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 |< < (295) of 778 > >|
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          <p>
            <s xml:id="echoid-s38763" xml:space="preserve">
              <pb o="295" file="0597" n="597" rhead="LIBER SEPTIMVS."/>
            enim linea e d:</s>
            <s xml:id="echoid-s38764" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s38765" xml:space="preserve"> ita, ut fiat e d b angulus acutus:</s>
            <s xml:id="echoid-s38766" xml:space="preserve"> ſit ergo q e l linea contingens ſectionem in
              <lb/>
            puncto e:</s>
            <s xml:id="echoid-s38767" xml:space="preserve"> & ſuper punctum ſectιonis b fiat circulus æquidιſtans baſibus ſpeculi per 102 th.</s>
            <s xml:id="echoid-s38768" xml:space="preserve"> 1 huius,
              <lb/>
            qui ſit b t o:</s>
            <s xml:id="echoid-s38769" xml:space="preserve"> cuius centrũ ſit d:</s>
            <s xml:id="echoid-s38770" xml:space="preserve"> & ducatur à pũcto e linea longitudinis ſpeculi per 101 th.</s>
            <s xml:id="echoid-s38771" xml:space="preserve"> 1 huius, quæ
              <lb/>
            ſit e t.</s>
            <s xml:id="echoid-s38772" xml:space="preserve"> A puncto quoq;</s>
            <s xml:id="echoid-s38773" xml:space="preserve"> d per 11 p 1 ducatur linea d g per pendicularis ſuper lineam b d in ipſa circu-
              <lb/>
            li ſuperficie.</s>
            <s xml:id="echoid-s38774" xml:space="preserve"> Palàm ergo quod ſuperficies h d g (cum per axem h k tranſeat, qui per 92 th.</s>
            <s xml:id="echoid-s38775" xml:space="preserve"> 1 huius
              <lb/>
            eſt erectus ſuper circuli ſuperficiem) perpendicularis eſt ſuper eandem circuli ſuperficiem per
              <lb/>
            18 p 11.</s>
            <s xml:id="echoid-s38776" xml:space="preserve"> Superficies uerò contingens ſpeculum in puncto b, erit æquidiſtans ſuperficiei h d g ſpecu-
              <lb/>
            lum ſecanti.</s>
            <s xml:id="echoid-s38777" xml:space="preserve"> Ideo enim quia linea longitudinis ſpeculi ducta à puncto b eſt æquidiſtans axi h k, & li
              <lb/>
            nea cιrculum b t o contιngens ſuper punctum b, eſt æquidiſtans lineæ g d per 29 p 1:</s>
            <s xml:id="echoid-s38778" xml:space="preserve"> angulus enim
              <lb/>
            g d b eſt rectus, ut patet ex pręmiſsis, & angulus contentus ſub linea d b & ſub linea contingente
              <lb/>
            cιrculum in pũcto b eſt rectus per 18 p 3:</s>
            <s xml:id="echoid-s38779" xml:space="preserve"> ergo
              <lb/>
              <figure xlink:label="fig-0597-01" xlink:href="fig-0597-01a" number="703">
                <variables xml:id="echoid-variables681" xml:space="preserve">n q t e l g o f m k d h c a s u b p z</variables>
              </figure>
            illæ ſuperficies æquidiſtant per 14 p 11.</s>
            <s xml:id="echoid-s38780" xml:space="preserve"> Igitur
              <lb/>
            ſuperficies, in qua ſunt lineæ l e & e t, non eſt
              <lb/>
            æquidιſtans ſuperficiei h d g:</s>
            <s xml:id="echoid-s38781" xml:space="preserve"> quod patet per
              <lb/>
            24 th.</s>
            <s xml:id="echoid-s38782" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s38783" xml:space="preserve"> quoniam ſuperficies contin-
              <lb/>
            gens ſectionem oxygoniam in puncto b non
              <lb/>
            eſt æquidiſtans ſuperficiei contingenti ean-
              <lb/>
            dem ſectionem in puncto e, in quo ſunt linea
              <lb/>
            l e q contingens ſectionem, & linea longitu-
              <lb/>
            dinis, quæ eſt e t.</s>
            <s xml:id="echoid-s38784" xml:space="preserve"> Angulus enim e d b, ut pa-
              <lb/>
            tet ex hypotheſi, eſt acutus:</s>
            <s xml:id="echoid-s38785" xml:space="preserve"> ſuperficies ergo
              <lb/>
            h d g non æquidiſtat ſuperficiei l e t:</s>
            <s xml:id="echoid-s38786" xml:space="preserve"> ergo con
              <lb/>
            curret cum illa:</s>
            <s xml:id="echoid-s38787" xml:space="preserve"> concurrat ergo in linea l g & ducatur linea g t:</s>
            <s xml:id="echoid-s38788" xml:space="preserve"> quæ neceſſariò erit contingens
              <lb/>
            circulum b t o:</s>
            <s xml:id="echoid-s38789" xml:space="preserve"> cum ſuperficies, in quá ducitur linea g t, ipſum ſpeculum ſit contingens.</s>
            <s xml:id="echoid-s38790" xml:space="preserve"> Ducta
              <lb/>
            autem linea t d, erit angulus g t d rectus per 18 p 3:</s>
            <s xml:id="echoid-s38791" xml:space="preserve"> quoniam linea t d eſt diameter circuli & li-
              <lb/>
            nea g t contingit ιllum cιrculum in puncto t.</s>
            <s xml:id="echoid-s38792" xml:space="preserve"> Fiat quo que, ut prius, ſuper e punctum ſectionis
              <lb/>
            circulus æquidiſtans baſibus ſpeculi, qui ſit e s z p:</s>
            <s xml:id="echoid-s38793" xml:space="preserve"> & centrum huius circuli ſit punctus axιs, qui
              <lb/>
            k:</s>
            <s xml:id="echoid-s38794" xml:space="preserve"> & ducatur linea k e:</s>
            <s xml:id="echoid-s38795" xml:space="preserve"> & ducatur etιam linea d l:</s>
            <s xml:id="echoid-s38796" xml:space="preserve"> quæ quidem ſecabit ſuperficιem circuli e s p:</s>
            <s xml:id="echoid-s38797" xml:space="preserve">
              <lb/>
            ſecet ergo illam in puncto f Quia ιtaque punctum d eſt in ſuperficie ſectιonis per 24 huius:</s>
            <s xml:id="echoid-s38798" xml:space="preserve"> cum
              <lb/>
            ipſa ſectionis ſuperficies ſit ſuperficies reflexionis, & punctum l, quod eſt punctum lineæ contin-
              <lb/>
            gentιs ſectιonem, eſtιn eadem ſuperficie ſectionis:</s>
            <s xml:id="echoid-s38799" xml:space="preserve"> ergo per 1 p 11 tota linea d l eſt in ſuperficie ſe-
              <lb/>
            ctionis:</s>
            <s xml:id="echoid-s38800" xml:space="preserve"> punctum ergo f eſt in ſuperficie ſectionis:</s>
            <s xml:id="echoid-s38801" xml:space="preserve"> ſed ipſum eſt in ſuperficie circuli e z p:</s>
            <s xml:id="echoid-s38802" xml:space="preserve"> eſt er-
              <lb/>
            goin communi ſectione illarum ſuperficierũ, circuli ſcilicet & ſectionis:</s>
            <s xml:id="echoid-s38803" xml:space="preserve"> ſed & punctum e eſt in am
              <lb/>
            babus eιſdem ſuperficιebus:</s>
            <s xml:id="echoid-s38804" xml:space="preserve"> ergo item per 1 p 11 linea e f ducta erit in ambabus illis ſuperficiebus:</s>
            <s xml:id="echoid-s38805" xml:space="preserve">
              <lb/>
            ergo per 19 th.</s>
            <s xml:id="echoid-s38806" xml:space="preserve"> 1 huius ſecundum lineam e f ſecant ſe ſuperficies ſectionis & circuli e z p.</s>
            <s xml:id="echoid-s38807" xml:space="preserve"> Duca-
              <lb/>
            tur ιtaque lιnea k f:</s>
            <s xml:id="echoid-s38808" xml:space="preserve"> & a puncto f ducatur perpendicularιs ſuper ſuperficiem circuli b t o per 11
              <lb/>
            p 11, quæ ſit f m:</s>
            <s xml:id="echoid-s38809" xml:space="preserve"> cadetq́;</s>
            <s xml:id="echoid-s38810" xml:space="preserve"> punctus m in linea d g, ut patet:</s>
            <s xml:id="echoid-s38811" xml:space="preserve"> & ducatur linea t m.</s>
            <s xml:id="echoid-s38812" xml:space="preserve"> Palàm quoniam
              <lb/>
            linea k d æquidιſtans & æqualis eſt lineæ f m per 25 th.</s>
            <s xml:id="echoid-s38813" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s38814" xml:space="preserve"> ſunt enim lineæ k d & f m am-
              <lb/>
            bæ perpendιculares ſuper ſuperficiem circuli b t o:</s>
            <s xml:id="echoid-s38815" xml:space="preserve"> quia illi circuli æquidιſtant per 24 th.</s>
            <s xml:id="echoid-s38816" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s38817" xml:space="preserve">
              <lb/>
            utraque enιm ipſarum æquidιſtat baſibus columnæ per 100 th.</s>
            <s xml:id="echoid-s38818" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s38819" xml:space="preserve"> Quoniam ergo linea f m
              <lb/>
            eſt æqualιs & æquidiſtans lineæ d k, quæ eſt pars axis:</s>
            <s xml:id="echoid-s38820" xml:space="preserve"> ergo per 33 p 1 linea k f æqualis & æ-
              <lb/>
            quιdiſtans eſt lineæ d m:</s>
            <s xml:id="echoid-s38821" xml:space="preserve"> & ſimiliter erit linea f m æqualis & æquidiſtans lineæ longitudinis,
              <lb/>
            quę eſt e t, per 33 p 1:</s>
            <s xml:id="echoid-s38822" xml:space="preserve"> quoniam linea e t eſt æqualis & æquidiſtans axi k d per 92 th.</s>
            <s xml:id="echoid-s38823" xml:space="preserve"> 1 huius,
              <lb/>
            cum ſιt lιnea longitudinis ſpeculi:</s>
            <s xml:id="echoid-s38824" xml:space="preserve"> & erit, ut prius, linea k e æqualιs & æquidiſtans lineæ d t, &
              <lb/>
            linea e f æqualis eſt & æquidiſtans lineæ t m per eandem 33 p 1.</s>
            <s xml:id="echoid-s38825" xml:space="preserve"> Verùm etiam ſuperficies k d l g
              <lb/>
            (quιa tranſit axem columnæ, & angulus g d b eſt rectus) orthogonalis eſt ſuper ſuperficiem ſe-
              <lb/>
            ctionis oxygoniæ, quę eſt a e b c per definitionem ſuperficiei erectę:</s>
            <s xml:id="echoid-s38826" xml:space="preserve"> & eadem ſuperficies k d l g
              <lb/>
            orthogonalιs eſt ſuper ſuperficiem circuli e s p:</s>
            <s xml:id="echoid-s38827" xml:space="preserve"> quoniam illa ſuperficies k d l tranſiens per axem,
              <lb/>
            per 18 p 11 erecta eſt ſuper baſes columnæ:</s>
            <s xml:id="echoid-s38828" xml:space="preserve"> ergo & ſuper ſuperficiem circuli e s p, æquιdiſtantis
              <lb/>
            baſibus erecta eſt eadem ſuperficies k d l.</s>
            <s xml:id="echoid-s38829" xml:space="preserve"> Quia itaque dicta ſuperficies k d l eſt erecta ſuper ſu-
              <lb/>
            perficiem ſectionis oxygonię & circuli e s p:</s>
            <s xml:id="echoid-s38830" xml:space="preserve"> eſt ergo orthogonalιs ſuper lineam communem di-
              <lb/>
            ctę ſectιonιs & circuli (quę eſt linea e f) per 19 p 11 Et quia linea e f eſt erecta ſuper ſuperficiem
              <lb/>
            k d l, in qua ducta eſt lιnea k f:</s>
            <s xml:id="echoid-s38831" xml:space="preserve"> igitur per definitionem lineę ſuper ſuperficiem erectę angulus
              <lb/>
            e f k eſt rectus:</s>
            <s xml:id="echoid-s38832" xml:space="preserve"> ergo & angulus t m d eſt rectus per 10 p 11:</s>
            <s xml:id="echoid-s38833" xml:space="preserve"> latera enim illos angulos continen-
              <lb/>
            tia in æquιdιſtantιbus circulorum ſuperficiebus protracta æqualia ſunt & æquidiſtantia, ut pa-
              <lb/>
            tet ex pręmiſsis.</s>
            <s xml:id="echoid-s38834" xml:space="preserve"> Cum ergo angulus d m t ſit rectus, & angulus g t d ſit rectus per 18 p 3:</s>
            <s xml:id="echoid-s38835" xml:space="preserve"> in trigono
              <lb/>
            ergo orthogonιo d t g ducta eſt ab angulo ad baſim perpendicularis t m:</s>
            <s xml:id="echoid-s38836" xml:space="preserve"> ergo per 8 & 17 p 6
              <lb/>
            illud, quod ſit ex ductu lιneæ d m in g m eſt æquale quadrato lineæ m t.</s>
            <s xml:id="echoid-s38837" xml:space="preserve"> Et quoniam linea g t
              <lb/>
            contingit cιrculum b t o, cum ſit in ſuperficιe contingente ducta ad punctum contingentię,
              <lb/>
            quod eſt t:</s>
            <s xml:id="echoid-s38838" xml:space="preserve"> palàm quòd linea l g eſt ęquidiſtans axi k d.</s>
            <s xml:id="echoid-s38839" xml:space="preserve"> Quoniam enim ſuperficies lecun-
              <lb/>
            dum lineam longιtudinis ſpeculum contingentes ſunt erectę ſuper baſium columnę ſuperfi-
              <lb/>
            cιes:</s>
            <s xml:id="echoid-s38840" xml:space="preserve"> ergo per 19 p 11 earum communis ſectio, quę in propoſito eſt linea l g, ſuper eaſdem ſu-
              <lb/>
            perficιes baſium perpendιcularis erιt:</s>
            <s xml:id="echoid-s38841" xml:space="preserve"> ęquidiſtabit ergo axi h k per 6 p 11:</s>
            <s xml:id="echoid-s38842" xml:space="preserve"> ergo etiam ęquidi-
              <lb/>
            ſtabιt lineæ f m per 30 p 1.</s>
            <s xml:id="echoid-s38843" xml:space="preserve"> Quia ergo in trigono l g d linea f m æquidiſtat baſi l g:</s>
            <s xml:id="echoid-s38844" xml:space="preserve"> patet per 2 p 6
              <lb/>
            </s>
          </p>
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