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

Table of figures

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[21] p k c z q x y b
[Figure 22]
[Figure 23]
[24] e d f a c b
[25] a s b c
[26] a k f s d m b g c h
[27] a e g c b d h f
[28] a b f g c d n
[29] b a f l g e k h n d c
[30] a b e c f h g r i d m
[31] a b h c
[32] a d b k ſ c
[33] b ſ a u f d c h n g r k s x q p
[34] f d d e r b g c h i p ſ q s n k
[35] f a r d e b g c h p ſ s n k
[36] ſ g d f h b a
[37] a d f t e b
[38] d b c e f g b d
[39] a f b c d e
[40] a f b c d e g
[41] h t a d ſ s g k b e
[42] a b h e d z
[43] b a g q t d z e h
[44] a g b e d z t q h
[45] b g a t z d h
[46] a h b e g p d z n q
[47] h a b e g p d z n q
[48] a h b e g p f d z n q
[49] x e g k z a d
[50] g m h z p b d a k
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          <p>
            <s xml:id="echoid-s7780" xml:space="preserve">
              <pb o="132" file="0138" n="138" rhead="ALHAZEN"/>
            angulus b d e æqualis eſt angulo a d c [per 10 n 4, & per 15 p 1 angulus b d e æqualis angulo g d c:</s>
            <s xml:id="echoid-s7781" xml:space="preserve"> er-
              <lb/>
            go per 1 ax:</s>
            <s xml:id="echoid-s7782" xml:space="preserve"> angulus a d c æquatur angulo g d c] & angulus a c d æ-
              <lb/>
              <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a" number="40">
                <variables xml:id="echoid-variables30" xml:space="preserve">a f b c d e g</variables>
              </figure>
            qualis angulo g c d [per 10 ax:</s>
            <s xml:id="echoid-s7783" xml:space="preserve">] & latus c d commune.</s>
            <s xml:id="echoid-s7784" xml:space="preserve"> Quare [per 26
              <lb/>
            p 1] triangulum æquale triangulo.</s>
            <s xml:id="echoid-s7785" xml:space="preserve"> Quare g c æqualis a c.</s>
            <s xml:id="echoid-s7786" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div293" type="section" level="0" n="0">
          <head xml:id="echoid-head315" xml:space="preserve" style="it">12. Viſu & uiſibili datis, in ſpeculo plano punctum reflexionis
            <lb/>
          inuenire. 46 p 5.</head>
          <p>
            <s xml:id="echoid-s7787" xml:space="preserve">ET ſi uoluerimus per perpendicularem inuenire locum reflexio
              <lb/>
            nis:</s>
            <s xml:id="echoid-s7788" xml:space="preserve"> ſecetur ex perpendiculari ultra ſpeculum pars, æqualis par
              <lb/>
            ti eius uſq;</s>
            <s xml:id="echoid-s7789" xml:space="preserve"> ad ſpeculum:</s>
            <s xml:id="echoid-s7790" xml:space="preserve"> & eſt, ut ſit g c æqualis a c:</s>
            <s xml:id="echoid-s7791" xml:space="preserve"> & ducatur li
              <lb/>
            nea à centro uiſus ad punctum g, quæ ſit b d g.</s>
            <s xml:id="echoid-s7792" xml:space="preserve"> Dico, quòd d, eſt pun-
              <lb/>
            ctum reflexionis.</s>
            <s xml:id="echoid-s7793" xml:space="preserve"> Quoniam enim [per fabricationem & 2 ax:</s>
            <s xml:id="echoid-s7794" xml:space="preserve">] a c &
              <lb/>
            c d ſunt æqualia c g & c d, & angulus angulo [a c d ipſi g c d per theſin
              <lb/>
            & 10 ax.</s>
            <s xml:id="echoid-s7795" xml:space="preserve">] Ergo [per 4 p 1] triangulum triangulo.</s>
            <s xml:id="echoid-s7796" xml:space="preserve"> Igitur angulus g d c
              <lb/>
            eſt æqualis angulo a d c:</s>
            <s xml:id="echoid-s7797" xml:space="preserve"> Sed g d c eſt æqualis angulo b d e [per 15 p 1]
              <lb/>
            reſtat ergo [per 1 ax] ut angulus b d e ſit æqualis angulo a d c.</s>
            <s xml:id="echoid-s7798" xml:space="preserve"> Et ita
              <lb/>
            [per 10 n 4] d eſt punctum reflexionis:</s>
            <s xml:id="echoid-s7799" xml:space="preserve"> & ita patet propoſitum.</s>
            <s xml:id="echoid-s7800" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div294" type="section" level="0" n="0">
          <head xml:id="echoid-head316" xml:space="preserve" style="it">13. Si recta linea ab uno uiſu ſit perpendicularis ſpeculo plano,
            <lb/>
          unum ipſi{us} punctũ; in quo uiſ{us} ſuperficiem ſecat, ab uno ſpeculi
            <lb/>
          puncto, in quod cadit, ad eundem uiſum reflectetur. 32 p 5.</head>
          <p>
            <s xml:id="echoid-s7801" xml:space="preserve">SIt a centrum uiſus:</s>
            <s xml:id="echoid-s7802" xml:space="preserve"> & a g perpendicularis ſuper ſpeculũ planũ:</s>
            <s xml:id="echoid-s7803" xml:space="preserve"> & d ſecet hanc perpendicularẽ in
              <lb/>
            ſuperficie oculi.</s>
            <s xml:id="echoid-s7804" xml:space="preserve"> Dico, quòd in hac perpendiculari non eſt punctũ, quod reflectatur ab hoc ſpe-
              <lb/>
            culo ad uiſum, præter d.</s>
            <s xml:id="echoid-s7805" xml:space="preserve"> Sin autem:</s>
            <s xml:id="echoid-s7806" xml:space="preserve"> ſumatur ultra uiſum punctum in hac perpendiculari:</s>
            <s xml:id="echoid-s7807" xml:space="preserve"> & ſit
              <lb/>
            h:</s>
            <s xml:id="echoid-s7808" xml:space="preserve"> Non iam perueniet forma eius ad ſpeculũ ſuper
              <lb/>
              <figure xlink:label="fig-0138-02" xlink:href="fig-0138-02a" number="41">
                <variables xml:id="echoid-variables31" xml:space="preserve">h t a d ſ s g k b e</variables>
              </figure>
            perpendicularẽ a h, propter ſolidi corporis inter-
              <lb/>
            poſitionem:</s>
            <s xml:id="echoid-s7809" xml:space="preserve"> & ita nõ reflectetur forma eius ſuper
              <lb/>
            perpendicularẽ.</s>
            <s xml:id="echoid-s7810" xml:space="preserve"> Et ſi dicatur, quòd ab alio puncto
              <lb/>
            ſpeculi poſsit reflecti:</s>
            <s xml:id="echoid-s7811" xml:space="preserve"> ſit illud b.</s>
            <s xml:id="echoid-s7812" xml:space="preserve"> Mouebitur quidẽ
              <lb/>
            forma eius ad punctũ b per lineã h b:</s>
            <s xml:id="echoid-s7813" xml:space="preserve"> & reflectetur
              <lb/>
            per lineam b a.</s>
            <s xml:id="echoid-s7814" xml:space="preserve"> Diuidatur angulus h b a [per 9 p 1]
              <lb/>
            per ęqualia, per lineã t b.</s>
            <s xml:id="echoid-s7815" xml:space="preserve"> Igitur erit perpẽdicularis
              <lb/>
            ſuper ſuperficiẽ ſpeculi.</s>
            <s xml:id="echoid-s7816" xml:space="preserve"> [Quia enim angulus h b c
              <lb/>
            æquatur angulo a b g ք theſin & 10 n 4, & h b t ipſi
              <lb/>
            a b t per fabricationẽ:</s>
            <s xml:id="echoid-s7817" xml:space="preserve"> totus t b c æquabitur toti t b
              <lb/>
            g.</s>
            <s xml:id="echoid-s7818" xml:space="preserve"> quare per 10 d 1 t b eſt perpendicularis ipſi g c cõ
              <lb/>
            muni ſectioni ſuperficierũ reflexionis & ſpeculi.</s>
            <s xml:id="echoid-s7819" xml:space="preserve">
              <lb/>
            Itaq;</s>
            <s xml:id="echoid-s7820" xml:space="preserve"> cũ reflexiõis ſuperficies, in qua eſt t b, ſit per
              <lb/>
            pendicularis ſuperficiei ſpeculi per 13 n 4:</s>
            <s xml:id="echoid-s7821" xml:space="preserve"> erit t b
              <lb/>
            քpẽdicularis ſuperficiei ſpeculi per cõuerſam 4 d
              <lb/>
            11] ſed [per hypotheſin] t g eſt perpẽdicularis ſuper
              <lb/>
            eandẽ.</s>
            <s xml:id="echoid-s7822" xml:space="preserve"> Quare ab eodẽ puncto eſt ducere duas per
              <lb/>
            pendiculares ad ſuperficiem ſpeculi, quod eſt im-
              <lb/>
            poſsibile:</s>
            <s xml:id="echoid-s7823" xml:space="preserve"> [ſic enim tres interiores anguli triangu-
              <lb/>
            li eſſent maiores duobus rectis, cõtra 32 p 1.</s>
            <s xml:id="echoid-s7824" xml:space="preserve">] Eadẽ
              <lb/>
            erit probatio, quòd forma puncti d nõ poteſt refle
              <lb/>
            cti ab alio ſpeculi puncto, quam à puncto g.</s>
            <s xml:id="echoid-s7825" xml:space="preserve"> Quare
              <lb/>
            non reflectitur, niſi ſuper perpendicularẽ d g.</s>
            <s xml:id="echoid-s7826" xml:space="preserve"> Pun
              <lb/>
            ctum aũt in hac perpendiculari ſumptum inter g & d:</s>
            <s xml:id="echoid-s7827" xml:space="preserve"> ſi dicatur formã per reflexionẽ ad uiſum mit-
              <lb/>
            tere:</s>
            <s xml:id="echoid-s7828" xml:space="preserve"> improbo.</s>
            <s xml:id="echoid-s7829" xml:space="preserve"> Quoniã aut erit corpus ſolidum, aut rarũ.</s>
            <s xml:id="echoid-s7830" xml:space="preserve"> Si ſolidum, procedet ſecundum perpendi-
              <lb/>
            cularem forma eius ad ſpeculum, & regredietur ſecundũ eandem uſq;</s>
            <s xml:id="echoid-s7831" xml:space="preserve"> ad ipſum, [per 11 n 4] & pro-
              <lb/>
            pter ſoliditatẽ non poterit tranſire, & ad uiſum peruenire.</s>
            <s xml:id="echoid-s7832" xml:space="preserve"> Si aũt punctum illud fuerit rarum:</s>
            <s xml:id="echoid-s7833" xml:space="preserve"> forma
              <lb/>
            eius regrediẽs à ſpeculo ſuper perpendicularẽ miſcebitur ei, & adhærebit, nec reflectetur ad uiſum.</s>
            <s xml:id="echoid-s7834" xml:space="preserve">
              <lb/>
            Quòd autem forma cuiuſcunq;</s>
            <s xml:id="echoid-s7835" xml:space="preserve"> puncti in hac perpendiculari inter g & d ſumpti non poſsit ab alio
              <lb/>
            puncto ſpeculi ad uiſum reflecti, modo ſuprà dicto poteſt probari.</s>
            <s xml:id="echoid-s7836" xml:space="preserve"> Similiter forma puncti inter a &
              <lb/>
            d ſumpti non reflectitur ad uiſum per perpendicularem, nec per aliam.</s>
            <s xml:id="echoid-s7837" xml:space="preserve"> Quoniã puncta inter centrũ
              <lb/>
            uiſus & ſuperficiem eius interpoſita ſunt ualde rara.</s>
            <s xml:id="echoid-s7838" xml:space="preserve"> Vnde nec mittitur eorum forma, nec reflecti-
              <lb/>
            tur, ut ſentiatur.</s>
            <s xml:id="echoid-s7839" xml:space="preserve"> Et quoniám quodlibet punctum, præter d in ſuperficie uiſus ſumptum:</s>
            <s xml:id="echoid-s7840" xml:space="preserve"> opponitur
              <lb/>
            ſpeculo, non ad rectum angulum, uidebitur quodlibet ſuper perpendicularem ab eo ad ſpeculum
              <lb/>
            ductam, & imago eius ultra ſpeculum æquè diſtans à ſuperficie, ſicut ipſum punctum [per 11 n.</s>
            <s xml:id="echoid-s7841" xml:space="preserve">] Et
              <lb/>
            quoniam d uidetur continuum cum alijs ſuperficiei uiſus punctis, & imago eius cõtinua cum alijs
              <lb/>
            imaginibus:</s>
            <s xml:id="echoid-s7842" xml:space="preserve"> uidebitur imago d tantùm diſtans à ſuperficiei ſpeculi, quantùm diſtat d ab eadem.</s>
            <s xml:id="echoid-s7843" xml:space="preserve"> Pa-
              <lb/>
            làm ergo, quòd cuiuſcunq;</s>
            <s xml:id="echoid-s7844" xml:space="preserve"> puncti in ſpeculo uiſi imago uidebitur ſuper perpendicularem:</s>
            <s xml:id="echoid-s7845" xml:space="preserve"> & elon-
              <lb/>
            gatio imaginis, & uiſi corporis à ſuperficie ſpeculi eſt eadem.</s>
            <s xml:id="echoid-s7846" xml:space="preserve"/>
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