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

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        <div xml:id="echoid-div726" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s21165" xml:space="preserve">
              <pb o="18" file="0320" n="320" rhead="VITELLONIS OPTICAE"/>
            æ quidiſtat lateri a b, & linea n p lateri a d, & linea p m lateri c d.</s>
            <s xml:id="echoid-s21166" xml:space="preserve"> Ergo ք 29 p 1 anguli ſuperficiei l m
              <lb/>
            p n ſunt æquales angulis datæ ſuperficiei a b c d, & latera eorum ſunt proportionalia per 4 p 6.</s>
            <s xml:id="echoid-s21167" xml:space="preserve"> Pa-
              <lb/>
            tet ergo, quòd illæ ſuperficies ſunt ſimiles:</s>
            <s xml:id="echoid-s21168" xml:space="preserve"> & hoc proponebatur faciendũ:</s>
            <s xml:id="echoid-s21169" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s21170" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div727" type="section" level="0" n="0">
          <head xml:id="echoid-head616" xml:space="preserve" style="it">42. Omnis angulus à diametro & quacun linea ſuper circumferentiam circuli contẽtus,
            <lb/>
          neceſſariò est acutus. Alhazen 60 n 5.</head>
          <p>
            <s xml:id="echoid-s21171" xml:space="preserve">Sit circulus a b c, cuius diameter a b, & ducatur linea a c, utcunq;</s>
            <s xml:id="echoid-s21172" xml:space="preserve"> contingit.</s>
            <s xml:id="echoid-s21173" xml:space="preserve"> Dico, quòd angulus
              <lb/>
            b a c neceſſariò eſt acutus.</s>
            <s xml:id="echoid-s21174" xml:space="preserve"> Producatur enim linea b c
              <lb/>
              <figure xlink:label="fig-0320-01" xlink:href="fig-0320-01a" number="306">
                <variables xml:id="echoid-variables290" xml:space="preserve">c a b</variables>
              </figure>
            ad peripheriam in pũctum c.</s>
            <s xml:id="echoid-s21175" xml:space="preserve"> Et quoniã angulus a c b
              <lb/>
            eſt rectus per 31 p 3, patet per 32 p 1, quia angulus b a c
              <lb/>
            eſt acutus:</s>
            <s xml:id="echoid-s21176" xml:space="preserve"> & ſimiliter angulus a b c.</s>
            <s xml:id="echoid-s21177" xml:space="preserve"> Patet itaq;</s>
            <s xml:id="echoid-s21178" xml:space="preserve"> propo
              <lb/>
            ſitum:</s>
            <s xml:id="echoid-s21179" xml:space="preserve"> & de hoc theoremate nõ ſeruimus intellectui,
              <lb/>
            ſed breuitati, quia hanc demonſtrationem toties, ut
              <lb/>
            occurrit, repetere, tædium fuit.</s>
            <s xml:id="echoid-s21180" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div729" type="section" level="0" n="0">
          <head xml:id="echoid-head617" xml:space="preserve" style="it">43. Omnes angulos æqualium ucl ſimilium por-
            <lb/>
          tionum eiuſdem circuli ſub arcu & recta contentos
            <lb/>
          æquales: angulos uerò cuiuſcun minoris portionis
            <lb/>
          minores, & maioris maiores eſſe neceſſe eſt. Ex quo
            <lb/>
          patet, omnes angulos ſemicir culorum æquales eſſe.</head>
          <p>
            <s xml:id="echoid-s21181" xml:space="preserve">Sit circulus, cuius centrum a, & diameter g f:</s>
            <s xml:id="echoid-s21182" xml:space="preserve"> & in
              <lb/>
            c o ſignentur arcus æquales, qui ſint b c & d e, produ-
              <lb/>
            ctis chordis b c & d e.</s>
            <s xml:id="echoid-s21183" xml:space="preserve"> Dico, quòd anguli g b c, & f d e,
              <lb/>
            ſub arcubus & chordis contenti ſunt æquales.</s>
            <s xml:id="echoid-s21184" xml:space="preserve"> Duca-
              <lb/>
            tur enim à puncto b linea contingens circulum per 17 p 3, quæ ſit b l, & à puncto d linea d m:</s>
            <s xml:id="echoid-s21185" xml:space="preserve"> & du-
              <lb/>
            cantur à centro lineę a b, a c, a d, a e, eruntq́;</s>
            <s xml:id="echoid-s21186" xml:space="preserve"> per 5 p 1 anguli a b c & a c b æquales:</s>
            <s xml:id="echoid-s21187" xml:space="preserve"> & anguli a d e &
              <lb/>
            a e d æquales:</s>
            <s xml:id="echoid-s21188" xml:space="preserve"> ſed trigona a b c & a d e ſunt æquiangula per 4 p 1:</s>
            <s xml:id="echoid-s21189" xml:space="preserve"> angulus enim b a c eſt æqualis an-
              <lb/>
            gulo d a e, per 27 p 3:</s>
            <s xml:id="echoid-s21190" xml:space="preserve"> angulus quoq;</s>
            <s xml:id="echoid-s21191" xml:space="preserve"> a b l eſt æqualis angulo
              <lb/>
              <figure xlink:label="fig-0320-02" xlink:href="fig-0320-02a" number="307">
                <variables xml:id="echoid-variables291" xml:space="preserve">ſ q r n g o b c s c a d e f m</variables>
              </figure>
            a d m, quoniam uterq;</s>
            <s xml:id="echoid-s21192" xml:space="preserve"> eorũ eſt rectus per 18 p 3:</s>
            <s xml:id="echoid-s21193" xml:space="preserve"> ſed angulus
              <lb/>
            contingentiæ l b g eſt æqualis angulo contingentiæ m d f:</s>
            <s xml:id="echoid-s21194" xml:space="preserve">
              <lb/>
            quoniam uterq;</s>
            <s xml:id="echoid-s21195" xml:space="preserve"> ipſorum eſt minimus acutorum per 18 p 3.</s>
            <s xml:id="echoid-s21196" xml:space="preserve">
              <lb/>
            Relin quitur ergo angulus g b c a b arcu b g, & recta b c con
              <lb/>
            tentus, æqualis angulo f d e, ab arcu f d, & recta d e conten-
              <lb/>
            to:</s>
            <s xml:id="echoid-s21197" xml:space="preserve"> ſed & angulus g c b eſt ęqualis angulo g b c eadem ratio-
              <lb/>
            ne:</s>
            <s xml:id="echoid-s21198" xml:space="preserve"> ſimiliter quoq;</s>
            <s xml:id="echoid-s21199" xml:space="preserve"> angulus f e d eſt æqualis angulo f d e.</s>
            <s xml:id="echoid-s21200" xml:space="preserve"> O-
              <lb/>
            mnes itaq;</s>
            <s xml:id="echoid-s21201" xml:space="preserve"> hi anguli ſunt æquales.</s>
            <s xml:id="echoid-s21202" xml:space="preserve"> Sit quoq;</s>
            <s xml:id="echoid-s21203" xml:space="preserve"> arcus minor ar
              <lb/>
            cu b c, quireſecetur ab arcu b c, qui ſit arcus n o, & ducãtur
              <lb/>
            lineæ a n, a o:</s>
            <s xml:id="echoid-s21204" xml:space="preserve"> ducatur quoq;</s>
            <s xml:id="echoid-s21205" xml:space="preserve"> chorda n o:</s>
            <s xml:id="echoid-s21206" xml:space="preserve"> & ducantur contin
              <lb/>
            gẽtes n q & o r.</s>
            <s xml:id="echoid-s21207" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21208" xml:space="preserve"> trigoni a n o anguli ad baſim ſunt
              <lb/>
            æquales per 5 p 1, & angulus o a n minor angulo c a b, per
              <lb/>
            33 p 6:</s>
            <s xml:id="echoid-s21209" xml:space="preserve"> erit per 32 p 1 quilibet angulorum a n o & a o n maior
              <lb/>
            quolibet angulorum a b c & a c b.</s>
            <s xml:id="echoid-s21210" xml:space="preserve"> Sit itaq;</s>
            <s xml:id="echoid-s21211" xml:space="preserve"> angulus o n a m a
              <lb/>
            ior angulo c b a:</s>
            <s xml:id="echoid-s21212" xml:space="preserve"> ſed angulus contingentię q n g eſt ęqualis
              <lb/>
            angulo cõtingentię l b g:</s>
            <s xml:id="echoid-s21213" xml:space="preserve"> relinquitur ergo angulus g n o mi-
              <lb/>
            nor angulo g b c, cum anguli l b a & q n a ſint æquales:</s>
            <s xml:id="echoid-s21214" xml:space="preserve"> quia
              <lb/>
            uterq;</s>
            <s xml:id="echoid-s21215" xml:space="preserve"> rectus per 18 p 3.</s>
            <s xml:id="echoid-s21216" xml:space="preserve"> Sit iam arcus maior arcu b c, qui ſit s c, & ducatur chorda s c:</s>
            <s xml:id="echoid-s21217" xml:space="preserve"> & quia angulus
              <lb/>
            c a s eſt maior angulo c a b per 33 p 6:</s>
            <s xml:id="echoid-s21218" xml:space="preserve"> patet tũc, quòd angulus a s c eſt minor angulo a b c:</s>
            <s xml:id="echoid-s21219" xml:space="preserve"> & ita con-
              <lb/>
            cludetur, ut prius, quoniã angulus g s c contentus ſub arcu g s, & chorda s c eſt maior angulo g b c:</s>
            <s xml:id="echoid-s21220" xml:space="preserve">
              <lb/>
            ergo & angulo g n o.</s>
            <s xml:id="echoid-s21221" xml:space="preserve"> Patet & hocidem de ſimilibus arcubus quibuſcunq;</s>
            <s xml:id="echoid-s21222" xml:space="preserve"> eorundem circulorum,
              <lb/>
            quoniam per definitionem ſimilium arcuũ ipſi angulos ſuſcipiunt æquales per 10 defin.</s>
            <s xml:id="echoid-s21223" xml:space="preserve"> 3.</s>
            <s xml:id="echoid-s21224" xml:space="preserve"> Ex quo
              <lb/>
            patet corollarium, quoniam omnes anguli ſemicirculorum ſunt æquales:</s>
            <s xml:id="echoid-s21225" xml:space="preserve"> omnes enim ſemicirculi
              <lb/>
            ſunt ſimiles:</s>
            <s xml:id="echoid-s21226" xml:space="preserve"> & eiuſdem circuli ſimiles & ęquales:</s>
            <s xml:id="echoid-s21227" xml:space="preserve"> hoc itaq;</s>
            <s xml:id="echoid-s21228" xml:space="preserve"> proponebatur.</s>
            <s xml:id="echoid-s21229" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div731" type="section" level="0" n="0">
          <head xml:id="echoid-head618" xml:space="preserve" style="it">44. Si idem angulus ſuper centrum unius æqualium circulorum, & ſuper peripheriam alte-
            <lb/>
          rius conſiſtat, arcus reſpondens angulo ſuper peripheriã conſtituto, reliquo arcui duplus erit. In
            <lb/>
          circulis uerò inæqualibus illorũ arcuum proportio ad ſuas totales peripherias duplicatur.</head>
          <p>
            <s xml:id="echoid-s21230" xml:space="preserve">Sint duo circuli æquales, unus a b c d, cuius centrum g:</s>
            <s xml:id="echoid-s21231" xml:space="preserve"> & alius e f g, cuius centrum b, punctum
              <lb/>
            peripheriæ circuli a b c d:</s>
            <s xml:id="echoid-s21232" xml:space="preserve"> & producantur lineę a b & c b, ſecantes circulum e g f in punctis e & f.</s>
            <s xml:id="echoid-s21233" xml:space="preserve"> Pa-
              <lb/>
            làm itaq;</s>
            <s xml:id="echoid-s21234" xml:space="preserve"> quoniam angulus a b c erit ſuper peripheriam circuli a b c & ſuper centrum circuli e g f.</s>
            <s xml:id="echoid-s21235" xml:space="preserve">
              <lb/>
            Dico, quòd arcus a d c capiens angulũ a b c ſuper circũferentiam ſui circuli, eſt duplus arcui e g f, ca
              <lb/>
            pienti eundẽ angulũ ſuper eius centrũ b.</s>
            <s xml:id="echoid-s21236" xml:space="preserve"> Sit enim, ut linea b a ſecet circulũ e g f in puncto e, & linea
              <lb/>
            b cin puncto f:</s>
            <s xml:id="echoid-s21237" xml:space="preserve"> ducatur quoq;</s>
            <s xml:id="echoid-s21238" xml:space="preserve"> linea e f, & ducta linea g h ſuper centrũ g, fiat per 23 p 1 angulus æqua
              <lb/>
            lis angulo a b c, qui ſit h g l, ductis lineis g h & g l ad circumferentiam circuli a b c d:</s>
            <s xml:id="echoid-s21239" xml:space="preserve"> & ducantur li-
              <lb/>
            neę b h, b l, h l.</s>
            <s xml:id="echoid-s21240" xml:space="preserve"> Palàm itaq;</s>
            <s xml:id="echoid-s21241" xml:space="preserve"> per 20 p 3, q́uoniam angulus h g l eſt duplus angulo h b l:</s>
            <s xml:id="echoid-s21242" xml:space="preserve"> ergo etiam an-
              <lb/>
            gulus a b c eſt duplus eidem:</s>
            <s xml:id="echoid-s21243" xml:space="preserve"> ergo per 33 p 6 arcus a d c eſt duplus arcui h d l:</s>
            <s xml:id="echoid-s21244" xml:space="preserve"> ſed arcus h d l
              <lb/>
            </s>
          </p>
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