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

< >
291
291 (285) 		292
292 (286)
293
293 (287) 		294
294 (288)
295
295 		296
296
297
297 		298
298
299
299 		300
300
< >
page |< < (285) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div638" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s19781" xml:space="preserve">
              <pb o="285" file="0291" n="291" rhead="DE CREPVSCVLIS LIBER."/>
            maioris circuli ad b c ſemidiametrum minoris b c:</s>
            <s xml:id="echoid-s19782" xml:space="preserve"> ſic a h ad b h.</s>
            <s xml:id="echoid-s19783" xml:space="preserve">] Et protraham à puncto h lineam
              <lb/>
            contingẽtem circulũ a g d [per 17 p 3] quæ ſit h c d.</s>
            <s xml:id="echoid-s19784" xml:space="preserve"> Dico ergo, quòd ipſa contingit etiã circulũ b e z:</s>
            <s xml:id="echoid-s19785" xml:space="preserve">
              <lb/>
            quod patet:</s>
            <s xml:id="echoid-s19786" xml:space="preserve"> quia cõtinuabo a cum d per lineam a d:</s>
            <s xml:id="echoid-s19787" xml:space="preserve"> ergo eſt perpendi-
              <lb/>
              <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a" number="250">
                <variables xml:id="echoid-variables237" xml:space="preserve">d a k g e c b z h</variables>
              </figure>
            cularis ſuper lineam h d [per 18 p 3] & protraham à puncto b perpen-
              <lb/>
            dicularem ſuper lineam h c d [per 11 p 1] quæ ſit b c.</s>
            <s xml:id="echoid-s19788" xml:space="preserve"> Et quoniam duæ
              <lb/>
            lineæ b c, a d ſunt perpendiculares ſuper lineam h d [è fabricatione &
              <lb/>
            concluſo] ſunt æ quidiſtantes [per 28 p 1.</s>
            <s xml:id="echoid-s19789" xml:space="preserve">] Et quia linea b c eſt æqui-
              <lb/>
            diſtans ipſi a d, quæ eſt baſis trianguli:</s>
            <s xml:id="echoid-s19790" xml:space="preserve"> erit ergo proportio a d ad b c,
              <lb/>
            ſicut ꝓportio a h ad h b [per 4 p 6:</s>
            <s xml:id="echoid-s19791" xml:space="preserve"> quia triangula a h d, b h c ſunt æqui-
              <lb/>
            angula per 29.</s>
            <s xml:id="echoid-s19792" xml:space="preserve"> 32 p 1] & iam poſuimus proportionem a h ad h b, ſicut
              <lb/>
            proportionem medietatis diametri circuli a g d, ad medietatẽ diame-
              <lb/>
            tri b e z:</s>
            <s xml:id="echoid-s19793" xml:space="preserve"> ergo linea b c eſt medietas diametri circuli b e z:</s>
            <s xml:id="echoid-s19794" xml:space="preserve"> ergo punctũ
              <lb/>
            c eſt ſuper circumferẽtiam circuli b e z [per 17 d 1] & duos angulos ad
              <lb/>
            d & c poſuimus rectos:</s>
            <s xml:id="echoid-s19795" xml:space="preserve"> ergo linea h c d contingit minorem circulum
              <lb/>
            [per conſectarium 16 p 3] nos uerò iam protraximus eam contingen-
              <lb/>
            tem maiorẽ:</s>
            <s xml:id="echoid-s19796" xml:space="preserve"> ergo ipſa eſt contingens utroſq;</s>
            <s xml:id="echoid-s19797" xml:space="preserve"> ſimul.</s>
            <s xml:id="echoid-s19798" xml:space="preserve"> Et protraham ſimi
              <lb/>
            liter ex puncto h lineam, contingentem duos circulos ſimiliter in par-
              <lb/>
            te z, quæ ſit linea h z k.</s>
            <s xml:id="echoid-s19799" xml:space="preserve"> Eſt ergo, quòd ex circulo a maiore uerſa facie
              <lb/>
            reſpicit circulum b minorem, portio d g k:</s>
            <s xml:id="echoid-s19800" xml:space="preserve"> & eſt minor medietate cir-
              <lb/>
            culi:</s>
            <s xml:id="echoid-s19801" xml:space="preserve"> quoniam angulus h a d eſt minor recto [per 32 p 1] quoniam ipſe
              <lb/>
            eſt in trian gulo uno, & eſt triangulum d a h cum angulo a d h recto.</s>
            <s xml:id="echoid-s19802" xml:space="preserve">
              <lb/>
            Ergo eſt portio d g minor quarta circuli [per 33 p 6] & ſimiliter por-
              <lb/>
            tio g k, æqualis e [quòd autem g k ſit æqualis d g, patet, ducta ſemidia-
              <lb/>
            metro a k.</s>
            <s xml:id="echoid-s19803" xml:space="preserve"> Quia enim rectæ d h, k h tangentes æquantur per conſecta-
              <lb/>
            rium 36 p 3 & ſemidiametri a d, a k per 15 d 1, eſtq́;</s>
            <s xml:id="echoid-s19804" xml:space="preserve"> communis a h:</s>
            <s xml:id="echoid-s19805" xml:space="preserve"> ęqua-
              <lb/>
            bitur angulus h a d angulo h a k per 8 p 1:</s>
            <s xml:id="echoid-s19806" xml:space="preserve"> quare per 26 p 3 peripheria d
              <lb/>
            g æquabitur peripheriæ g k.</s>
            <s xml:id="echoid-s19807" xml:space="preserve">] Ergo portio d g k eſt minor medietate
              <lb/>
            circuli.</s>
            <s xml:id="echoid-s19808" xml:space="preserve"> Et quoniam linea b c eſt æquidiſtans lineæ a d [è concluſo] eſt angulus c b h æqualis an-
              <lb/>
            gulo d a h [per 29 p 1] ergo erit portio c l ſimilis portioni d g, & tota portio c l z ſimilis portioni d g
              <lb/>
            k [per 33 p 6.</s>
            <s xml:id="echoid-s19809" xml:space="preserve">] Ergo unaquęq;</s>
            <s xml:id="echoid-s19810" xml:space="preserve"> earũ eſt minor medietate circuli:</s>
            <s xml:id="echoid-s19811" xml:space="preserve"> remanet ergo portio c e z maior me
              <lb/>
            dietate circuli:</s>
            <s xml:id="echoid-s19812" xml:space="preserve"> & illud eſt, quod ex circulo minore uerſa facie reſpicit circulum maiorem.</s>
            <s xml:id="echoid-s19813" xml:space="preserve"> Ergo duę
              <lb/>
            portiones c e z, & d g k ſunt ex duobus circulis, qui uerſa facie ſe reſpiciũt.</s>
            <s xml:id="echoid-s19814" xml:space="preserve"> Et ſignifico quidem per
              <lb/>
            hoc, quòd aliquid portionis unius nõ cooperitur ex circulo altero:</s>
            <s xml:id="echoid-s19815" xml:space="preserve"> & portio c e z eſt maior medie-
              <lb/>
            tate circuli, & portio d g k minor.</s>
            <s xml:id="echoid-s19816" xml:space="preserve"> Etillud eſt, quod uoluimus declarare.</s>
            <s xml:id="echoid-s19817" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div640" type="section" level="0" n="0">
          <figure number="251">
            <variables xml:id="echoid-variables238" xml:space="preserve">e d a n b g m q t k z h l</variables>
          </figure>
          <head xml:id="echoid-head550" xml:space="preserve" style="it">4. Si peripheri{as} duorum circulorum æqualium duæ rectæ lιneæ tangant: punct a ſemiperi-
            <lb/>
          pheriarum cõuexis partib{us} ſe reſpicientium ſingula ſingulis appa-
            <lb/>
          rent, reliquarum uerò ſemiperipheriarum conuexis partib{us} ſenon reſpicientium latent.</head>
          <p>
            <s xml:id="echoid-s19818" xml:space="preserve">ET dico, quòd quando ſunt duo circuli æquales, & protrahuntur
              <lb/>
            duæ lineæ, quarum unaquæq;</s>
            <s xml:id="echoid-s19819" xml:space="preserve"> contingit duos circulos ſimul, ſe-
              <lb/>
            cundum formam, quam præmiſimus:</s>
            <s xml:id="echoid-s19820" xml:space="preserve"> tunc in unaquaq;</s>
            <s xml:id="echoid-s19821" xml:space="preserve"> duarum
              <lb/>
            portionum, quarum una uerſa facie reſpicit alteram, non eſt locus, qui
              <lb/>
            abſcõdat aliquid ex circulo uno circulo alteri:</s>
            <s xml:id="echoid-s19822" xml:space="preserve"> & quòd in reliquis dua-
              <lb/>
            bus portionibus duorum circulorum, quę non facie ad faciem ſe reſpi-
              <lb/>
            ciunt, non eſt locus, qui appareat circulo alteri.</s>
            <s xml:id="echoid-s19823" xml:space="preserve"> Cuius exemplum eſt,
              <lb/>
            quòd ſint duo circuli a b g d e, & z h t k l:</s>
            <s xml:id="echoid-s19824" xml:space="preserve"> & protrahantur duę lineæ b h,
              <lb/>
            & d k contingentes duos circulos ſimul:</s>
            <s xml:id="echoid-s19825" xml:space="preserve"> ergo duæ portiones b g d, &
              <lb/>
            h t k ſunt, quæ ſe facie ad faciem reſpiciunt:</s>
            <s xml:id="echoid-s19826" xml:space="preserve"> earum portiones b e d, & h
              <lb/>
            l k ſunt, quæ ſe non facie ad faciem reſpiciunt.</s>
            <s xml:id="echoid-s19827" xml:space="preserve"> Dico ergo, quòd non eſt
              <lb/>
            in portione b g d punctum, quod aliquid ex circulo z h abſcondat cir-
              <lb/>
            culo a b:</s>
            <s xml:id="echoid-s19828" xml:space="preserve"> & quòd non eſt in portione b e d punctum, quod appareat pe-
              <lb/>
            nitus circulo z h:</s>
            <s xml:id="echoid-s19829" xml:space="preserve"> & quòd tota ipſa portio eſt abſcondita circulo z h:</s>
            <s xml:id="echoid-s19830" xml:space="preserve"> &
              <lb/>
            quòd neq;</s>
            <s xml:id="echoid-s19831" xml:space="preserve"> eſt in portione h l k punctũ, quod appareat circulo a b.</s>
            <s xml:id="echoid-s19832" xml:space="preserve"> Cu-
              <lb/>
            ius demonſtratio eſt:</s>
            <s xml:id="echoid-s19833" xml:space="preserve"> quòd ego continuabo a cum z, per lineam a g z,
              <lb/>
            & ſignabo ſuper arcum b g d punctum, qualiter uelim, quod ſit punctũ
              <lb/>
            m.</s>
            <s xml:id="echoid-s19834" xml:space="preserve"> Si ergo fuerit punctum m à puncto g ad partem b:</s>
            <s xml:id="echoid-s19835" xml:space="preserve"> tunc protraham
              <lb/>
            ex puncto m lineã æquidiſtantem lineæ b h [per 31 p 1] & ſi fuerit pun-
              <lb/>
            ctum m à puncto g ad partem d:</s>
            <s xml:id="echoid-s19836" xml:space="preserve"> tunc protraham ex puncto m lineam
              <lb/>
            æquidiſtãtem lineæ d k:</s>
            <s xml:id="echoid-s19837" xml:space="preserve"> ſit ergo m t.</s>
            <s xml:id="echoid-s19838" xml:space="preserve"> Dico igitur quòd linea m t tota eſt
              <lb/>
            extra circulũ b m g d e, de qua nõ cadit aliquid in eo.</s>
            <s xml:id="echoid-s19839" xml:space="preserve"> Cuius demõſtratio eſt:</s>
            <s xml:id="echoid-s19840" xml:space="preserve"> quòd ego cõtinuabo a
              <lb/>
            cũ b, & protrahã lineã m t ſecundũ rectitudinẽ, donec cõcurrat cũ linea b a ſuper punctũ n [cõcur-
              <lb/>
            ret aũt per lẽma Procli ad 29 p1:</s>
            <s xml:id="echoid-s19841" xml:space="preserve"> ꝗa m t parallela ducta eſt ipſi b h, quę cõcurrit cũ a b in b] ergo duo
              <lb/>
            rũ angulorũ ad n unuſquiſq;</s>
            <s xml:id="echoid-s19842" xml:space="preserve"> eſt rectus [ꝗa enim angulus n b h rectus eſt ք 18 p 3, & ipſi b h parallela
              <lb/>
            ducta eſt t m n:</s>
            <s xml:id="echoid-s19843" xml:space="preserve"> ęquabitur per 29 p 1 angulus t n b angulo n b h, ideoq́;</s>
            <s xml:id="echoid-s19844" xml:space="preserve"> rectus, & per 13 p 1 an t rectus]
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
Impressum