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-s15581" xml:space="preserve">
              <pb o="220" file="0226" n="226" rhead="ALHAZEN"/>
            ergo t eſt inter duo pũcta f, h:</s>
            <s xml:id="echoid-s15582" xml:space="preserve"> erit linea z t inter duas lineas z f, z h.</s>
            <s xml:id="echoid-s15583" xml:space="preserve"> linea ergo z t ſecat lineam k l:</s>
            <s xml:id="echoid-s15584" xml:space="preserve"> ſe-
              <lb/>
            cet ergo lineam ipſam in i:</s>
            <s xml:id="echoid-s15585" xml:space="preserve"> i igitur eſt imago t [per 6 n 5] & t nullam habet imaginem niſi i.</s>
            <s xml:id="echoid-s15586" xml:space="preserve"> [quia ab
              <lb/>
            uno tantùm puncto peripheriæ f h fit reflexio per 73 n 5.</s>
            <s xml:id="echoid-s15587" xml:space="preserve">] Et ſic declarabitur, quòd imago cuiusli-
              <lb/>
            bet puncti lineę g r eſt punctum lineæ k l:</s>
            <s xml:id="echoid-s15588" xml:space="preserve"> k l ergo eſt imago g r:</s>
            <s xml:id="echoid-s15589" xml:space="preserve"> & k l eſt linea recta:</s>
            <s xml:id="echoid-s15590" xml:space="preserve"> quia eſt pars ſe-
              <lb/>
            midiametri circuli, a e:</s>
            <s xml:id="echoid-s15591" xml:space="preserve"> & g r eſt linea recta, quia eſt pars ſemidiametri circuli, o e.</s>
            <s xml:id="echoid-s15592" xml:space="preserve"> Ergo comprehen
              <lb/>
            dit formam g r rectè in ſpeculo ſphærico a b.</s>
            <s xml:id="echoid-s15593" xml:space="preserve"> Et hoc eſt quod uoluimus.</s>
            <s xml:id="echoid-s15594" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div526" type="section" level="0" n="0">
          <head xml:id="echoid-head462" xml:space="preserve" style="it">46. In ſpeculo ſphærico cauo imagines linearum: conuexæ, cauæ, aliquando uidentur cõuexæ,
            <lb/>
          cauæ: eadem́ obliquitate uiſum, qua ipſæ lineæ ſpeculum, reſpiciunt. 55 p 8.</head>
          <p>
            <s xml:id="echoid-s15595" xml:space="preserve">ET iteremus figurã, & conſtituamus ſuper lineam g r à duobus lateribus duos arcus, quomo-
              <lb/>
            docunq;</s>
            <s xml:id="echoid-s15596" xml:space="preserve"> ſint, ſcilicet g n r, g q r:</s>
            <s xml:id="echoid-s15597" xml:space="preserve"> & ſit arcus g n r non ſecans lineam g h:</s>
            <s xml:id="echoid-s15598" xml:space="preserve"> & ponamus in linea g r
              <lb/>
            punctum m, quomodocunq;</s>
            <s xml:id="echoid-s15599" xml:space="preserve"> ſit.</s>
            <s xml:id="echoid-s15600" xml:space="preserve"> Forma ergo m reflectitur ad z ex pũcto aliquo arcus f h [per
              <lb/>
            proximum numerum.</s>
            <s xml:id="echoid-s15601" xml:space="preserve">] reflectatur ergo ex t:</s>
            <s xml:id="echoid-s15602" xml:space="preserve"> & con
              <lb/>
              <figure xlink:label="fig-0226-01" xlink:href="fig-0226-01a" number="197">
                <variables xml:id="echoid-variables186" xml:space="preserve">t f h a p k l i d e z b n r m o g q</variables>
              </figure>
            tinuemus lineas z t, & m t.</s>
            <s xml:id="echoid-s15603" xml:space="preserve"> Duo ergo anguli z t e,
              <lb/>
            e t m ſunt æquales [per theſin & 12 n 4.</s>
            <s xml:id="echoid-s15604" xml:space="preserve">] Linea ergo
              <lb/>
            m t ſecabit arcũ g n r:</s>
            <s xml:id="echoid-s15605" xml:space="preserve"> ſecet ergo in n:</s>
            <s xml:id="echoid-s15606" xml:space="preserve"> & extrahamus
              <lb/>
            lineã t m in parte m:</s>
            <s xml:id="echoid-s15607" xml:space="preserve"> ſecabit ergo g q r:</s>
            <s xml:id="echoid-s15608" xml:space="preserve"> ſecet ergo in
              <lb/>
            pũcto q:</s>
            <s xml:id="echoid-s15609" xml:space="preserve"> & cõtinuemus n e:</s>
            <s xml:id="echoid-s15610" xml:space="preserve"> & extrahatur rectè:</s>
            <s xml:id="echoid-s15611" xml:space="preserve"> ſeca
              <lb/>
            bit ergo z t ſub linea k l:</s>
            <s xml:id="echoid-s15612" xml:space="preserve"> ſecet ergo illã in i:</s>
            <s xml:id="echoid-s15613" xml:space="preserve"> & cõtinue
              <lb/>
            mus q e:</s>
            <s xml:id="echoid-s15614" xml:space="preserve"> & extrahamus ipſam rectè:</s>
            <s xml:id="echoid-s15615" xml:space="preserve"> ſecabit ergo z t
              <lb/>
            ſupra k l:</s>
            <s xml:id="echoid-s15616" xml:space="preserve"> ſecet ergo ipſam in p.</s>
            <s xml:id="echoid-s15617" xml:space="preserve"> Quia ergo duo angu
              <lb/>
            li ad t ſunt æquales:</s>
            <s xml:id="echoid-s15618" xml:space="preserve"> [per theſin & 12 n 4] erit i ima
              <lb/>
            go n:</s>
            <s xml:id="echoid-s15619" xml:space="preserve"> [per 6 n 5] & duo puncta k, l imagines duo-
              <lb/>
            rum punctorum g, r.</s>
            <s xml:id="echoid-s15620" xml:space="preserve"> Imago ergo arcus g n r, eſt linea
              <lb/>
            tranſiens per puncta k, i, l, ut linea k i l.</s>
            <s xml:id="echoid-s15621" xml:space="preserve"> Sed linea k i l
              <lb/>
            eſt conuexa ex parte uiſus z:</s>
            <s xml:id="echoid-s15622" xml:space="preserve"> & arcus g n r eſt con-
              <lb/>
            uexus ex parte ſpeculi.</s>
            <s xml:id="echoid-s15623" xml:space="preserve"> Ergo uiſus z comprehendet
              <lb/>
            formam lineæ g n r conuexæ, lineam conuexam.</s>
            <s xml:id="echoid-s15624" xml:space="preserve"> Et
              <lb/>
            quia duo anguli apud t ſunt æquales [nimirũ p t e,
              <lb/>
            q t e per theſin & 12 n 4] erit p etiam imago q [per
              <lb/>
            6 n 5] & erit linea l p k ex parte uiſus cõcaua:</s>
            <s xml:id="echoid-s15625" xml:space="preserve"> & eſt imago arcus g q r, cõcaui ex parte ſuperficiei ſp
              <gap/>
              <lb/>
            culi.</s>
            <s xml:id="echoid-s15626" xml:space="preserve"> Ergo uiſus z comprehendet formam arcus g q r concaui, lineam concauam.</s>
            <s xml:id="echoid-s15627" xml:space="preserve"> In ſpeculis ergo
              <lb/>
            concauis ex quibuſdam ſitibus comprehenditur linea conuexa, conuexa:</s>
            <s xml:id="echoid-s15628" xml:space="preserve"> & concaua, concaua.</s>
            <s xml:id="echoid-s15629" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div528" type="section" level="0" n="0">
          <head xml:id="echoid-head463" xml:space="preserve" style="it">47. In ſpeculo ſphærico cauo lineæ: recta, & curua conuexa parte ſpeculum reſpiciens, habent
            <lb/>
          aliquando imagines curuas: recta quatuor: curua unam: omnes́ caua parte uiſum reſpi-
            <lb/>
          ciunt. 56 p 8.</head>
          <p>
            <s xml:id="echoid-s15630" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15631" xml:space="preserve"> ſit ſpeculum concauum:</s>
            <s xml:id="echoid-s15632" xml:space="preserve"> in quo ſit circulus a b d maximus:</s>
            <s xml:id="echoid-s15633" xml:space="preserve"> & centrum g:</s>
            <s xml:id="echoid-s15634" xml:space="preserve"> & extrahamus
              <lb/>
            lineam b g, quomodocunq;</s>
            <s xml:id="echoid-s15635" xml:space="preserve"> ſit:</s>
            <s xml:id="echoid-s15636" xml:space="preserve"> & diuidamus ex ipſa lineam g t maiorem medietate:</s>
            <s xml:id="echoid-s15637" xml:space="preserve"> & extraha-
              <lb/>
            mus ext lineam e t z perpendicularẽ ſuper b g:</s>
            <s xml:id="echoid-s15638" xml:space="preserve"> & ſit utraq;</s>
            <s xml:id="echoid-s15639" xml:space="preserve"> e t, t z æqualis t g [per 3 p 1.</s>
            <s xml:id="echoid-s15640" xml:space="preserve">] Et cõti-
              <lb/>
            nuemus e g, g z:</s>
            <s xml:id="echoid-s15641" xml:space="preserve"> & deſcribamus circa triangulũ e g z circulũ:</s>
            <s xml:id="echoid-s15642" xml:space="preserve"> [per 5 p 4] ſecabit ergo circulũ a b d in
              <lb/>
            duobus punctis:</s>
            <s xml:id="echoid-s15643" xml:space="preserve"> [per 10 p 3] nam punctũ t eſt centrũ huius circuli [per 9 p 3:</s>
            <s xml:id="echoid-s15644" xml:space="preserve"> æquatæ enim ſunt
              <lb/>
            rectæ e t, t z, t g] & t g eſt maior t b.</s>
            <s xml:id="echoid-s15645" xml:space="preserve"> Secet ergo circulus iſte circulum a b d in punctis a, d:</s>
            <s xml:id="echoid-s15646" xml:space="preserve"> & conti-
              <lb/>
            nuemus lineas g a, g d, e a, e b, e d, z a, z b, z d.</s>
            <s xml:id="echoid-s15647" xml:space="preserve"> Quia ergo duæ lineæ e t, t z ſunt æquales:</s>
            <s xml:id="echoid-s15648" xml:space="preserve"> erunt duæ
              <lb/>
            lineæ e g, g z æquales:</s>
            <s xml:id="echoid-s15649" xml:space="preserve"> [per 4 p 1:</s>
            <s xml:id="echoid-s15650" xml:space="preserve"> quia t g communis eſt, & anguli ad t per fabricationem recti ſunt]
              <lb/>
            & ſimiliter e b, b z æquales.</s>
            <s xml:id="echoid-s15651" xml:space="preserve"> Et quia duo arcus e g, g z ſunt æquales:</s>
            <s xml:id="echoid-s15652" xml:space="preserve"> [per 28 p 3:</s>
            <s xml:id="echoid-s15653" xml:space="preserve"> quia ſubtenduntur
              <lb/>
            a b æqualibus rectis e g, g z] duæ lineæ e a, a z reflectentur inter ſe propter angulos æquales [nam
              <lb/>
            anguli e a g, z a g per 27 p 3 æquantur] & duæ lineæ e b, b z reflectentur inter ſe propter angulos [
              <gap/>
              <lb/>
            b g, z b g] æquales [per 27 p 3.</s>
            <s xml:id="echoid-s15654" xml:space="preserve">] Et quia g t eſt maior quàm t b:</s>
            <s xml:id="echoid-s15655" xml:space="preserve"> [ex theſi] erit g e maior quàm e b.</s>
            <s xml:id="echoid-s15656" xml:space="preserve">
              <lb/>
            [Quia enim anguli ad t ſunt recti per fabricationem, æquabitur per 47 p 1 quadratum e g quadra-
              <lb/>
            tis g t, e t:</s>
            <s xml:id="echoid-s15657" xml:space="preserve"> item quadratum e b quadratis b t, e t:</s>
            <s xml:id="echoid-s15658" xml:space="preserve"> itaque cum quadratum g t ſit maius quadrato t b:</s>
            <s xml:id="echoid-s15659" xml:space="preserve">
              <lb/>
            quia g t maior eſt t b ex theſi:</s>
            <s xml:id="echoid-s15660" xml:space="preserve"> ſubducto communi e t:</s>
            <s xml:id="echoid-s15661" xml:space="preserve"> erit per 5 ax.</s>
            <s xml:id="echoid-s15662" xml:space="preserve"> quadratum e g maius quadra-
              <lb/>
            to e b:</s>
            <s xml:id="echoid-s15663" xml:space="preserve"> ideoq́ue latus e g maius latere e b.</s>
            <s xml:id="echoid-s15664" xml:space="preserve">] Angulus ergo e b g eſt maior angulo e g b [per 18 p 1] &
              <lb/>
            angulus e g b eſt ſemirectus.</s>
            <s xml:id="echoid-s15665" xml:space="preserve"> [Quia enim angulus ad t rectus eſt per fabricationem, & t e g, t g e æ-
              <lb/>
            quales per 5 p 1:</s>
            <s xml:id="echoid-s15666" xml:space="preserve"> quia e t, g t æquales poſitæ ſunt:</s>
            <s xml:id="echoid-s15667" xml:space="preserve"> erit eorum quilibet dimidius unius recti per 32 p 1.</s>
            <s xml:id="echoid-s15668" xml:space="preserve">]
              <lb/>
            Igitur duo anguli e g b, e b g ſimul ſunt maiores recto:</s>
            <s xml:id="echoid-s15669" xml:space="preserve"> ergo angulus b e g eſt recto minor:</s>
            <s xml:id="echoid-s15670" xml:space="preserve"> [ք 32 p 1]
              <lb/>
            & angulus e g z eſt rectus [ք 31 p 3.</s>
            <s xml:id="echoid-s15671" xml:space="preserve">] Ergo duæ lineæ e b, g z cõcurrẽt extra circulũ in parte b z [ք 11
              <lb/>
            ax.</s>
            <s xml:id="echoid-s15672" xml:space="preserve">] Cõcurrant ergo in l.</s>
            <s xml:id="echoid-s15673" xml:space="preserve"> Et quia e d eſt intra triangulũ l e g:</s>
            <s xml:id="echoid-s15674" xml:space="preserve"> cõcurret cũlinea g m:</s>
            <s xml:id="echoid-s15675" xml:space="preserve"> cõcurrat ergo in
              <lb/>
            m.</s>
            <s xml:id="echoid-s15676" xml:space="preserve"> Et quia g b trãſit per centrũ z e g circuli:</s>
            <s xml:id="echoid-s15677" xml:space="preserve"> erit portio a g minor ſemicirculo:</s>
            <s xml:id="echoid-s15678" xml:space="preserve"> ergo [ք 31 p 3] angulus
              <lb/>
            a e g eſt obtuſus, & angulus e g z eſt rectus.</s>
            <s xml:id="echoid-s15679" xml:space="preserve"> Ergo illæ duæ lineæ a e, z g cõcurrẽt in parte e g [erunt
              <lb/>
            enim anguli ad e & g dictis angulis deinceps, minores duobus rectis per 13 p 1.</s>
            <s xml:id="echoid-s15680" xml:space="preserve"> Quare cõcurrent ex
              <lb/>
            parte e g per 11 ax.</s>
            <s xml:id="echoid-s15681" xml:space="preserve">] Concurrant ergo in f.</s>
            <s xml:id="echoid-s15682" xml:space="preserve"> Si ergo uiſus fuerit in e, & z in aliquo uiſibili:</s>
            <s xml:id="echoid-s15683" xml:space="preserve"> tunc puncta
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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