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-div322" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s8282" xml:space="preserve">
              <pb o="140" file="0146" n="146" rhead="ALHAZEN"/>
            tur diameter b q:</s>
            <s xml:id="echoid-s8283" xml:space="preserve"> & concurrat cum cõtingente in puncto p [concurrit enim per lemma Procli ad 29
              <lb/>
            p 1:</s>
            <s xml:id="echoid-s8284" xml:space="preserve">] & ducatur linea a u q ſecãs ſphęram in puncto u.</s>
            <s xml:id="echoid-s8285" xml:space="preserve"> Iam dictum eſt, quòd m o eſt æqualis o b [per
              <lb/>
            theſin communẽ 20.</s>
            <s xml:id="echoid-s8286" xml:space="preserve"> 21.</s>
            <s xml:id="echoid-s8287" xml:space="preserve"> 22.</s>
            <s xml:id="echoid-s8288" xml:space="preserve"> 23.</s>
            <s xml:id="echoid-s8289" xml:space="preserve"> 24.</s>
            <s xml:id="echoid-s8290" xml:space="preserve"> 25.</s>
            <s xml:id="echoid-s8291" xml:space="preserve"> 26.</s>
            <s xml:id="echoid-s8292" xml:space="preserve"> 27 n.</s>
            <s xml:id="echoid-s8293" xml:space="preserve">] Sed [per 15 p 3] u q eſt
              <lb/>
              <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a" number="57">
                <variables xml:id="echoid-variables47" xml:space="preserve">a d u m b g o e q s z h p</variables>
              </figure>
            maior m o:</s>
            <s xml:id="echoid-s8294" xml:space="preserve"> quare u q eſt maior o b, id eſt b q.</s>
            <s xml:id="echoid-s8295" xml:space="preserve"> Et linea ducta à circum-
              <lb/>
            ferentia ad diametrum p b, ęqualis parti p b, interiacenti inter ipſam
              <lb/>
            & centrum:</s>
            <s xml:id="echoid-s8296" xml:space="preserve"> non cadet inter q & b.</s>
            <s xml:id="echoid-s8297" xml:space="preserve"> Si enim ceciderit:</s>
            <s xml:id="echoid-s8298" xml:space="preserve"> ſecundũ ſupra-
              <lb/>
            dictam probationem [23 & præcedente numeris] erit u q minor q b.</s>
            <s xml:id="echoid-s8299" xml:space="preserve">
              <lb/>
            Reſtat ergo, ut linea ęqualis cadat inter p & q.</s>
            <s xml:id="echoid-s8300" xml:space="preserve"> Et quòd non cadatin
              <lb/>
            punctũ p:</s>
            <s xml:id="echoid-s8301" xml:space="preserve"> palàm per hoc:</s>
            <s xml:id="echoid-s8302" xml:space="preserve"> quia angulus p g b eſt rectus [per 18 p 3.</s>
            <s xml:id="echoid-s8303" xml:space="preserve">] I-
              <lb/>
            gitur [per 19 p 1] p b maius eſt p g.</s>
            <s xml:id="echoid-s8304" xml:space="preserve"> Cadet ergo citra punctum p:</s>
            <s xml:id="echoid-s8305" xml:space="preserve"> Sit
              <lb/>
            punctum, in quod cadit:</s>
            <s xml:id="echoid-s8306" xml:space="preserve"> s.</s>
            <s xml:id="echoid-s8307" xml:space="preserve"> Erit ergo s meta locorum imaginum [per
              <lb/>
            23 n:</s>
            <s xml:id="echoid-s8308" xml:space="preserve">] & quodlibet punctũ inter p & s erit locus imaginum.</s>
            <s xml:id="echoid-s8309" xml:space="preserve"> Et eadẽ
              <lb/>
            eſt probatio, quæ ſuprà [25.</s>
            <s xml:id="echoid-s8310" xml:space="preserve"> 26 n.</s>
            <s xml:id="echoid-s8311" xml:space="preserve">] Palàm ex his, quòd imagines dia-
              <lb/>
            metrorum arcus h o, omnes ſunt extra ſuperficiem ſpeculi:</s>
            <s xml:id="echoid-s8312" xml:space="preserve"> imaginũ
              <lb/>
            diametri f y, una in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s8313" xml:space="preserve"> quę eſt in l:</s>
            <s xml:id="echoid-s8314" xml:space="preserve"> aliæ intra, ſcilicet in
              <lb/>
            i l:</s>
            <s xml:id="echoid-s8315" xml:space="preserve"> aliæ omnes extra, ſcilicet in l e.</s>
            <s xml:id="echoid-s8316" xml:space="preserve"> Omniũ aũt imaginum diametri ar-
              <lb/>
            cus o g, quædam intra ſpeculum:</s>
            <s xml:id="echoid-s8317" xml:space="preserve"> quędã extra:</s>
            <s xml:id="echoid-s8318" xml:space="preserve"> quędam in ſuperficie.</s>
            <s xml:id="echoid-s8319" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div324" type="section" level="0" n="0">
          <head xml:id="echoid-head331" xml:space="preserve" style="it">28. Perpendicularis incidentiæ ſecans occult ãperipheriam cir
            <lb/>
          culι (quieſt communis ſectio ſuperficierum reflexionis & ſpeculi
            <lb/>
          ſphærici conuexi) inter terminos rectæ per centra uiſ{us} ac ſpeculi
            <lb/>
          ductæ, & quadrantis peripheriæ, à puncto tact{us} rectæ à uiſu ſpe-
            <lb/>
          culum tangentis, inchoati: imaginem nullam habet. 33 p 6.</head>
          <p>
            <s xml:id="echoid-s8320" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8321" xml:space="preserve"> in arcu h z non poteſt ſumi diameter, in qua eſt locus imaginis.</s>
            <s xml:id="echoid-s8322" xml:space="preserve"> Quoniam nulla dia-
              <lb/>
            meter ibi ſumpta concurrit cũ contingente a p.</s>
            <s xml:id="echoid-s8323" xml:space="preserve"> [Quia enim g h eſt quadrans totius periphe
              <lb/>
            riæ ex theſi:</s>
            <s xml:id="echoid-s8324" xml:space="preserve"> rectus eſt angulus h b g per 33 p 6:</s>
            <s xml:id="echoid-s8325" xml:space="preserve"> & ſimiliter b g
              <lb/>
              <figure xlink:label="fig-0146-02" xlink:href="fig-0146-02a" number="58">
                <variables xml:id="echoid-variables48" xml:space="preserve">a d u m c g b o t q p n z h</variables>
              </figure>
            p per 18 p 3.</s>
            <s xml:id="echoid-s8326" xml:space="preserve"> Quare perpẽdicularis incidentię, cadens in peripheriã
              <lb/>
            h z, facit cum b g angulũ obtuſum:</s>
            <s xml:id="echoid-s8327" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s8328" xml:space="preserve"> cũ tangente a g p non cõ
              <lb/>
            curret ad partes h & p:</s>
            <s xml:id="echoid-s8329" xml:space="preserve"> ſecus duæ rectæ ſpatium cõprehenderẽt cõ-
              <lb/>
            tra 12 ax.</s>
            <s xml:id="echoid-s8330" xml:space="preserve">] Et à quocunq;</s>
            <s xml:id="echoid-s8331" xml:space="preserve"> puncto illius talis diametri ducatur linea
              <lb/>
            ad ſphærã:</s>
            <s xml:id="echoid-s8332" xml:space="preserve"> cadet quidem in portionẽ g z c, & nulla in portionẽ g d
              <lb/>
            c, niſi ſecando ſphęram.</s>
            <s xml:id="echoid-s8333" xml:space="preserve"> Quare nulla forma alicuius puncti talis dia
              <lb/>
            metri ueniet ad portionem uiſui apparentem.</s>
            <s xml:id="echoid-s8334" xml:space="preserve"> Quod aũt dictum eſt
              <lb/>
            in arcu g h z:</s>
            <s xml:id="echoid-s8335" xml:space="preserve"> poteſt eodem modo demonſtrari in parte arcus c z eã
              <lb/>
            reſpiciente.</s>
            <s xml:id="echoid-s8336" xml:space="preserve"> Et ſumpto arcu citra z, æquali h z:</s>
            <s xml:id="echoid-s8337" xml:space="preserve"> in nulla diametro il-
              <lb/>
            lius arcus erit imaginis locus.</s>
            <s xml:id="echoid-s8338" xml:space="preserve"> Idẽ eſt demonſtrandi modus in quo-
              <lb/>
            cunq;</s>
            <s xml:id="echoid-s8339" xml:space="preserve"> circulo.</s>
            <s xml:id="echoid-s8340" xml:space="preserve"> Quare ſi linea h b moueatur, eodem manente angu-
              <lb/>
            lo h b z:</s>
            <s xml:id="echoid-s8341" xml:space="preserve"> ſignabit motu ſuo portionem ſphæræ, in cuius diametris
              <lb/>
            nullus ſit imaginis locus.</s>
            <s xml:id="echoid-s8342" xml:space="preserve"> Si uerò h b immota, moueatur o h:</s>
            <s xml:id="echoid-s8343" xml:space="preserve"> deſcri-
              <lb/>
            betur portio, cuius oẽs imagines extra ſpeculum ſunt.</s>
            <s xml:id="echoid-s8344" xml:space="preserve"> Moto aũt ar
              <lb/>
            cu o g:</s>
            <s xml:id="echoid-s8345" xml:space="preserve"> fiet portio, cuius quędam imagines ſuntin ſuperficie:</s>
            <s xml:id="echoid-s8346" xml:space="preserve"> quędã
              <lb/>
            extra ſpeculum:</s>
            <s xml:id="echoid-s8347" xml:space="preserve"> quędam intra.</s>
            <s xml:id="echoid-s8348" xml:space="preserve"> Verũ uiſus nõ comprehendit, quæ
              <lb/>
            imagines ſint in ſuperficie ſphęræ, aut quę extra:</s>
            <s xml:id="echoid-s8349" xml:space="preserve"> nec certificatur in
              <lb/>
            comprehenſione earum:</s>
            <s xml:id="echoid-s8350" xml:space="preserve"> niſi quòd ſint ultra portionem apparentẽ.</s>
            <s xml:id="echoid-s8351" xml:space="preserve">
              <lb/>
            Iam ergo determinata ſunt in his ſpeculis imaginum loca.</s>
            <s xml:id="echoid-s8352" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div326" type="section" level="0" n="0">
          <head xml:id="echoid-head332" xml:space="preserve" style="it">29. Ab uno ſpeculi ſphærici conuexi puncto, unum uiſibilis punctum adunũ uiſum reflecti-
            <lb/>
          tur. Ita uni{us} punctiuna uidetur imago. 16 p 6.</head>
          <p>
            <s xml:id="echoid-s8353" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8354" xml:space="preserve"> Puncti uiſi forma nõ poteſt in hoc ſpeculo ad unũ uiſum reflecti, niſi ab uno ſolo pũ
              <lb/>
            cto ſpeculi.</s>
            <s xml:id="echoid-s8355" xml:space="preserve"> Sit enim punctũ uiſum b:</s>
            <s xml:id="echoid-s8356" xml:space="preserve"> a centrũ uiſus:</s>
            <s xml:id="echoid-s8357" xml:space="preserve"> & nõ ſit a in perpẽdiculari ducta ad cẽtrũ
              <lb/>
            ſphęrę.</s>
            <s xml:id="echoid-s8358" xml:space="preserve"> Dico, quòd b reflectitur ad a ab uno ſolo ſpeculi puncto:</s>
            <s xml:id="echoid-s8359" xml:space="preserve"> & unã ſolã oſtendit uiſui ima
              <lb/>
            ginẽ in hoc ſpeculo.</s>
            <s xml:id="echoid-s8360" xml:space="preserve"> Palàm [per 25 n 4] quòd ab aliquo puncto poteſt reflecti forma eius:</s>
            <s xml:id="echoid-s8361" xml:space="preserve"> ſit illud g:</s>
            <s xml:id="echoid-s8362" xml:space="preserve">
              <lb/>
            & ducantur b g, a g:</s>
            <s xml:id="echoid-s8363" xml:space="preserve"> & ſit n centrum ſphęrę:</s>
            <s xml:id="echoid-s8364" xml:space="preserve"> & ducatur diameter b n, ſecans ſuperficiem ſphæræ in
              <lb/>
            puncto l:</s>
            <s xml:id="echoid-s8365" xml:space="preserve"> & termini portionis uiſui oppoſitæ ſint d, e:</s>
            <s xml:id="echoid-s8366" xml:space="preserve"> & ſecet linea a g perpẽdicularem in puncto q:</s>
            <s xml:id="echoid-s8367" xml:space="preserve">
              <lb/>
            quod eſt locus imaginum [per 3 uel 16 n.</s>
            <s xml:id="echoid-s8368" xml:space="preserve">] Palàm, quòd a, n, b ſint in eadẽ ſuperficie orthogonali ſuք
              <lb/>
            ſphæram [per 13.</s>
            <s xml:id="echoid-s8369" xml:space="preserve"> 23 n 4.</s>
            <s xml:id="echoid-s8370" xml:space="preserve">] Et cum omnes ſuperficies orthogonales ſuper ſphærã, in quibus fuerint b,
              <lb/>
            n, ſecent ſe ſuper b n:</s>
            <s xml:id="echoid-s8371" xml:space="preserve"> & nõ poſsit ſuperficies, in qua b n linea, extendi ad punctũ a, niſi una tantũ:</s>
            <s xml:id="echoid-s8372" xml:space="preserve"> [ꝗa
              <lb/>
            punctum a indiuiduũ eſt.</s>
            <s xml:id="echoid-s8373" xml:space="preserve">] Palàm, quòd a, & b, & n ſunt in una ſuperficie tantùm, orthogonali ſuper
              <lb/>
            ſphęrã, non in plurib.</s>
            <s xml:id="echoid-s8374" xml:space="preserve"> & cũ neceſſe ſit, [per 13.</s>
            <s xml:id="echoid-s8375" xml:space="preserve"> 23 n 4] ut omne punctũ uiſum, & a ſint in eadẽ ſuperfi-
              <lb/>
            cie orthogonali ſuper punctũ reflexionis:</s>
            <s xml:id="echoid-s8376" xml:space="preserve"> palàm, quòd non fiet reflexio puncti b ad uiſum, niſi in cir
              <lb/>
            culo ſphęrę, qui eſt in ſuperficie a n b.</s>
            <s xml:id="echoid-s8377" xml:space="preserve"> Sit ergo circulus d g e.</s>
            <s xml:id="echoid-s8378" xml:space="preserve"> Dico igitur, quòd à nullo puncto huius
              <lb/>
            circuli pręterꝗ̃ à g, fiet reflexio.</s>
            <s xml:id="echoid-s8379" xml:space="preserve"> Si enim dicatur, quòd à pũcto l:</s>
            <s xml:id="echoid-s8380" xml:space="preserve"> cum b n ſit ſuք ſuքficiẽ ſpeculi per-
              <lb/>
            pendicularis:</s>
            <s xml:id="echoid-s8381" xml:space="preserve"> [ut oſtẽſum eſt 25 n 4] & a l nõ ſit perpẽdicularis:</s>
            <s xml:id="echoid-s8382" xml:space="preserve"> [ꝗa nõ tranſit per centrü:</s>
            <s xml:id="echoid-s8383" xml:space="preserve">] & forma
              <lb/>
            per perpẽdicularẽ ueniens, neceſſariò ք perpendicularẽ reflectatur:</s>
            <s xml:id="echoid-s8384" xml:space="preserve"> [ք 11 n 4:</s>
            <s xml:id="echoid-s8385" xml:space="preserve">] palã, quòd non refle
              <lb/>
            ctetur b ad a à puncto l, Planum etiam eſt, quòd non reflectetur ab alio puncto arcus l e:</s>
            <s xml:id="echoid-s8386" xml:space="preserve"> quìa ad
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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