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

Table of figures

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[341] a c e h d b
Page: 332
[342] l h g b e c k a d f
Page: 334
[343] f c c l a
Page: 334
[344] b c a
Page: 335
[345] c f b e d a
Page: 335
[346] a b d c
Page: 335
[347] a b d c e
Page: 336
[348] a d b c
Page: 336
[349] a b d c
Page: 337
[350] d e a b c
Page: 337
[351] g a m e n b h i c p f o d k l
Page: 338
[352] a e d c g b
Page: 338
[353] d f f f g g b h h d c h e e c
Page: 339
[354] a e h f g b d c
Page: 340
[355] a f e g h b d c
Page: 341
[356] a k f l e m h g b d c
Page: 341
[357] a l f e h k g b d c
Page: 342
[358] d a b c
Page: 342
[359] a x e i b g d h c k f o l n m p
Page: 342
[360] a g g e b d c f
Page: 343
[361] a k b d c
Page: 343
[362] f e h g
Page: 343
[363] a m k n b d c
Page: 344
[364] f o l p p h g
Page: 344
[365] a e t g o f z h d c p y k b r q
Page: 345
[366] n q e t o l g f m d K d h c a s u p z b
Page: 346
[367] e b h a f c l m k d g
Page: 347
[368] b d a c e f g
Page: 348
[369] a b c d e f
Page: 348
[370] a h b z d g
Page: 348
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        <div xml:id="echoid-div820" type="section" level="0" n="0">
          <pb o="39" file="0341" n="341" rhead="LIBER PRIMVS."/>
          <p>
            <s xml:id="echoid-s22599" xml:space="preserve">Eſto pyramis, cuius uertex punctũ a, axis uerò a d:</s>
            <s xml:id="echoid-s22600" xml:space="preserve"> in quo ſit datus punctus e, à quo debemus cir
              <lb/>
            culum totali ſuperficiei conicæ circunducere.</s>
            <s xml:id="echoid-s22601" xml:space="preserve"> Sit itaq;</s>
            <s xml:id="echoid-s22602" xml:space="preserve">, ut ſuperficies plana ſecet pyramidẽ ſecundũ
              <lb/>
            axem a d trans punctũ e:</s>
            <s xml:id="echoid-s22603" xml:space="preserve"> cõmunis itaq;</s>
            <s xml:id="echoid-s22604" xml:space="preserve"> ſectio illius ſuperficiei planæ & ſuperficiei conicæ erit trigo
              <lb/>
            num per 90 huius:</s>
            <s xml:id="echoid-s22605" xml:space="preserve"> cuius baſis ſit b c, quę erit diameter baſis pyrami-
              <lb/>
              <figure xlink:label="fig-0341-01" xlink:href="fig-0341-01a" number="355">
                <variables xml:id="echoid-variables339" xml:space="preserve">a f e g h b d c</variables>
              </figure>
            dis.</s>
            <s xml:id="echoid-s22606" xml:space="preserve"> In hac itaq;</s>
            <s xml:id="echoid-s22607" xml:space="preserve"> ſuperficie per 11 p 1 ducatur à puncto e linea perpendi
              <lb/>
            culariter ſuper axem a d, quæ producta ad conicã ſuperficiem ſit e f:</s>
            <s xml:id="echoid-s22608" xml:space="preserve">
              <lb/>
            & item ab eodẽ puncto e ducatur linea e g perpendiculariter ſuper
              <lb/>
            axẽ a d:</s>
            <s xml:id="echoid-s22609" xml:space="preserve"> cadatq́;</s>
            <s xml:id="echoid-s22610" xml:space="preserve"> punctũ g in conica pyramidis ſuperficie:</s>
            <s xml:id="echoid-s22611" xml:space="preserve"> & ſimiliter
              <lb/>
            ducatur linea e h perpendiculariter ſuper axem a d:</s>
            <s xml:id="echoid-s22612" xml:space="preserve"> cadatq́;</s>
            <s xml:id="echoid-s22613" xml:space="preserve"> punctus
              <lb/>
            h in conica ſuperficie.</s>
            <s xml:id="echoid-s22614" xml:space="preserve"> Quia ergo linea a e ſuper cõmunem terminum
              <lb/>
            linearũ e f, e g, e h orthogonaliter inſiſtit, palàm per 5 p 11, quoniã illæ
              <lb/>
            lineę ſunt in una ſuperficie:</s>
            <s xml:id="echoid-s22615" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s22616" xml:space="preserve"> per 4 p 11 linea a e perpẽdiculariter
              <lb/>
            erecta ſuper illã ſuperficiẽ f g h.</s>
            <s xml:id="echoid-s22617" xml:space="preserve"> Et quoniã linea a d erecta eſt perpen-
              <lb/>
            diculariter ſuper baſim pyramidis per 89 huius, & per definitionẽ p y
              <lb/>
            ramidis:</s>
            <s xml:id="echoid-s22618" xml:space="preserve"> patet per 14 p 11, quoniã ſuperficies f g h æquidiſtat baſi pyra
              <lb/>
            midis.</s>
            <s xml:id="echoid-s22619" xml:space="preserve"> Eſt ergo per 100 huius f g h circulus.</s>
            <s xml:id="echoid-s22620" xml:space="preserve"> Quòd ſi pũctus datus ſit
              <lb/>
            in ſuperficie conica, ſit ille punctus f:</s>
            <s xml:id="echoid-s22621" xml:space="preserve"> & ducatur à puncto f perpendi-
              <lb/>
            cularis ſuper axem a d, quę ſit f e, per 12 p 1:</s>
            <s xml:id="echoid-s22622" xml:space="preserve"> educanturq́;</s>
            <s xml:id="echoid-s22623" xml:space="preserve"> à puncto e li-
              <lb/>
            neæ e g & e h perpendiculares ſuper axem a d per 11 p 1:</s>
            <s xml:id="echoid-s22624" xml:space="preserve"> & deinde, ut
              <lb/>
            prius, compleatur demonſtratio.</s>
            <s xml:id="echoid-s22625" xml:space="preserve"> Patet itaq;</s>
            <s xml:id="echoid-s22626" xml:space="preserve"> propoſitum:</s>
            <s xml:id="echoid-s22627" xml:space="preserve"> quoniã ſim
              <lb/>
            pliciter eodem modo negotiandum eſt in columnis.</s>
            <s xml:id="echoid-s22628" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div822" type="section" level="0" n="0">
          <head xml:id="echoid-head677" xml:space="preserve" style="it">103. Omnis ſuperficiei ſecantis pyramidem uel columnã rotun-
            <lb/>
          dam trans axem non æquidiſtanter baſibus, & ſuperficiei curuæ
            <lb/>
          communem ſectionem circulum eſſe eſt impoßibile. 5 theo. 1 Conicorum Apollonij. item 9 theor.
            <lb/>
          Cylindricorum Sereni.</head>
          <p>
            <s xml:id="echoid-s22629" xml:space="preserve">Sit pyramis, cuius uertex a, diameter baſis b c:</s>
            <s xml:id="echoid-s22630" xml:space="preserve"> & centrum baſis d, & axis a d:</s>
            <s xml:id="echoid-s22631" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s22632" xml:space="preserve"> ipſam ſuper-
              <lb/>
            ficies plana trans axem a d in puncto e, nõ æquidiſtanter baſi:</s>
            <s xml:id="echoid-s22633" xml:space="preserve"> & ſit cõmunis ſectio huius ſuperficiei
              <lb/>
            planæ & ſuperficiei conicæ linea f g h k.</s>
            <s xml:id="echoid-s22634" xml:space="preserve"> Dico quòd hæc ſectio non eſt poſsibile, ut ſit circulus.</s>
            <s xml:id="echoid-s22635" xml:space="preserve"> Eſto
              <lb/>
            enim, ut circa punctum e in pyramidis conica ſuperficie ducatur circulus per præmiſſam:</s>
            <s xml:id="echoid-s22636" xml:space="preserve"> hic itaq;</s>
            <s xml:id="echoid-s22637" xml:space="preserve">
              <lb/>
            æquidiſtabit baſi per 100 huius:</s>
            <s xml:id="echoid-s22638" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s22639" xml:space="preserve"> f g l m:</s>
            <s xml:id="echoid-s22640" xml:space="preserve"> & ſignentur lineę longi-
              <lb/>
              <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a" number="356">
                <variables xml:id="echoid-variables340" xml:space="preserve">a k f l e m h g b d c</variables>
              </figure>
            tudinis pyramidis a f, a g, a l, a m.</s>
            <s xml:id="echoid-s22641" xml:space="preserve"> Eæ itaq;</s>
            <s xml:id="echoid-s22642" xml:space="preserve"> omnes erunt æquales per
              <lb/>
            89 huius, ideo quòd ſuperficies æquidiſtans baſi pyramidis nouã py
              <lb/>
            ramidem abſcindit per 100 huius.</s>
            <s xml:id="echoid-s22643" xml:space="preserve"> Et quoniã ſectio f g h k nõ æquidi-
              <lb/>
            ſtat baſi pyramidis, patet quòd non æqualiter diſtat à uertice pyrami
              <lb/>
            dis, qui eſt punctus a:</s>
            <s xml:id="echoid-s22644" xml:space="preserve"> ſit itaq;</s>
            <s xml:id="echoid-s22645" xml:space="preserve"> punctus h remotior à uertice a, & cadat
              <lb/>
            in linea a l producta, & punctus k ſit propinquior uertici a, & cadat
              <lb/>
            in linea a m.</s>
            <s xml:id="echoid-s22646" xml:space="preserve"> Erit itaq;</s>
            <s xml:id="echoid-s22647" xml:space="preserve"> linea a h maior quàm linea a l, & linea a k mi-
              <lb/>
            nor eſt quàm linea a m:</s>
            <s xml:id="echoid-s22648" xml:space="preserve"> & continuentur lineę h e, k e, f e, g e, & lineæ
              <lb/>
            e l, e m.</s>
            <s xml:id="echoid-s22649" xml:space="preserve"> Et quoniã angulus a l e eſt acutus per 89 huius, erit angulus
              <lb/>
            h l e obtuſus per 13 p 1.</s>
            <s xml:id="echoid-s22650" xml:space="preserve"> Ergo per 19 p 1 latus h e trigoni h e l eſt maius
              <lb/>
            latere e l:</s>
            <s xml:id="echoid-s22651" xml:space="preserve"> ſed latus e l eſt æquale lateri e f per definitionẽ circuli.</s>
            <s xml:id="echoid-s22652" xml:space="preserve"> Li-
              <lb/>
            nea uerò e f uenit à puncto axis ad punctũ ſectionis:</s>
            <s xml:id="echoid-s22653" xml:space="preserve"> quia eſt cõmu-
              <lb/>
            nis ſectio circuli & ſuperficiei obliquè pyramidem ſecantis:</s>
            <s xml:id="echoid-s22654" xml:space="preserve"> inæ qua-
              <lb/>
            les itaq;</s>
            <s xml:id="echoid-s22655" xml:space="preserve"> lineę ab hoc puncto e producuntur ad peripheriã ſectionis.</s>
            <s xml:id="echoid-s22656" xml:space="preserve">
              <lb/>
            Non eſt ergo ſectio illa circulus per circuli definitionẽ.</s>
            <s xml:id="echoid-s22657" xml:space="preserve"> Dicemus er-
              <lb/>
            go illam ſectionẽ in pyramidibus pyramidalem, & in columnis colu
              <lb/>
            mnalem.</s>
            <s xml:id="echoid-s22658" xml:space="preserve"> Eſt tamẽ illa ſectio in pyramidibus in 98 huius prius dicta
              <lb/>
            ſectio oxygonia uel ellipſis.</s>
            <s xml:id="echoid-s22659" xml:space="preserve"> Et quoniam talis ſectio eſt figuræ oblon
              <lb/>
            gæ, patet quòd ipſa habet diametros plurimas omnes inæquales, &
              <lb/>
            per idem punctum axis ſecti corporis tranſeuntes, ipſam quoq;</s>
            <s xml:id="echoid-s22660" xml:space="preserve"> ſectionem per æqualia diuidentes:</s>
            <s xml:id="echoid-s22661" xml:space="preserve">
              <lb/>
            quarum maxima eſt, quæ tranſit longitudinem ſectionis, minima uerò eſt, quæ pertranſit latitudi-
              <lb/>
            nem:</s>
            <s xml:id="echoid-s22662" xml:space="preserve"> & eſt ſuper maximam diametrum orthogonaliter erecta.</s>
            <s xml:id="echoid-s22663" xml:space="preserve"> Patet itaq;</s>
            <s xml:id="echoid-s22664" xml:space="preserve"> propoſitum.</s>
            <s xml:id="echoid-s22665" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div824" type="section" level="0" n="0">
          <head xml:id="echoid-head678" xml:space="preserve" style="it">104. Omnium duarum planarum ſuperficierũ ſecantium pyramidem uel columnam rotun-
            <lb/>
          dam trans idem punctum axis, ſi una æquidiſtanter baſi, & alia nõ æquidiſtanter ſecuerit: com
            <lb/>
          munis ſectio eſt linea recta tranſiens pyramidem uel columnam, orhogonalis ſuper axem. Ex
            <lb/>
          quo patet, quòd ſiue circulι peripheria, ſiue ſectio alia quæcun non in eadem ſuperficie, quam-
            <lb/>
          cun ſecuerit ſectionem, in duobus tantùm punctis ipſam interſecabit.</head>
          <p>
            <s xml:id="echoid-s22666" xml:space="preserve">Sit, ut pyramis, cuius uertex a:</s>
            <s xml:id="echoid-s22667" xml:space="preserve"> & axis a d ſecetur ſecundum punctum axis e, per
              <gap/>
            duas planas ſu-
              <lb/>
            perficies, quarum una ſecet æquidiſtanter baſi, ut f g h, alia uerò non æquidiſtanter, ut f g k l.</s>
            <s xml:id="echoid-s22668" xml:space="preserve"> Di-
              <lb/>
            co, quòd communis ſectio iſtarum ſuperficierum eſt linea tranſiens pyramidem, orthogonalis ſu-
              <lb/>
            per axem, ut eſt linea f e g.</s>
            <s xml:id="echoid-s22669" xml:space="preserve"> Quòd enim illæ ſuperficies ſe interſecent, patet per hoc, quòd aliquæ li-
              <lb/>
            </s>
          </p>
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