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

Page concordance

< >
Scan Original
231 225
232 226
233 227
234 228
235 229
236 230
237 231
238 232
239 233
240 234
241 235
242 236
243 237
244 238
245 239
246 240
247 241
248 242
249 243
250 244
251 245
252 246
253 247
254 248
255 249
256 250
257 251
258 252
259 253
260 254
< >
page |< < (34) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div797" type="section" level="0" n="0">
          <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>
        </div>
      </text>
    </echo>
Impressum