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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[331] a e g b d c f
Page: 329
[332] g f h k b l a c e m d n
Page: 329
[333] g b c a f d e
Page: 329
[334] g f c b d a
Page: 329
[335] e a d b c
Page: 330
[336] b z g a e d
Page: 330
[337] b f c a d g e
Page: 330
[338] d f b c e d
Page: 331
[339] c a d b
Page: 331
[340] g h e b f d a
Page: 332
[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
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          <p>
            <s xml:id="echoid-s22294" xml:space="preserve">
              <pb o="34" file="0336" n="336" rhead="VITELLONIS OPTICAE"/>
            tio, ſicut angulorum contentorum ſub lineis à centro d ad ipſorum terminos productis ad 4 rectos
              <lb/>
            ſuperficiales per 33 p 6.</s>
            <s xml:id="echoid-s22295" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22296" xml:space="preserve"> Et etiam poteſt patere ex hoc, quoniam ſicut ille an-
              <lb/>
            gulus correſpõdet illi parti ſuperficiei ſphæricæ:</s>
            <s xml:id="echoid-s22297" xml:space="preserve"> ſic reſiduum 8 ſolidorum angulorũ rectorũ totali
              <lb/>
            reſiduo ſuperficiei illius ſphæræ reſpondet:</s>
            <s xml:id="echoid-s22298" xml:space="preserve"> ergo per 16 p 5 erit permutatim anguli ad angulum, ſi-
              <lb/>
            cut ſuperficiei ad ſuperficiem, & per 18 p 5 coniunctim, & per 5 huius è contrario patet propoſitũ.</s>
            <s xml:id="echoid-s22299" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div798" type="section" level="0" n="0">
          <head xml:id="echoid-head662" xml:space="preserve" style="it">88. Si inter duas quartas circulorũ æqualium in ſphæræ ſuperficie ſe ſecantium, ad extremi-
            <lb/>
          tates arcuum æqualium lineæ rectæ ducantur: illæ erũt æquidiſtantes: & remotior à puncto ſe-
            <lb/>
          ctionis erit longior. È
            <unsure/>
          14 p 12 ele. in Campano.</head>
          <p>
            <s xml:id="echoid-s22300" xml:space="preserve">Sint arcus magnorum circulorũ in ſuperficie alicuius ſphæræ ſe ſecantiũ, qui a b c & a d e, ſecan-
              <lb/>
            tes ſe in puncto a:</s>
            <s xml:id="echoid-s22301" xml:space="preserve"> in quibus ſignentur arcus æquales, ita, ut arcus a b ſit æqualis arcui a d, & arcus b
              <lb/>
            c ſit æqualis arcui d e, & cõtinuentur lineæ rectę, quę
              <lb/>
              <figure xlink:label="fig-0336-01" xlink:href="fig-0336-01a" number="347">
                <variables xml:id="echoid-variables331" xml:space="preserve">a b d c e</variables>
              </figure>
            ſint b d & c e.</s>
            <s xml:id="echoid-s22302" xml:space="preserve"> Dico, quòd lineæ c e & b d ſunt æquidi-
              <lb/>
            ſtantes:</s>
            <s xml:id="echoid-s22303" xml:space="preserve"> & quòd linea c e eſt maior ꝗ̃ linea b d.</s>
            <s xml:id="echoid-s22304" xml:space="preserve"> Quia
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s22305" xml:space="preserve"> arcus a b eſt æqualis arcui a d:</s>
            <s xml:id="echoid-s22306" xml:space="preserve"> palàm ք 29 p 3 &
              <lb/>
            per 65 huius, quoniã punctus a eſt polus circuli trãſ-
              <lb/>
            euntis per pũcta d & b:</s>
            <s xml:id="echoid-s22307" xml:space="preserve"> ideo quòd rectę lineę, quę a d
              <lb/>
            & a b, ſunt æquales:</s>
            <s xml:id="echoid-s22308" xml:space="preserve"> & ſimiliter eſt de circulo trãſeũte
              <lb/>
            per pũcta c & e.</s>
            <s xml:id="echoid-s22309" xml:space="preserve"> Circũducatur ergo ſuperficiei ſphęrę
              <lb/>
            per puncta d, b circulus erectus ſuper diametrũ ſphæ
              <lb/>
            rę per 69 huius, & ſimiliter per puncta e & c.</s>
            <s xml:id="echoid-s22310" xml:space="preserve"> Erũt er-
              <lb/>
            go illi circuli æquidiſtãtes per 14 p 11.</s>
            <s xml:id="echoid-s22311" xml:space="preserve"> Erunt ergo li-
              <lb/>
            neæ c e & b d æquidiſtantes per 16 p 11, imaginata ſu-
              <lb/>
            perficie plana, in qua ſunt puncta b, c, d, e, circulos ſe-
              <lb/>
            cundum illas lineas ſecãte.</s>
            <s xml:id="echoid-s22312" xml:space="preserve"> Sed & linea c e eſt maior
              <lb/>
            quàm linea d b.</s>
            <s xml:id="echoid-s22313" xml:space="preserve"> Si enim ſit æqualis, cũ ſit æquidiſtãs:</s>
            <s xml:id="echoid-s22314" xml:space="preserve">
              <lb/>
            palàm, quia circuli a b c & a e d æquidiſtantes erunt:</s>
            <s xml:id="echoid-s22315" xml:space="preserve">
              <lb/>
            quod eſt cõtra hypotheſim:</s>
            <s xml:id="echoid-s22316" xml:space="preserve"> ſupponũtur enim ſe ſeca
              <lb/>
            re in puncto a:</s>
            <s xml:id="echoid-s22317" xml:space="preserve"> aut ſequetur circulum tranſeuntẽ per
              <lb/>
            puncta b & d æqualem fieri circulo tranſeunti per puncta c & d, quorum circulorum polus eſt pun
              <lb/>
            ctum a:</s>
            <s xml:id="echoid-s22318" xml:space="preserve"> quod iterum eſt impoſsibile.</s>
            <s xml:id="echoid-s22319" xml:space="preserve"> Et ſi linea c e ſit minor quàm linea b d, concurrent circuli a b c
              <lb/>
            & a d e ultra lineam c e potius quàm ultra lineam b d.</s>
            <s xml:id="echoid-s22320" xml:space="preserve"> Eſt ergo linea b d remotior à puncto ſectiõis.</s>
            <s xml:id="echoid-s22321" xml:space="preserve">
              <lb/>
            Quod eſt propoſitum hypotheſis.</s>
            <s xml:id="echoid-s22322" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div800" type="section" level="0" n="0">
          <head xml:id="echoid-head663" xml:space="preserve" style="it">89. Omnes lineæ longitudinis unius pyramidis rotundæ, ſunt æquales: & cum ſemidiametris
            <lb/>
          baſis æquales, ſed acutos angulos continentes. Ex quo patet omnem pũctum uerticis pyramidis
            <lb/>
          eſſe polum circuli ſuæ b a ſis: omnem́ lineam longitudinis eſſe in eadẽ ſuperficie cum axe: ipſum
            <lb/>
          quo axem centrum circuli baſis orthogonaliter attingere. È
            <unsure/>
          18 defin. 11 element.</head>
          <p>
            <s xml:id="echoid-s22323" xml:space="preserve">Quoniã enim per principium 11 Euclidis pyramis rotunda fit per trãſitum trianguli rectanguli,
              <lb/>
            alterutro ſuorum laterum rectum angulum continentiũ fixo, donec
              <lb/>
              <figure xlink:label="fig-0336-02" xlink:href="fig-0336-02a" number="348">
                <variables xml:id="echoid-variables332" xml:space="preserve">a d b c</variables>
              </figure>
            ad locum ſuum, unde in cœpit, redeat, triangulo ipſo circumducto:</s>
            <s xml:id="echoid-s22324" xml:space="preserve">
              <lb/>
            qui triangulus, ſi fuerit duorum laterum æqualium:</s>
            <s xml:id="echoid-s22325" xml:space="preserve"> & unum laterũ
              <lb/>
            æqualium rectum angulum continentium fuerit fixum, cauſſabitur
              <lb/>
            pyramis rectãgula:</s>
            <s xml:id="echoid-s22326" xml:space="preserve"> ideo, quòd angulus duplicati ſui trianguli ad uer
              <lb/>
            ticem pyramidis eſt rectus per 5 & 32 p 1.</s>
            <s xml:id="echoid-s22327" xml:space="preserve"> Et ſi fixũ latus fuerit minus
              <lb/>
            latere moto, erit pyramis amblygonia:</s>
            <s xml:id="echoid-s22328" xml:space="preserve"> quoniã per 19 p 1 angulus ad
              <lb/>
            uerticem fit obtuſus.</s>
            <s xml:id="echoid-s22329" xml:space="preserve"> Et ſi latus fixum fuerit maius latere moto, erit
              <lb/>
            pyramis oxygonia:</s>
            <s xml:id="echoid-s22330" xml:space="preserve"> quia per eandem 19 p 1 angulus eius ad uerticem
              <lb/>
            remanet acutus, adiuuãte ſemper 32 p 1.</s>
            <s xml:id="echoid-s22331" xml:space="preserve"> Sic ergo diuerſantur formæ
              <lb/>
            pyramidum ſecundum diuerſitatem proportionis lateris fixi ad alte
              <lb/>
            rum latus motum rectum angulum cõtinens cum fixo.</s>
            <s xml:id="echoid-s22332" xml:space="preserve"> Et quia latus
              <lb/>
            ſubtenſum angulo recto, cauſſat omnes lineas longitudinis in quali
              <lb/>
            bet pyramide:</s>
            <s xml:id="echoid-s22333" xml:space="preserve"> palàm, quòd omnes lineæ longitudinis totius rotun-
              <lb/>
            dæ pyramidis uni lineæ ſunt æquales ei, ſcilicet q̃ in trigono rectan-
              <lb/>
            gulo opponitur angulo recto.</s>
            <s xml:id="echoid-s22334" xml:space="preserve"> Ergo & oẽs inter ſe ſunt æquales.</s>
            <s xml:id="echoid-s22335" xml:space="preserve"> Si
              <lb/>
            ergo trigonũ orthogoniũ cauſſans pyramidẽ, ſit a b c, cuius angulus
              <lb/>
            a b c ſit rectus:</s>
            <s xml:id="echoid-s22336" xml:space="preserve"> erit per 32 p 1 angulus a c b acutus:</s>
            <s xml:id="echoid-s22337" xml:space="preserve"> & eſt a c b angulus,
              <lb/>
            cui omnes anguli cõtenti à lineis lõgitudinis & ſemidiametris baſis,
              <lb/>
            ſunt æquales:</s>
            <s xml:id="echoid-s22338" xml:space="preserve"> & hoc proponebatur.</s>
            <s xml:id="echoid-s22339" xml:space="preserve"> Patet etiã ex ijs, quoniã punctus
              <lb/>
            uerticis pyramidis cuiuslibet eſt polus circuli ſuę baſis per 65 huius.</s>
            <s xml:id="echoid-s22340" xml:space="preserve"> Et quoniã linea a c eſt in eadẽ
              <lb/>
            ſuperficie trigona cum linea a b, patet, quoniam omnes lineæ longitudinis ſuntin eadem ſuperficie
              <lb/>
            cum axe a b.</s>
            <s xml:id="echoid-s22341" xml:space="preserve"> Et quoniam linea b c motu ſuo deſcribit circulum baſis, patet, quòd axis a b centrum
              <lb/>
            circuli baſis orthogonaliter attingit per 8 p 1:</s>
            <s xml:id="echoid-s22342" xml:space="preserve"> quia ex circuli definitione & prima parte præſen-
              <lb/>
            </s>
          </p>
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