/>
            cta, quę eſt æquidiſtãs dia-
              <lb/>
            metro baſis pyramidis, &
              <lb/>
            punctũ f ſignatum eſt in ſu-
              <lb/>
            perficie conica in ſupremo,
              <lb/>
            ſuperficies e f g ſecat coni-
              <lb/>
            cam ſuperficiẽ.</s>
            <s xml:id="echoid-s22481" xml:space="preserve"> Si itaq;</s>
            <s xml:id="echoid-s22482" xml:space="preserve"> ſe-
              <lb/>
            ctio e f g ſit trigonũ ſcilicet
              <lb/>
            rectilineum:</s>
            <s xml:id="echoid-s22483" xml:space="preserve"> patet, quoniã duæ lineæ longitudinis pyramidis, quæ ſunt e f & g f, concurrunt in pun-
              <lb/>
            cto f, præter uerticem pyramidis, quod eſt impoſsibile & cõtra 91 huius.</s>
            <s xml:id="echoid-s22484" xml:space="preserve"> Trigonũ quoq;</s>
            <s xml:id="echoid-s22485" xml:space="preserve"> curuilineũ
              <lb/>
            fieri eſt impoſsibile:</s>
            <s xml:id="echoid-s22486" xml:space="preserve"> quoniã ſuperficies ſecans ſupponitur eſſe plana, & ſuperficies illius trigoni eſt
              <lb/>
            curua, ut patet ex definitione.</s>
            <s xml:id="echoid-s22487" xml:space="preserve"> Erit ergo linea e f g linea una.</s>
            <s xml:id="echoid-s22488" xml:space="preserve"> Cum itaq;</s>
            <s xml:id="echoid-s22489" xml:space="preserve"> illa ſectio ſit linea una:</s>
            <s xml:id="echoid-s22490" xml:space="preserve"> dica-
              <lb/>
            tur ſectio conica uel pyramidalis.</s>
            <s xml:id="echoid-s22491" xml:space="preserve"> Si itaq́ axis pyramidis, qui eſt a d, ſit æqualis ſemidiametro baſis,
              <lb/>
            quæ eſt d b:</s>
            <s xml:id="echoid-s22492" xml:space="preserve"> palàm, quia pyramis a b c eſt orthogonia, quoniam angulus b a c trigoni a b c eſt rectus.</s>
            <s xml:id="echoid-s22493" xml:space="preserve">
              <lb/>
            Si ergo linea f h, quæ eſt communis ſectio ſuperficiei e f g, & trigoni a b c æquidiſtet lineæ a c, quæ
              <lb/>
            eſt latus trigoni, & linea longitudinis pyramidis:</s>
            <s xml:id="echoid-s22494" xml:space="preserve"> palàm per 29 p 1, cum angulus b a c ſit rectus, quòd
              <lb/>
            etiam angulus b f h erit rectus:</s>
            <s xml:id="echoid-s22495" xml:space="preserve"> & ſimiliter angulus h f a:</s>
            <s xml:id="echoid-s22496" xml:space="preserve"> tunc itaq;</s>
            <s xml:id="echoid-s22497" xml:space="preserve"> ſectio e f g dicetur ſectio rectangu
              <lb/>
            la, uel parabola:</s>
            <s xml:id="echoid-s22498" xml:space="preserve"> & eſt illa, quam Arabes dicunt mukefi.</s>
            <s xml:id="echoid-s22499" xml:space="preserve"> Si uerò lineæ h f & a c non æquidiſtent, ſed
              <lb/>
            concurrant:</s>
            <s xml:id="echoid-s22500" xml:space="preserve"> ſi concurſus fiat ad partem puncti a, qui eſt uertex pyramidis:</s>
            <s xml:id="echoid-s22501" xml:space="preserve"> tunc patet per 14 huius,
              <lb/>
            quòd angulus h f a erit obtuſus:</s>
            <s xml:id="echoid-s22502" xml:space="preserve"> & tunc ſectio e f g dicetur amblygonia uel hyperbole uel mukefi
              <lb/>
            addita.</s>
            <s xml:id="echoid-s22503" xml:space="preserve"> Si uerò lineæ h f & a c concurrant uerſus punctum c, qui non eſt uertex pyramidis:</s>
            <s xml:id="echoid-s22504" xml:space="preserve"> tunc per
              <lb/>
            14 huius erit angulus h f a acutus:</s>
            <s xml:id="echoid-s22505" xml:space="preserve"> & tunc ſectio e f g dicetur oxygonia, uel ellipſis uel mukefi dimi-
              <lb/>
            nuta.</s>
            <s xml:id="echoid-s22506" xml:space="preserve"> Et ſecundum hunc modum iſtæ ſectiones & earum paſsiones ampliſsimè uariantur.</s>
            <s xml:id="echoid-s22507" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div816" type="section" level="0" n="0">
          <head xml:id="echoid-head673" xml:space="preserve" style="it">99. Omnis ſuperficiei planæ ſecantis pyramidem uel columnam lateratã trans axem æquidi-
            <lb/>
          stanter baſi & ſuperficiei pyramidalis uel columnaris cõmunis ſectio eſt ſimilis peripheriæ baſis:
            <lb/>
          & ſi illa ſectio peripheriæ baſis eſt ſimilis, ſuperficies ſecans æquidistat baſi pyramidis uel colũnæ.</head>
          <p>
            <s xml:id="echoid-s22508" xml:space="preserve">Si enim illa ſectio baſi æquidiſtat, omnes trigoni laterales totius pyramidis & partiales trigoni
              <lb/>
            ſunt æquianguli per 29 p 1.</s>
            <s xml:id="echoid-s22509" xml:space="preserve"> Patet ergo per 4 p 6, quòd tota peripheria ſectionis eſt ſimilis baſi pyra-
              <lb/>
            midis, quoniam omnia latera trigonorum totalium & partialium erunt proportionalia.</s>
            <s xml:id="echoid-s22510" xml:space="preserve"> Et ſi illa
              <lb/>
            ſectio eſt baſi ſimilis, eſt etiam baſi æquidiſtans.</s>
            <s xml:id="echoid-s22511" xml:space="preserve"> Quoniam ſi nõ eſt æquidiſtans, erit alia ſecundum
              <lb/>
            idem punctum ſecans axem, æquidiſtans baſi, ſimilis peripheriæ baſis per præmiſſa, Sequitur itaq;</s>
            <s xml:id="echoid-s22512" xml:space="preserve">
              <lb/>
            ut una ſimilis, alia quoq;</s>
            <s xml:id="echoid-s22513" xml:space="preserve"> non ſimilis, ſecundum idem punctum ſecent axem pyramidis.</s>
            <s xml:id="echoid-s22514" xml:space="preserve"> Alia uerò
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
Impressum