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-div745" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s21514" xml:space="preserve">
              <pb o="23" file="0325" n="325" rhead="LIBER PRIMVS."/>
            ipſam a c in duo æqualia per 3 p 3, producaturq́;</s>
            <s xml:id="echoid-s21515" xml:space="preserve"> linea o g, ut ſecet lineam d e in puncto f:</s>
            <s xml:id="echoid-s21516" xml:space="preserve"> & ductis li-
              <lb/>
            neis o a, o c, o d, o e:</s>
            <s xml:id="echoid-s21517" xml:space="preserve"> palàm per 4 p 1, cum in trigonis a g o & c g o duo latera a g & g c ſint æqualia, &
              <lb/>
            latus g o commune, & anguli ad g recti ex hypotheſi:</s>
            <s xml:id="echoid-s21518" xml:space="preserve"> quòd angulus a o g eſt æqualis angulo c o g:</s>
            <s xml:id="echoid-s21519" xml:space="preserve">
              <lb/>
            ſed angulus a o d æqualis eſt angulo c o e per 27 p 3:</s>
            <s xml:id="echoid-s21520" xml:space="preserve"> relin quitur ergo angulus d o f æqualis angulo
              <lb/>
            f o e:</s>
            <s xml:id="echoid-s21521" xml:space="preserve"> ſed latus d o æquale lateri e o, & latus o f commune:</s>
            <s xml:id="echoid-s21522" xml:space="preserve"> erit ergo per 4 p 1 angulus o f d æqualis an
              <lb/>
            gulo o fe:</s>
            <s xml:id="echoid-s21523" xml:space="preserve"> uterq;</s>
            <s xml:id="echoid-s21524" xml:space="preserve"> ergo eſt rectus.</s>
            <s xml:id="echoid-s21525" xml:space="preserve"> Eſt ergo angulus o f d æqualis angulo o g a:</s>
            <s xml:id="echoid-s21526" xml:space="preserve"> ergo per 28 p 1 lineę d e
              <lb/>
            & a c ſunt æquidiſtantes:</s>
            <s xml:id="echoid-s21527" xml:space="preserve"> quod eſt propoſitum primum.</s>
            <s xml:id="echoid-s21528" xml:space="preserve"> Quòd ſi una illarum duarum linearum ſe-
              <lb/>
            cet circulum, & alia ipſum contingat:</s>
            <s xml:id="echoid-s21529" xml:space="preserve"> ſi ſecans tranſit centrũ, & ſit diameter, quæ h k, & linea l m con
              <lb/>
            tingat in puncto n:</s>
            <s xml:id="echoid-s21530" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s21531" xml:space="preserve"> arcus n h æqualis arcui n k:</s>
            <s xml:id="echoid-s21532" xml:space="preserve"> palàm, quòd illorum arcuum quilibet eſt quar-
              <lb/>
            ta circuli:</s>
            <s xml:id="echoid-s21533" xml:space="preserve"> ducatur ita que linea n o:</s>
            <s xml:id="echoid-s21534" xml:space="preserve"> ergo per 18 p 3 angulus l n o eſt rectus:</s>
            <s xml:id="echoid-s21535" xml:space="preserve"> ſed & angulus n o h eſt re-
              <lb/>
            ctus:</s>
            <s xml:id="echoid-s21536" xml:space="preserve"> ergo per 28 p 1 lineæ l m & h k ęquidiſtant:</s>
            <s xml:id="echoid-s21537" xml:space="preserve"> quod eſt ſecundũ propoſitum.</s>
            <s xml:id="echoid-s21538" xml:space="preserve"> Quòd ſi linea l m cir-
              <lb/>
            culum contingente in puncto n, linea d e ſecet circulum nõ per centrũ:</s>
            <s xml:id="echoid-s21539" xml:space="preserve"> ducantur lineę o d l & o e m,
              <lb/>
            & à centro o ad punctum contactus, quod eſt n, ducatur linea o n ſecãs lineam d e in puncto f.</s>
            <s xml:id="echoid-s21540" xml:space="preserve"> Quia
              <lb/>
            ita que arcus n d eſt æqualis arcui n e:</s>
            <s xml:id="echoid-s21541" xml:space="preserve"> erit per 27 p 3 angulus l o n ęqualis angulo m o n:</s>
            <s xml:id="echoid-s21542" xml:space="preserve"> ſed per 18 p 3
              <lb/>
            angulus o n l eſt æqualis angulo o n m:</s>
            <s xml:id="echoid-s21543" xml:space="preserve"> quia ambo ſunt recti.</s>
            <s xml:id="echoid-s21544" xml:space="preserve"> Item per 4 p 1 angulus o f d eſt æqualis
              <lb/>
            angulo o f e:</s>
            <s xml:id="echoid-s21545" xml:space="preserve"> ſunt ergo recti.</s>
            <s xml:id="echoid-s21546" xml:space="preserve"> Ergo per 28 p 1 patet propoſitum tertium.</s>
            <s xml:id="echoid-s21547" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div747" type="section" level="0" n="0">
          <head xml:id="echoid-head627" xml:space="preserve" style="it">53. Lineas æquidiſt antes trans circuli ſuperficiem product{as}, ſiue ambæ ſecent, ſiue ambæ cõ-
            <lb/>
          tingant, ſiue una ſecet & alia contingat, arcus interiacent æquales.</head>
          <p>
            <s xml:id="echoid-s21548" xml:space="preserve">Sit circulus a c b d, cuius centrum e:</s>
            <s xml:id="echoid-s21549" xml:space="preserve"> contingantq́;</s>
            <s xml:id="echoid-s21550" xml:space="preserve"> ipſum duæ lineæ ęquidiſtãtes f g in puncto d,
              <lb/>
            & h q in puncto c:</s>
            <s xml:id="echoid-s21551" xml:space="preserve"> & à puncto contingentiæ, quod eſt d,
              <lb/>
              <figure xlink:label="fig-0325-01" xlink:href="fig-0325-01a" number="322">
                <variables xml:id="echoid-variables306" xml:space="preserve">f m a h k d p e o c l g n b q</variables>
              </figure>
            ducatur linea d e ad centrum e.</s>
            <s xml:id="echoid-s21552" xml:space="preserve"> Eſt ergo per 18 p 3 linea
              <lb/>
            d e perpendicularis ſuper lineam in illo puncto contin-
              <lb/>
            gentem, quæ f g.</s>
            <s xml:id="echoid-s21553" xml:space="preserve"> Ducatur quoque linea c e à puncto cõ
              <lb/>
            tingentiæ ad centrum e:</s>
            <s xml:id="echoid-s21554" xml:space="preserve"> erit ergo linea c e perpendicu-
              <lb/>
            laris ſuper lineam h q contingentem in puncto c.</s>
            <s xml:id="echoid-s21555" xml:space="preserve"> Duca
              <lb/>
            tur quoq;</s>
            <s xml:id="echoid-s21556" xml:space="preserve"> à centro e linea ęquidiſtans lineę f g per 31 p 1,
              <lb/>
            quæ ſit n m:</s>
            <s xml:id="echoid-s21557" xml:space="preserve"> hæc quoq;</s>
            <s xml:id="echoid-s21558" xml:space="preserve"> etiam æquidiſtabit lineæ h q per
              <lb/>
            30 p 1:</s>
            <s xml:id="echoid-s21559" xml:space="preserve"> ergo per 29 p 1 angulus m e d eſt æqualis angulo
              <lb/>
            m e c:</s>
            <s xml:id="echoid-s21560" xml:space="preserve"> ergo per 14 p 1 lineæ d e & e c cõiunctæ, ſunt linea
              <lb/>
            una:</s>
            <s xml:id="echoid-s21561" xml:space="preserve"> eſt ergo linea d c diameter circuli, cum trãſeat per
              <lb/>
            centrum e:</s>
            <s xml:id="echoid-s21562" xml:space="preserve"> arcus itaque d a c eſt ſemicirculus æqualis
              <lb/>
            ſemicirculo d b c.</s>
            <s xml:id="echoid-s21563" xml:space="preserve"> Sed & ſi linea a b ſecet circulum æqui
              <lb/>
            diſtans lineæ h q contingenti in puncto e, erit iterum ar
              <lb/>
            cus a c æqualis arcui c b.</s>
            <s xml:id="echoid-s21564" xml:space="preserve"> Quia enim ſemidiameter e c
              <lb/>
            ſecat lineam contingẽtem, quæ h q:</s>
            <s xml:id="echoid-s21565" xml:space="preserve"> palàm per 2 huius,
              <lb/>
            quoniam ſecabit & eius æquidiſtantem, quæ eſt linea
              <lb/>
            a b:</s>
            <s xml:id="echoid-s21566" xml:space="preserve"> ſit, ut ſecet ipſam in puncto o.</s>
            <s xml:id="echoid-s21567" xml:space="preserve"> Et quia angulus h c e
              <lb/>
            eſtrectus per 18 p 3, palàm per 29 p 1, quoniam angulus
              <lb/>
            b o e eſt rectus:</s>
            <s xml:id="echoid-s21568" xml:space="preserve"> ergo per 3 p 3 linea a b diuiditur per æqualia in puncto o.</s>
            <s xml:id="echoid-s21569" xml:space="preserve"> Ducantur itaq;</s>
            <s xml:id="echoid-s21570" xml:space="preserve"> lineę a c &
              <lb/>
            c b:</s>
            <s xml:id="echoid-s21571" xml:space="preserve"> palamq́;</s>
            <s xml:id="echoid-s21572" xml:space="preserve"> per 4 p 1, quoniã illę erunt æquales:</s>
            <s xml:id="echoid-s21573" xml:space="preserve"> ergo per 28 p 3 arcus a c eſt æqualis arcui b c.</s>
            <s xml:id="echoid-s21574" xml:space="preserve"> Quòd
              <lb/>
            ſi linea æquidiſtans lineę b a ſecet circulũ:</s>
            <s xml:id="echoid-s21575" xml:space="preserve"> quæ ſit k l:</s>
            <s xml:id="echoid-s21576" xml:space="preserve"> palàm, quoniam ſemidiameter e c producta ſe-
              <lb/>
            cabit lineam k l per ęqualia per 29 p 1.</s>
            <s xml:id="echoid-s21577" xml:space="preserve"> 3 p 3:</s>
            <s xml:id="echoid-s21578" xml:space="preserve"> ſecet ergo ipſam per æqualia & orthogonaliter in puncto
              <lb/>
            p:</s>
            <s xml:id="echoid-s21579" xml:space="preserve"> & ducãtur lineæ p a, p b, k a, l b:</s>
            <s xml:id="echoid-s21580" xml:space="preserve"> erit ergo in trigonis p a c, p b c ք præmiſſa, & 4 p 1 latus p a ęquale
              <lb/>
            lateri p b:</s>
            <s xml:id="echoid-s21581" xml:space="preserve"> & angulus p b c æqualis angulo a p c:</s>
            <s xml:id="echoid-s21582" xml:space="preserve"> relin quitur ergo angulus k p a æqualis angulo b p l:</s>
            <s xml:id="echoid-s21583" xml:space="preserve">
              <lb/>
            ſed linea k p eſt æqualis lineæ p l:</s>
            <s xml:id="echoid-s21584" xml:space="preserve"> erit ergo per 4 p 1 linea k a æqualis lineæ l b.</s>
            <s xml:id="echoid-s21585" xml:space="preserve"> Ergo per 28 p 3 erit ar-
              <lb/>
            cus k a æqualis arcui l b:</s>
            <s xml:id="echoid-s21586" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s21587" xml:space="preserve"/>
          </p>
          <figure number="323">
            <variables xml:id="echoid-variables307" xml:space="preserve">a h b g e f d c
              <gap/>
            </variables>
          </figure>
        </div>
        <div xml:id="echoid-div749" type="section" level="0" n="0">
          <head xml:id="echoid-head628" xml:space="preserve" style="it">54. Duabus chordis in aliquo circulo ſe ſecanti-
            <lb/>
          bus: erit quilibet angulus ſectionis æqualis angulo
            <lb/>
          apud circumferentiam, cadenti in arcum æqua-
            <lb/>
          lem duobus arcubus ſcilicet eidem angulo & ſuo cõ
            <lb/>
          trapoſito ſubtenſis. Albazen 24 n 7.</head>
          <p>
            <s xml:id="echoid-s21588" xml:space="preserve">Sit circulus a b c d, in quo ſecẽt ſe duę chordę a c &
              <lb/>
            b d:</s>
            <s xml:id="echoid-s21589" xml:space="preserve"> & ſit pũctũ ſectionis e.</s>
            <s xml:id="echoid-s21590" xml:space="preserve"> Dico, quòd angulus a e b
              <lb/>
            eſt æqualis angulo, qui eſt in circumferentia, quam
              <lb/>
            ſubtẽdunt duo arcus a b & c d:</s>
            <s xml:id="echoid-s21591" xml:space="preserve"> & quòd angulus b e c
              <lb/>
            eſt ęqualis angulo in circumferẽtia, quã ſubtendunt
              <lb/>
            duo arcus d g a & b z c.</s>
            <s xml:id="echoid-s21592" xml:space="preserve"> Ducatur enim à puncto b li-
              <lb/>
            nea b z ęquidiſtanter lineę a c per 31 p 1.</s>
            <s xml:id="echoid-s21593" xml:space="preserve"> Si ergo linea
              <lb/>
            b z ſecat circulum, palã, quia arcus c z eſt ęqualis ar-
              <lb/>
            cui a b per præcedentem:</s>
            <s xml:id="echoid-s21594" xml:space="preserve"> arcus itaq;</s>
            <s xml:id="echoid-s21595" xml:space="preserve"> z d æqualis eſt
              <lb/>
            ambobus arcubus a b & d c:</s>
            <s xml:id="echoid-s21596" xml:space="preserve"> quoniam arcus d c utro-
              <lb/>
            biq;</s>
            <s xml:id="echoid-s21597" xml:space="preserve"> eſt cõmunis:</s>
            <s xml:id="echoid-s21598" xml:space="preserve"> fed arcus d z reſpicit angulũ d b z,
              <lb/>
            </s>
          </p>
        </div>
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