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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          <p>
            <s xml:id="echoid-s8386" xml:space="preserve">
              <pb o="141" file="0147" n="147" rhead="OPTICAE LIBER V."/>
            quodcunq;</s>
            <s xml:id="echoid-s8387" xml:space="preserve"> punctum illius arcus ducatur linea à puncto b:</s>
            <s xml:id="echoid-s8388" xml:space="preserve"> tenebit cũ contingente illius puncti an-
              <lb/>
            gulum obtuſum ex parte e:</s>
            <s xml:id="echoid-s8389" xml:space="preserve"> [Nam ſemidiameter circuli ad rectam lineam per punctum illud, ſpecu
              <lb/>
            lum tangẽtem educta, declinat à puncto b uerſus
              <lb/>
              <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a" number="59">
                <variables xml:id="echoid-variables49" xml:space="preserve">b k a p f m e l z g t r o q h n d</variables>
              </figure>
            e, & facit cum tangente angulum rectum per 18 p
              <lb/>
            3.</s>
            <s xml:id="echoid-s8390" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s8391" xml:space="preserve"> angulus tangentis & lineæ rectę à puncto
              <lb/>
            b ad pũctum tactus ductę, maior eſt recto:</s>
            <s xml:id="echoid-s8392" xml:space="preserve"> ide o q́;</s>
            <s xml:id="echoid-s8393" xml:space="preserve">
              <lb/>
            per 11 d 1 obtuſus:</s>
            <s xml:id="echoid-s8394" xml:space="preserve">] & linea ducta à puncto a ad il.</s>
            <s xml:id="echoid-s8395" xml:space="preserve">
              <lb/>
            lud punctum, tenebit cũ contingente illa angulũ
              <lb/>
            acutum ex parte l:</s>
            <s xml:id="echoid-s8396" xml:space="preserve"> [quia angulus ſemidiametri &
              <lb/>
            tangentis rectus eſt per 18 p 3:</s>
            <s xml:id="echoid-s8397" xml:space="preserve"> & recta à puncto a
              <lb/>
            ad illud punctũ ducta ſecat circulũ d e g.</s>
            <s xml:id="echoid-s8398" xml:space="preserve">] Quare
              <lb/>
            ſi ab illo puncto fieret reflexio:</s>
            <s xml:id="echoid-s8399" xml:space="preserve"> eſſet angulus acu-
              <lb/>
            tus æqualis obtuſo [per 12 n 4.</s>
            <s xml:id="echoid-s8400" xml:space="preserve">] Iterũ à nullo pun
              <lb/>
            cto arcus g l poteſt fieri reflexio.</s>
            <s xml:id="echoid-s8401" xml:space="preserve"> Sumatur enim
              <lb/>
            punctum quodcunq;</s>
            <s xml:id="echoid-s8402" xml:space="preserve">: & ſit z:</s>
            <s xml:id="echoid-s8403" xml:space="preserve"> & ducatur linea a z o,
              <lb/>
            ſecans perpendicularẽ in puncto o:</s>
            <s xml:id="echoid-s8404" xml:space="preserve"> & ducatur li-
              <lb/>
            nea contingens circulum in puncto z:</s>
            <s xml:id="echoid-s8405" xml:space="preserve"> [per 17 p 3]
              <lb/>
            quæ cadit neceſſariò inter b g, & b l:</s>
            <s xml:id="echoid-s8406" xml:space="preserve"> [quia punctũ
              <lb/>
            z eſt inter puncta g & l] & ſit m z:</s>
            <s xml:id="echoid-s8407" xml:space="preserve"> & f g circulũ cõ-
              <lb/>
            tingat in puncto g.</s>
            <s xml:id="echoid-s8408" xml:space="preserve"> Palã ex ſuperiorib.</s>
            <s xml:id="echoid-s8409" xml:space="preserve"> [18 n] quòd
              <lb/>
            proportio b n ad n q, ſicut b f ad f q.</s>
            <s xml:id="echoid-s8410" xml:space="preserve"> Eodem modo
              <lb/>
            proportio b n ad n o, ſicut proportio b m ad m o:</s>
            <s xml:id="echoid-s8411" xml:space="preserve">
              <lb/>
            Sed [per 9 ax.</s>
            <s xml:id="echoid-s8412" xml:space="preserve"> 8 p 5] maior eſt proportio b n ad n
              <lb/>
            q, quã b n ad n o.</s>
            <s xml:id="echoid-s8413" xml:space="preserve"> Igitur [per 11 p 5] maior eſt pro-
              <lb/>
            portio b fad f q, quàm b m ad m o.</s>
            <s xml:id="echoid-s8414" xml:space="preserve"> Quod planè im
              <lb/>
            poſsibile:</s>
            <s xml:id="echoid-s8415" xml:space="preserve"> cum [per 9 ax] b f ſit minor b m, & f q
              <lb/>
            maior m o.</s>
            <s xml:id="echoid-s8416" xml:space="preserve"> [ideoq́;</s>
            <s xml:id="echoid-s8417" xml:space="preserve"> ratio b f ad f q minor eſt ratio-
              <lb/>
            ne b m ad m o, ut patet ex 8 p 5.</s>
            <s xml:id="echoid-s8418" xml:space="preserve">] Reſtat ergo, ut à puncto z non fiat reflexio.</s>
            <s xml:id="echoid-s8419" xml:space="preserve"> Verùm quòd ab aliquo
              <lb/>
            puncto arcus g d non fiat reflexio:</s>
            <s xml:id="echoid-s8420" xml:space="preserve"> ſic conſtabit.</s>
            <s xml:id="echoid-s8421" xml:space="preserve"> Sumatur quodcunq;</s>
            <s xml:id="echoid-s8422" xml:space="preserve"> punctum:</s>
            <s xml:id="echoid-s8423" xml:space="preserve"> & ſit t:</s>
            <s xml:id="echoid-s8424" xml:space="preserve"> & ducatur li-
              <lb/>
            nea b t:</s>
            <s xml:id="echoid-s8425" xml:space="preserve"> & linea a t h, ſecans b n in pũcto h & [per 17 p 3] ducatur cõtingens circulũ in pũcto t:</s>
            <s xml:id="echoid-s8426" xml:space="preserve"> quę ſit
              <lb/>
            p t.</s>
            <s xml:id="echoid-s8427" xml:space="preserve"> Erit ergo ex ſuperiorib.</s>
            <s xml:id="echoid-s8428" xml:space="preserve"> [18 n] ꝓportio b n ad n h, ſicut b p ad p h:</s>
            <s xml:id="echoid-s8429" xml:space="preserve"> & b n ad n q, ſicut b fad f q:</s>
            <s xml:id="echoid-s8430" xml:space="preserve"> Sed
              <lb/>
            [per 9 ax.</s>
            <s xml:id="echoid-s8431" xml:space="preserve"> 8 p 5] b n ad n h maior eſt, quá b n ad n q:</s>
            <s xml:id="echoid-s8432" xml:space="preserve"> ergo [per 11 p 5] maior eſt proportio b p ad p h, ꝗ̃
              <lb/>
            b fad f q:</s>
            <s xml:id="echoid-s8433" xml:space="preserve"> quod planè falſum:</s>
            <s xml:id="echoid-s8434" xml:space="preserve"> cum [per 9 ax] b f ſit maior b p, & p h maiorf q.</s>
            <s xml:id="echoid-s8435" xml:space="preserve"> [ideoq́;</s>
            <s xml:id="echoid-s8436" xml:space="preserve"> ratio b p ad
              <lb/>
            p h minor eſt ratione b f ad f q, ut conſtat ex 8 p 5.</s>
            <s xml:id="echoid-s8437" xml:space="preserve">] Reſtat ergo, ut à nullo puncto arcus g d fiat refle-
              <lb/>
            xio puncti b.</s>
            <s xml:id="echoid-s8438" xml:space="preserve"> Quare quodlibet punctum ab uno ſolo puncto ſpeculi reflectitur ad uiſum.</s>
            <s xml:id="echoid-s8439" xml:space="preserve"> Ergo una
              <lb/>
            ſola erit linea reflexionis cuiuslibet puncti uiſi.</s>
            <s xml:id="echoid-s8440" xml:space="preserve"> Quare unica unius puncti imago.</s>
            <s xml:id="echoid-s8441" xml:space="preserve"> Si aũt punctum b
              <lb/>
            fuerit in perpendiculari uiſuali:</s>
            <s xml:id="echoid-s8442" xml:space="preserve"> palàm [per 11 n 4] quòd reflectetur ab uno ſolo puncto, per quod
              <lb/>
            perpendicularis, tantũ:</s>
            <s xml:id="echoid-s8443" xml:space="preserve"> & unica erit eius imago:</s>
            <s xml:id="echoid-s8444" xml:space="preserve"> & erit propter continuitatem aliorum punctorum,
              <lb/>
            in loco imaginis proprio.</s>
            <s xml:id="echoid-s8445" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div328" type="section" level="0" n="0">
          <figure number="60">
            <variables xml:id="echoid-variables50" xml:space="preserve">b k u a p e g t q n d</variables>
          </figure>
          <head xml:id="echoid-head333" xml:space="preserve" style="it">30. Siduo perpendicularis incidentiæ pun-
            <lb/>
          cta, à ſpeculo ſphærico conuexo ad unum uiſum reflectantur: loc{us} tum imaginis tum reflexio- nis, puncti centro ſpeculi propinquioris erit re- motior: imaginis ab eodem centro: reflexionis à uiſu. 17 p 6.</head>
          <p>
            <s xml:id="echoid-s8446" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8447" xml:space="preserve"> ſi in aliqua diametro ſumãtur duo
              <lb/>
            puncta ex parte centri eadem:</s>
            <s xml:id="echoid-s8448" xml:space="preserve"> locus ima-
              <lb/>
            ginis centro propinquioris, erit remotior
              <lb/>
            à centro ſphęrę, loco imaginis puncti remotioris
              <lb/>
            à cẽtro ſphęrę.</s>
            <s xml:id="echoid-s8449" xml:space="preserve"> Verbi gratia dico, quòd locus ima-
              <lb/>
            ginis puncti p, remotior eſt à centro, loco imagi-
              <lb/>
            nis puncti b:</s>
            <s xml:id="echoid-s8450" xml:space="preserve"> & punctum reflexionis puncti p re-
              <lb/>
            motius ab a puncto uiſus, puncto reflexionis pun
              <lb/>
            cti b, quod eſt punctum g.</s>
            <s xml:id="echoid-s8451" xml:space="preserve"> Dico, quòd punctum p
              <lb/>
            non reflectitur, niſi ab aliquo puncto arcus g l.</s>
            <s xml:id="echoid-s8452" xml:space="preserve">
              <lb/>
            Palàm enim, quòd non reflectitur ab aliquo pun-
              <lb/>
            cto arcus le, niſi à puncto l:</s>
            <s xml:id="echoid-s8453" xml:space="preserve"> [ſicq́;</s>
            <s xml:id="echoid-s8454" xml:space="preserve"> per 11 n 4 refle-
              <lb/>
            ctetur per perpendicularem b l n, nõ ad uiſum, in
              <lb/>
            a poſitum] nec à puncto g:</s>
            <s xml:id="echoid-s8455" xml:space="preserve"> cũ b reflectatur ab eo,
              <lb/>
            [ad uiſum ſcilicet a ex theſi:</s>
            <s xml:id="echoid-s8456" xml:space="preserve"> ideoq́ue nullum ali-
              <lb/>
            ud punctum, ut p, ab eodem puncto g reflectetur
              <lb/>
            ad eundẽ uiſum a ք pręcedẽtẽ numerũ.</s>
            <s xml:id="echoid-s8457" xml:space="preserve">] Et ſi dica
              <lb/>
            tur, qđ ab aliquo pũcto arcus g d:</s>
            <s xml:id="echoid-s8458" xml:space="preserve"> ſit illud pũctũ t:</s>
            <s xml:id="echoid-s8459" xml:space="preserve">
              <lb/>
            </s>
          </p>
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