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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[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
[51] t g p b h i z d a k s
[52] g z f h a b d c q e k ſ r
[53] m t n q h b f e z p d a g
[54] b ſ d h f r g z q t e a
[55] a d q c m x b g p o k t f z h
[56] a d k u m r h b g i l f e o z t y
[57] a d u m b g o e q s z h p
[58] a d u m c g b o t q p n z h
[59] b k a p f m e l z g t r o q h n d
[60] b k u a p e g t q n d
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page |< < (122) of 778 > >|
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        <div xml:id="echoid-div262" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s7154" xml:space="preserve">
              <pb o="122" file="0128" n="128" rhead="ALHAZEN"/>
            pyramidem illã in puncto contingentẽ, & fiet reflexio formæ per eandẽ lineam [per 11 n.</s>
            <s xml:id="echoid-s7155" xml:space="preserve">] Si aute
              <gap/>
              <lb/>
            deuiet à proceſſu perpẽdicularis:</s>
            <s xml:id="echoid-s7156" xml:space="preserve"> faciet quidẽ angulum cum perpendiculari acutũ in puncto ſum-
              <lb/>
            pto:</s>
            <s xml:id="echoid-s7157" xml:space="preserve"> & poterit produci in ſuperficie eius lineæ uiſualis, alia linea à puncto illo, quæ æqualẽ angulũ
              <lb/>
            huic teneat cum perpendiculari:</s>
            <s xml:id="echoid-s7158" xml:space="preserve"> cum perpendicularis orthogonalis ſit ſuper ſuperficiẽ contingen
              <lb/>
            tem.</s>
            <s xml:id="echoid-s7159" xml:space="preserve"> Linea autẽ quæcunq;</s>
            <s xml:id="echoid-s7160" xml:space="preserve"> ſuper ſuperficiem, contingentem in puncto ſumpto orthogonaliter ca-
              <lb/>
            dens, tranſit ad axem [per 11 a x:</s>
            <s xml:id="echoid-s7161" xml:space="preserve"> eſt enim perpẽdicularis conico lateri:</s>
            <s xml:id="echoid-s7162" xml:space="preserve"> quia, cum ex theſi ſit perpen-
              <lb/>
            dicularis plano conum tangenti in latere per 6 uel 35 n:</s>
            <s xml:id="echoid-s7163" xml:space="preserve"> erit per 3 d 11 ipſi lateri perpendicularis] &
              <lb/>
            ſi ab axe ducatur orthogonalis ad hanc ſuperficiem, efficient perpendiculares, interior & exterior,
              <lb/>
            lineã unam [per 14 p 1:</s>
            <s xml:id="echoid-s7164" xml:space="preserve">] quòd ſi non:</s>
            <s xml:id="echoid-s7165" xml:space="preserve"> cũ perpendicularis interior, extrà producta, ſit etiã perpendi-
              <lb/>
            cularis ſuper ſuperficiem:</s>
            <s xml:id="echoid-s7166" xml:space="preserve"> accidet ab eodẽ puncto ſuper aliquam ſuperficiem, erigi duas perpendi-
              <lb/>
            culares in eandẽ partem [contra 13 p 11.</s>
            <s xml:id="echoid-s7167" xml:space="preserve">] Palàm igitur, quòd à quocunq;</s>
            <s xml:id="echoid-s7168" xml:space="preserve"> puncto ſuperficiei pyrami-
              <lb/>
            dis uiſo, poteſt fieri reflexio ad paritatem angulorum.</s>
            <s xml:id="echoid-s7169" xml:space="preserve"> Et cum linea declinata occurrerit:</s>
            <s xml:id="echoid-s7170" xml:space="preserve"> forma ue-
              <lb/>
            niet ad ſpeculum ſuper lineam hanc, & reflectetur ad uiſum ſuper aliam:</s>
            <s xml:id="echoid-s7171" xml:space="preserve"> & ſunt hæ lineæ in eadem
              <lb/>
            ſuperficie orthogonali, ſuper ſuperficiem contingentem pyramidem in puncto reflexionis [per 6.</s>
            <s xml:id="echoid-s7172" xml:space="preserve">
              <lb/>
            13 n.</s>
            <s xml:id="echoid-s7173" xml:space="preserve">] Et hæc eſt ſuperficies reflexionis, in qua ſemper fit comprehenſio quatuor punctorũ, ſcilicet,
              <lb/>
            centri uiſus, puncti uiſi, puncti reflexionis, termini perpendicularis.</s>
            <s xml:id="echoid-s7174" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div264" type="section" level="0" n="0">
          <head xml:id="echoid-head288" xml:space="preserve" style="it">41. Communis ſectio ſuperficierum reflexionis & ſpeculi conici cõuexi eſt lat{us} conicum uel
            <lb/>
          ellipſis: nunquam uerò circul{us}. 12 p 7.</head>
          <p>
            <s xml:id="echoid-s7175" xml:space="preserve">DIuerſificantur autẽ lineæ cõmunes ſuperficiei reflexionis, & ſuperficiei pyramidis.</s>
            <s xml:id="echoid-s7176" xml:space="preserve"> Cũ enim
              <lb/>
            radius uiſualis continuus fuerit axi pyramidis, ſcilicet, cũ in ſuperficie reflexionis fuerit to-
              <lb/>
            tus axis, & perpendicularis ad axem tranſiens:</s>
            <s xml:id="echoid-s7177" xml:space="preserve"> erit ſuperficici reflexionis & ſuperficiei py-
              <lb/>
            ramidis cõmunis linea, linea longitudinis in hoc ſitu.</s>
            <s xml:id="echoid-s7178" xml:space="preserve"> Quoniã quælibet ſuperficies, in qua eſt totus
              <lb/>
            axis, hanchabet lineam communem cum ſuperficie pyramidis [ut patet è 18 d 11.</s>
            <s xml:id="echoid-s7179" xml:space="preserve">] Et in omni alio
              <lb/>
            ſitu unica longitudinis pyramidis linea erit communis, illa ſcilicet, quę fuerit in ſuperficie uiſus cẽ-
              <lb/>
            trum & axẽ continẽte.</s>
            <s xml:id="echoid-s7180" xml:space="preserve"> Et quãdo centrũ uiſus nõ erit in directo axis, una tantùm erit ſuperficies ta-
              <lb/>
            lis:</s>
            <s xml:id="echoid-s7181" xml:space="preserve"> & omnis alia cõmunis linea, erit ſectio pyramidalis, nõ circulus.</s>
            <s xml:id="echoid-s7182" xml:space="preserve"> Si enim fuerit circul
              <emph style="sub">9</emph>
            :</s>
            <s xml:id="echoid-s7183" xml:space="preserve"> erit ſuք-
              <lb/>
            ficies illi
              <emph style="sub">9</emph>
            circuli in ſuքficie reflexiõis.</s>
            <s xml:id="echoid-s7184" xml:space="preserve"> Et ꝗa axis orthogonalis eſt ſuք illũ circulũ [ք 18 d 11, & cõuer-
              <lb/>
            ſam 14 p 11] cũ quilibet circulus pyramidis ſit æquidiſtãs baſi [ք 4 th.</s>
            <s xml:id="echoid-s7185" xml:space="preserve"> 1 conicorũ Apollonij] erũt la-
              <lb/>
            tera pyramidis declinata ſuք circulũ:</s>
            <s xml:id="echoid-s7186" xml:space="preserve"> & ita ſuք ſuքficiẽ reflexiõis.</s>
            <s xml:id="echoid-s7187" xml:space="preserve"> Quare in ſuperficie illa nõ poteſt
              <lb/>
            duci քpẽdicularis ſuք lineã lõgitudinis pyramidis:</s>
            <s xml:id="echoid-s7188" xml:space="preserve"> ſed [ք 6.</s>
            <s xml:id="echoid-s7189" xml:space="preserve">13 n] քpendicularis ducta ſuք ſuքficiẽ,
              <lb/>
            cõtingẽtẽ locũ reflexiõis, eſt in ſuքficie reflexiõis, & քpẽdicularis ſuք lineã lõgitudinis [ut oſtẽſum
              <lb/>
            eſt ꝓximo numero] cũ q̃libet ſuքficies tãgẽs tãgat in linea lõgitudinis [ք 6.</s>
            <s xml:id="echoid-s7190" xml:space="preserve"> 35 n.</s>
            <s xml:id="echoid-s7191" xml:space="preserve">] Accidit igitur im
              <lb/>
            poſsibile [cõtra 13 n.</s>
            <s xml:id="echoid-s7192" xml:space="preserve">] Quare reſtat oẽs alias cõmunes reflexiõis lineas, ſectiões pyramidales eſſe.</s>
            <s xml:id="echoid-s7193" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div265" type="section" level="0" n="0">
          <head xml:id="echoid-head289" xml:space="preserve" style="it">42. Si communis ſectio ſuperficierum reflexionis & ſpeculi conici conuexi, fuerit lat{us} co-
            <lb/>
          nicum: reflexio à quocun ipſi{us} puncto facta, in eadem ſuperficie ſemper fiet. 19 p 7.</head>
          <p>
            <s xml:id="echoid-s7194" xml:space="preserve">ET cũ fuerit linea cõmunis, linea lõgitudinis, ex quocũ q;</s>
            <s xml:id="echoid-s7195" xml:space="preserve"> pũcto illius lineæ fiat reflexio:</s>
            <s xml:id="echoid-s7196" xml:space="preserve"> erit in
              <lb/>
            eadẽ ſuperficie cũ cuiuſcunq;</s>
            <s xml:id="echoid-s7197" xml:space="preserve"> alterius pũcti reflexione.</s>
            <s xml:id="echoid-s7198" xml:space="preserve"> Quoniã à quolibet huius lineæ pũcto
              <lb/>
            ducta perpẽdicularis cõtin get axẽ [ut oſtẽſum eſt 40 n:</s>
            <s xml:id="echoid-s7199" xml:space="preserve">] & erũt in ſuքficie reflexiõis cẽtrum
              <lb/>
            uiſus:</s>
            <s xml:id="echoid-s7200" xml:space="preserve"> & punctũ reflexionis:</s>
            <s xml:id="echoid-s7201" xml:space="preserve"> & punctũ axis.</s>
            <s xml:id="echoid-s7202" xml:space="preserve"> Quare in hac ſuperficie fit reflexio à quocunq;</s>
            <s xml:id="echoid-s7203" xml:space="preserve"> puncto.</s>
            <s xml:id="echoid-s7204" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div266" type="section" level="0" n="0">
          <head xml:id="echoid-head290" xml:space="preserve" style="it">43. Si cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi conici cõuexi fuerit ellipſis: ab uno uel
            <lb/>
          duob. cõſpicuæ ſuperficiei pũctis quib{us}libet, in eadẽ ſuքficie ad uiſum reflexio fieri poteſt. 34 p 7.</head>
          <p>
            <s xml:id="echoid-s7205" xml:space="preserve">SI uerò cõmunis linea nõ fuerit linea lõgitudinis:</s>
            <s xml:id="echoid-s7206" xml:space="preserve"> dico quòd uel
              <lb/>
              <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a" number="34">
                <variables xml:id="echoid-variables24" xml:space="preserve">f d d e r b g c h i p ſ q s n k</variables>
              </figure>
            ab uno cõmunis lineæ pũcto, in eadẽ ſuperficie fiat reflexio, uel
              <lb/>
            à duobus tantùm.</s>
            <s xml:id="echoid-s7207" xml:space="preserve"> Quoniã ducta perpendiculari à puncto refle-
              <lb/>
            xionis:</s>
            <s xml:id="echoid-s7208" xml:space="preserve"> perueniet ad axẽ, & cadet in aliquod punctũ eius [ut patuit
              <lb/>
            40 n:</s>
            <s xml:id="echoid-s7209" xml:space="preserve">] & intellecto circulo ſuper punctũ reflexionis, orthogonaliter
              <lb/>
            ſecabit circulus axem [Quia enim circulus parallelus eſt baſi per 4
              <lb/>
            th 1 conico.</s>
            <s xml:id="echoid-s7210" xml:space="preserve"> Apollonij:</s>
            <s xml:id="echoid-s7211" xml:space="preserve"> erit axis ad ipſum perpendicularis per 18 d, &
              <lb/>
            conuerſam 14 p 11.</s>
            <s xml:id="echoid-s7212" xml:space="preserve">] Et quia perpendicularis tenet angulum acutum
              <lb/>
            cum axe:</s>
            <s xml:id="echoid-s7213" xml:space="preserve"> erit perpẽdicularis declinata ſuper circulũ;</s>
            <s xml:id="echoid-s7214" xml:space="preserve"> & circumquaq;</s>
            <s xml:id="echoid-s7215" xml:space="preserve">
              <lb/>
            ducta, ſemper erit æqualis.</s>
            <s xml:id="echoid-s7216" xml:space="preserve"> Vnde fiet pyramis, cuius baſis circulus,
              <lb/>
            acumen punctum axis, in quod cadit perpendicularis.</s>
            <s xml:id="echoid-s7217" xml:space="preserve"> Igitur ſuper-
              <lb/>
            ficies reflexionis aut tanget hanc pyramidẽ, aut ſecabit.</s>
            <s xml:id="echoid-s7218" xml:space="preserve"> Si tangat:</s>
            <s xml:id="echoid-s7219" xml:space="preserve"> di-
              <lb/>
            co quòd à puncto reflexionis ſumpto poſsit tantùm fieri in eadẽ ſu-
              <lb/>
            perficie reflexio.</s>
            <s xml:id="echoid-s7220" xml:space="preserve"> Planũ enim, quòd ſuperficies reflexionis continget
              <lb/>
            hanc pyramidẽ ſuper perpendicularẽ, quæ eſt linea orthogonalis in
              <lb/>
            ſuperficie reflexionis [per 6.</s>
            <s xml:id="echoid-s7221" xml:space="preserve"> 35 n.</s>
            <s xml:id="echoid-s7222" xml:space="preserve">] Et ſi ab acumine totalis pyrami-
              <lb/>
            dis ducantur lineæ ad ſectionem communẽ ſuperficiei reflexionis &
              <lb/>
            pyramidis totalis, prius cadent in circulũ, qui eſt baſis pyramidιs in-
              <lb/>
            tellectæ, quàm in ſectionẽ:</s>
            <s xml:id="echoid-s7223" xml:space="preserve"> præter unã, quæ in punctũ reflexionis ca-
              <lb/>
            dit.</s>
            <s xml:id="echoid-s7224" xml:space="preserve"> Si ergo ab alio puncto cõmunis ſectionis fieret reflexio:</s>
            <s xml:id="echoid-s7225" xml:space="preserve"> linea ab
              <lb/>
            illo puncto ad acumen intellectæ pyramidis ducta:</s>
            <s xml:id="echoid-s7226" xml:space="preserve"> erit perpendicularis ſuper lineam longitudinis
              <lb/>
            </s>
          </p>
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