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

< >
[681] g y f l r k h p a c l d
[682] f y d b k t p x r z y g i e a
[683] d q f l y o m e d
[684] f h d m b k t p g i e o a
[685] f h d m u b k l s p t g i e o q z
[686] a d s f h f h b e g c b c
[687] x f h d m b k l t z s p u g i e n a q
[688] a q k b f g l n c e i d h
[689] a l f k b h d z g n q e o t s m i p
[690] t n q g z m b l f h r a d e k o
[691] b l a u f d c h n g r k s x q p
[692] l d a e p t m f k h i g a q o n b
[693] l d a e f z x y t u p r k o h y x m n q m i b c
[694] f a d e r c b y i h p s l n f q
[695] f a d e r c b g h p l s n k
[696] g m n b f q k l t s e p o h r a
[697] g m q n t e b r a
[698] z y v p d q b m n g t e f r h
[699] m n g f p i b a h e q t d k
[700] y z m q p a n y t e f r h
[701] a s t d k e h i o p u g m n b
[702] a o u m h z t s b c n d l e q f p
[703] n q t e l g o f m k d h c a s u b p z
[704] a e t b o f z h d g y k p b q
[705] a h l z x m o k e q d y p f b g
[706] e g d t m b u k h f q a c
[707] s f n h q x r p l z u t m a b o g e k d
[708] t n q g z m b f f h r a d e k o
[709] t i y n g z x q m b c l f h s a d p e k o u
[710] f d b g t e h e
< >
page |< < (282) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div1546" type="section" level="0" n="0">
          <head xml:id="echoid-head1176" xml:space="preserve" style="it">
            <pb o="282" file="0584" n="584" rhead="VITELLONIS OPTICAE"/>
          uexi: dati punctiuiſi ad datum centrum uiſus punctum reflexionis inuenire. Alhazen 48 n 9.</head>
          <p>
            <s xml:id="echoid-s37734" xml:space="preserve">Communi ſectione ſuperficiei reflexionis & ſpeculi propoſiti exiſtente linea longitudinis ſpe-
              <lb/>
            culi, punctus reflexionis poterit faciliter inueniri, ſicut in ſpeculis planis per 46 th.</s>
            <s xml:id="echoid-s37735" xml:space="preserve"> 5 huius oſten-
              <lb/>
            ſum eſt.</s>
            <s xml:id="echoid-s37736" xml:space="preserve"> Siuerò illa communis ſectio fuerit circulus:</s>
            <s xml:id="echoid-s37737" xml:space="preserve"> tunc punctus reflexionis poterit faciliter in-
              <lb/>
            ueniri, ſicut in ſpeculis ſphæricis conuexis oſtenſum eſt per 20 uel 22 th.</s>
            <s xml:id="echoid-s37738" xml:space="preserve"> 6 huius.</s>
            <s xml:id="echoid-s37739" xml:space="preserve"> Si autem illa com-
              <lb/>
            munis ſectio ſit oxygonia ſectio, qualis proponitur:</s>
            <s xml:id="echoid-s37740" xml:space="preserve"> ſit rei uiſæ datus punctus b, qui reflectatur ab
              <lb/>
            aliquo puncto ſectionis oxygoniæ ad a centrum uiſus.</s>
            <s xml:id="echoid-s37741" xml:space="preserve"> Dico quòd poſsibile eſt inueniri punctum
              <lb/>
            reflexionis.</s>
            <s xml:id="echoid-s37742" xml:space="preserve"> Ducatur enim à puncto a, utin præcedente propoſitione, ſuperficies æquidiftans baſi-
              <lb/>
            bus columnæ:</s>
            <s xml:id="echoid-s37743" xml:space="preserve"> quæ ſecabit columnam ſuper circulum, qui ſit e zi:</s>
            <s xml:id="echoid-s37744" xml:space="preserve"> & ducatur à puncto b perpendi-
              <lb/>
            cularis ſuper hanc ſuperficiem per 11 p 11, qu æ ſit b h, & per 20 uel 22 t 6 huius, ſicut in ſpeculis ſphę-
              <lb/>
            ricis conuexis oſtenſum eſt, inueniatur in hac ſuperficie punctus, àquo reflectitur forma puncti h
              <lb/>
            ad uiſum a, qui ſit pũctus z:</s>
            <s xml:id="echoid-s37745" xml:space="preserve"> & à puncto z per 101 th.</s>
            <s xml:id="echoid-s37746" xml:space="preserve"> 1 huius ducatur linea longitudinis, quæ ſit z g:</s>
            <s xml:id="echoid-s37747" xml:space="preserve"> &
              <lb/>
            ducatur linea h a:</s>
            <s xml:id="echoid-s37748" xml:space="preserve"> & à pũ cto z ducatur perpendicularis ſuper lineam h a per 12 p 1, quæ ſit z l:</s>
            <s xml:id="echoid-s37749" xml:space="preserve"> & huic
              <lb/>
            ducatur æquidiſtans à puncto a per 31 p 1, qu æ ſit a m:</s>
            <s xml:id="echoid-s37750" xml:space="preserve"> & linea h z producatur uſque quò concurrat
              <lb/>
            cum linea a m:</s>
            <s xml:id="echoid-s37751" xml:space="preserve"> & ſit cõcurſus in puncto m:</s>
            <s xml:id="echoid-s37752" xml:space="preserve"> & à puncto m ducatur linea ad punctum b, quæ neceſſa-
              <lb/>
            riò ſecabit lineam z g, cum ſit in eadem ſuperficie cum illa:</s>
            <s xml:id="echoid-s37753" xml:space="preserve"> quoniam cum linea b h ſit æquidiſtans
              <lb/>
            lineæ g z per 6 p 11, eò quòd amb æ lineæ b h & g z ſunt perpendiculares ſuper eandem ſuperficiem
              <lb/>
            e zi æ quidiſtantem baſibus column æ erit ergo linea h m in ſuperficie illarum per 7 p 11:</s>
            <s xml:id="echoid-s37754" xml:space="preserve"> & ita linea
              <lb/>
            m b erit in eadem ſuperficie:</s>
            <s xml:id="echoid-s37755" xml:space="preserve"> quæ ſi ſecuerit lineam z g in puncto g:</s>
            <s xml:id="echoid-s37756" xml:space="preserve"> palàm ex his, quę in præcedẽte
              <lb/>
            propoſitione præmiſſa ſunt, quòd punctus g erit punctus reflexionis formæ puncti b ad a uiſum.</s>
            <s xml:id="echoid-s37757" xml:space="preserve">
              <lb/>
            Hæc omnia pluraq́;</s>
            <s xml:id="echoid-s37758" xml:space="preserve"> alia patent per ea, quæ dicta ſunt in præcedente demõſtratione.</s>
            <s xml:id="echoid-s37759" xml:space="preserve"> Et hoc eſt pro-
              <lb/>
            poſitum:</s>
            <s xml:id="echoid-s37760" xml:space="preserve"> quoniam ſecundum hunc modum cuiuslibet dati punctiad datum uiſum punctus refle-
              <lb/>
            xionis poterit inueniri.</s>
            <s xml:id="echoid-s37761" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1547" type="section" level="0" n="0">
          <head xml:id="echoid-head1177" xml:space="preserve" style="it">30. Linea rectæ æquidiſt antis axi ſpeculi columnaris conuexi, uiſu non exiſtente in eadem
            <lb/>
          ſuperficie, reflexio fit à linea longitudinis ſpeculi ad uiſum. Alhazen 26 n 6.</head>
          <p>
            <s xml:id="echoid-s37762" xml:space="preserve">Eſto axis ſpeculi columnaris conuexi linea 3 k:</s>
            <s xml:id="echoid-s37763" xml:space="preserve"> & ſit linea uiſa axi æquidiſtans, quæ t h:</s>
            <s xml:id="echoid-s37764" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s37765" xml:space="preserve"> cen-
              <lb/>
            trum uiſus e extra ſuperſiciem t h z k.</s>
            <s xml:id="echoid-s37766" xml:space="preserve"> Dico quòd formà lineæ t h reflectitur ad uiſum e à linea lon-
              <lb/>
            gitudinis ſpeculi, quæ eſt communis fectio ſuperficiei t h z k, & ſuperficiel ſpeculi.</s>
            <s xml:id="echoid-s37767" xml:space="preserve"> Et quia uiſus e
              <lb/>
            non eſt in ſuperficie t h z k:</s>
            <s xml:id="echoid-s37768" xml:space="preserve"> ſit ſuperficies per ipſum uiſum tranſiens, ſecas columnam ſpeculi æqui-
              <lb/>
            diſtanter baſibus:</s>
            <s xml:id="echoid-s37769" xml:space="preserve"> eritq́, hæc ſuperficies ſecans columnam ſecundum circulum per 100 th 1 huius:</s>
            <s xml:id="echoid-s37770" xml:space="preserve">
              <lb/>
            qui circulus ſit b f.</s>
            <s xml:id="echoid-s37771" xml:space="preserve"> Palàm ergo cum linea h t ex hypotheſi æquidiſtet axi z k, quòd aliquis eius pun-
              <lb/>
            ctus reflectitur ad uiſum e ab aliquo puncto circuli b f:</s>
            <s xml:id="echoid-s37772" xml:space="preserve"> ſit ergo hoc à pũcto b.</s>
            <s xml:id="echoid-s37773" xml:space="preserve"> Punctus quoq;</s>
            <s xml:id="echoid-s37774" xml:space="preserve"> lineæ
              <lb/>
            th, qui reflectitur ad uiſum e à puncto ſpeculi b, ſit q:</s>
            <s xml:id="echoid-s37775" xml:space="preserve"> & ducantur lineæ q b, e b, q e:</s>
            <s xml:id="echoid-s37776" xml:space="preserve"> & ducatur per
              <lb/>
            101th.</s>
            <s xml:id="echoid-s37777" xml:space="preserve"> 1 huius à puncto b linea longitudinis columnæ quæ ſit a b g:</s>
            <s xml:id="echoid-s37778" xml:space="preserve"> & ducatur à puncto b perpen-
              <lb/>
            dicularis cadens ſuper axem z k in punctum 1:</s>
            <s xml:id="echoid-s37779" xml:space="preserve"> quæ producta ad lineam q e, ſecabit ipſam per 2 th.</s>
            <s xml:id="echoid-s37780" xml:space="preserve"> 1
              <lb/>
            huius:</s>
            <s xml:id="echoid-s37781" xml:space="preserve"> quoniam illæ duæ lineæ æquidiſtant, ut patet ex præmiſsis.</s>
            <s xml:id="echoid-s37782" xml:space="preserve"> Et quoniam ſuperſicies e q b eſt
              <lb/>
            ſuperficies reflexionis:</s>
            <s xml:id="echoid-s37783" xml:space="preserve"> patet quòd punctum b cum linea e q eſt in eadem ſuperficie.</s>
            <s xml:id="echoid-s37784" xml:space="preserve"> Secet ergo li-
              <lb/>
            nea b l producta ipſam lineam q e in puncto m:</s>
            <s xml:id="echoid-s37785" xml:space="preserve"> & ſit lineam l:</s>
            <s xml:id="echoid-s37786" xml:space="preserve"> ducaturq́;</s>
            <s xml:id="echoid-s37787" xml:space="preserve"> à puncto e linea æquidi-
              <lb/>
            ſtans lineæ m l per 31 p 1, quæſit e o:</s>
            <s xml:id="echoid-s37788" xml:space="preserve"> & produ-
              <lb/>
              <figure xlink:label="fig-0584-01" xlink:href="fig-0584-01a" number="690">
                <variables xml:id="echoid-variables668" xml:space="preserve">t n q g z m b l f h r a d e k o</variables>
              </figure>
            catur linea q b ultra punctũ b:</s>
            <s xml:id="echoid-s37789" xml:space="preserve"> quæ quia corl-
              <lb/>
            currit cũ linea m l:</s>
            <s xml:id="echoid-s37790" xml:space="preserve"> palàm per 2th.</s>
            <s xml:id="echoid-s37791" xml:space="preserve"> 1 huius quià
              <lb/>
            ipſa cõcurret cum eius æquidiſtante, quæ eſt
              <lb/>
            lineà e o:</s>
            <s xml:id="echoid-s37792" xml:space="preserve"> ſit ergo punctus cõcurſus o.</s>
            <s xml:id="echoid-s37793" xml:space="preserve"> Palàm
              <lb/>
            aũt per 20 th.</s>
            <s xml:id="echoid-s37794" xml:space="preserve"> 5 huius quoniam angulus inci-
              <lb/>
            dentiæ, qui eſt q b g, eſt æ qualis angulo refle-
              <lb/>
            xionis, qui eſt e b atanguli uerò m b g & m b a
              <lb/>
            funt æquales, quia recti:</s>
            <s xml:id="echoid-s37795" xml:space="preserve"> relin quitur ergo an-
              <lb/>
            gulus q b mæ qualis angulo reliquo, qui eſt e
              <lb/>
            b m:</s>
            <s xml:id="echoid-s37796" xml:space="preserve"> ſed per 29 p 21 angulus q b m eſt æ qualis
              <lb/>
            angulo b o e:</s>
            <s xml:id="echoid-s37797" xml:space="preserve"> quoniam extrinſecus intrinſe-
              <lb/>
            co eſt æ qualis:</s>
            <s xml:id="echoid-s37798" xml:space="preserve"> ſed & angulus m b e æqualis
              <lb/>
            eſt angulo b e o:</s>
            <s xml:id="echoid-s37799" xml:space="preserve"> quia coalternus eſt:</s>
            <s xml:id="echoid-s37800" xml:space="preserve"> ergo an-
              <lb/>
            gulus b o e æ qualis angulo b e o:</s>
            <s xml:id="echoid-s37801" xml:space="preserve"> ergo per 6 p
              <lb/>
            1 in trigono b e o latus b e eſt æ quale lateri b o.</s>
            <s xml:id="echoid-s37802" xml:space="preserve"> Sumatur autẽ & alius pũctus in linea th, qui ſit pun-
              <lb/>
            ctus t:</s>
            <s xml:id="echoid-s37803" xml:space="preserve"> & ducatur linea t o.</s>
            <s xml:id="echoid-s37804" xml:space="preserve"> Quia ergo linea th æquidiſtat lineæ longitudinis ſpeculi, quæ eſt a g per
              <lb/>
            30 p 1:</s>
            <s xml:id="echoid-s37805" xml:space="preserve"> ideòd quòd utraq;</s>
            <s xml:id="echoid-s37806" xml:space="preserve"> illarum eſt æquidiſtans axi z k:</s>
            <s xml:id="echoid-s37807" xml:space="preserve"> palàm ergo per 1 th.</s>
            <s xml:id="echoid-s37808" xml:space="preserve"> 1 huius quòd lineæ th &
              <lb/>
            a g ſunt in eadem ſuperficie, cum etiam linea t h & z k axis ſint in eadem ſuperficie.</s>
            <s xml:id="echoid-s37809" xml:space="preserve"> Ergo per 7 p 11
              <lb/>
            linea q b o ſecans illas lineas æ quidiſtantes, quę ſunt t h & a g, eſt cum illis in eadem ſuperficie:</s>
            <s xml:id="echoid-s37810" xml:space="preserve"> &
              <lb/>
            fimiliter linea t o eſt in eadem ſuperficie cum illis per 1 p 11:</s>
            <s xml:id="echoid-s37811" xml:space="preserve"> ſunt enim puncta t & o in dicta ſuper-
              <lb/>
            ficie:</s>
            <s xml:id="echoid-s37812" xml:space="preserve"> ſecabit ergo linea t o lineam a g:</s>
            <s xml:id="echoid-s37813" xml:space="preserve"> ſit punctus ſectionis g:</s>
            <s xml:id="echoid-s37814" xml:space="preserve"> & ducantur lineæ e g & e t.</s>
            <s xml:id="echoid-s37815" xml:space="preserve"> Quia itaq:</s>
            <s xml:id="echoid-s37816" xml:space="preserve">
              <lb/>
            a g, quę eſt linea longitudinis ſpeculis, eſt perpendicularis ſuper ſuperficiem circuli b f per 8 p 11:</s>
            <s xml:id="echoid-s37817" xml:space="preserve">
              <lb/>
            ideo quòd axis z k, cui æquidiſtat linea a g, perpendicularis eſt ſuper eandem circuli ſuper-
              <lb/>
            ficiem per 23 th.</s>
            <s xml:id="echoid-s37818" xml:space="preserve"> 1 huius, cum ipſa ſit perpendicularis ſuper baſim columnæ per 92 th:</s>
            <s xml:id="echoid-s37819" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s37820" xml:space="preserve">
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>