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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[111] h l m t k g e b f d p q o z a
[112] e p o l g h n d m t b q a z
[113] o e k m f l g h d t b q a z
[114] b u a x r o i c p e d z s h g q
[115] l m c k p q o f n y
[116] b a m h e f t d z n p l g q
[117] b a t h e p d z n l k g q
[118] l e p d a b g
[119] h d t b q g
[120] e o f n p d a b g
[121] e o f t p d a b g k
[122] e o f t p k d a b g
[123] t z e b a g h d
[124] t z e b a g h d
[125] z t n q p i b k f e l a n m g h d
[126] z t n q b k f a e o g h d
[127] k e t o z r l g b x n p f m q d s n a
[128] b o p n g k e f d a q l m
[129] b t o u p n g k e f d a q z m
[130] b u t o p n g k e f d a q z m
[131] u t b p n o g k e f d l a q m z
[132] s g z k t e f d o b r a
[133] t f i k e d m q z x h
[134] k e d q h z
[135] l b k d o
[136] a b n m k l q g d h e
[137] b a b a m f g d n
[138] m t h f b p a g d n
[139] m t h b a g d n
[140] a b l m l t a b m g n d n d
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          <head xml:id="echoid-head465" xml:space="preserve" style="it">49. In ſpeculo ſphærico cauo imago lineæ rectæ aliquando uidetur conuexa. 57 p 8.</head>
          <p>
            <s xml:id="echoid-s15938" xml:space="preserve">HIs præoſtenſis, iteremus circulum, & perficiamus demonſtrationem, ne multiplicentur & li-
              <lb/>
            neæ, & dubitentur literæ.</s>
            <s xml:id="echoid-s15939" xml:space="preserve"> Sit ergo circulus in ſecunda figura a b g:</s>
            <s xml:id="echoid-s15940" xml:space="preserve"> & centrum d:</s>
            <s xml:id="echoid-s15941" xml:space="preserve"> & extraha-
              <lb/>
            mus lineam d q:</s>
            <s xml:id="echoid-s15942" xml:space="preserve"> & ſit d b æqualis d b in prima figura:</s>
            <s xml:id="echoid-s15943" xml:space="preserve"> & d o æqualis d o in prima figura:</s>
            <s xml:id="echoid-s15944" xml:space="preserve"> & d q
              <lb/>
            ſit compar ſibi in prima figura:</s>
            <s xml:id="echoid-s15945" xml:space="preserve"> & ſimiliter d u:</s>
            <s xml:id="echoid-s15946" xml:space="preserve"> & extrahamus ſuper d q perpendicularem ſuper ſu-
              <lb/>
            perficiem circuli [per 12 p 11] & ſit d h æqualis ſibi in prima figura.</s>
            <s xml:id="echoid-s15947" xml:space="preserve"> Angulus ergo h d q erit rectus:</s>
            <s xml:id="echoid-s15948" xml:space="preserve">
              <lb/>
            [per 3 d 11] & circulus, quem facit h d q in ſpeculo, erit ex circulis, ex quibus forma punctorum o, u
              <lb/>
            reflectitur:</s>
            <s xml:id="echoid-s15949" xml:space="preserve"> & erit arcus, quem menſurant lineæ h d, d q, æqualis arcui a g in primo circulo:</s>
            <s xml:id="echoid-s15950" xml:space="preserve"> [per 33 p
              <lb/>
            6:</s>
            <s xml:id="echoid-s15951" xml:space="preserve"> quia uterque ſubtendit angulum rectum] & ex duobus punctis iſtius arcus, comparibus duobus
              <lb/>
            punctis b, f, reflectentur duo puncta lineæ u p ad duo puncta n, q æqualiter.</s>
            <s xml:id="echoid-s15952" xml:space="preserve"> Erit ergo q imago o, &
              <lb/>
            n imago u.</s>
            <s xml:id="echoid-s15953" xml:space="preserve"> Et extrahamus ex u perpendicularem lineam in ſuperficie circuli a b g, ſuper lineam d u
              <lb/>
            [per 11 p 1] & ſit z u e:</s>
            <s xml:id="echoid-s15954" xml:space="preserve"> & ſit d centrum:</s>
            <s xml:id="echoid-s15955" xml:space="preserve"> & in longitudine d o faciamus arcum circuli:</s>
            <s xml:id="echoid-s15956" xml:space="preserve"> ſecabit ergo li-
              <lb/>
            neam z u e in duobus punctis:</s>
            <s xml:id="echoid-s15957" xml:space="preserve"> [quia punctum o
              <lb/>
              <figure xlink:label="fig-0230-01" xlink:href="fig-0230-01a" number="200">
                <variables xml:id="echoid-variables189" xml:space="preserve">k q t ſ n ſ g b o e u z d h a</variables>
              </figure>
            altius eſt puncto u, ex prima theſi] ſecet ergo in
              <lb/>
            z, e:</s>
            <s xml:id="echoid-s15958" xml:space="preserve"> & ſit arcus z o e:</s>
            <s xml:id="echoid-s15959" xml:space="preserve"> & continuemus d z, d e:</s>
            <s xml:id="echoid-s15960" xml:space="preserve"> &
              <lb/>
            extrahamus extra circulum:</s>
            <s xml:id="echoid-s15961" xml:space="preserve"> & à d & in longitu-
              <lb/>
            dine d q faciamus arcum t q:</s>
            <s xml:id="echoid-s15962" xml:space="preserve"> ſecabit ergo duas
              <lb/>
            lineas d z, d e in t, k:</s>
            <s xml:id="echoid-s15963" xml:space="preserve"> & continuemus t k:</s>
            <s xml:id="echoid-s15964" xml:space="preserve"> ſecabit
              <lb/>
            ergo lineam d q in l.</s>
            <s xml:id="echoid-s15965" xml:space="preserve"> Quia ergo h d eſt perpendi-
              <lb/>
            cularis ſuper ſuperficiem circuli:</s>
            <s xml:id="echoid-s15966" xml:space="preserve"> uterque angu-
              <lb/>
            lus h d t, h d k erit rectus:</s>
            <s xml:id="echoid-s15967" xml:space="preserve"> [per 3 d 11] & utraque
              <lb/>
            ſuperficies h d t, h d k faciet in ſuperficie ſpecu-
              <lb/>
            li circulum [per 1 th.</s>
            <s xml:id="echoid-s15968" xml:space="preserve"> 1 ſphær.</s>
            <s xml:id="echoid-s15969" xml:space="preserve">] & arcus, qui eſt in-
              <lb/>
            ter duas lineas h d, d t erit æqualis arcui, qui eſt
              <lb/>
            inter duas lineas h d, d q:</s>
            <s xml:id="echoid-s15970" xml:space="preserve"> & ſimiliter arcus, qui
              <lb/>
            eſt inter duas lineas h d, d k & utraque linea d z,
              <lb/>
            d e eſt æqualis lineę d o [per 15 d 1.</s>
            <s xml:id="echoid-s15971" xml:space="preserve">] Ergo hi duo
              <lb/>
            arcus ſunt huiuſmodi, quòd ex illis reflectentur
              <lb/>
            ſecundum angulos æquales duo puncta z, e:</s>
            <s xml:id="echoid-s15972" xml:space="preserve"> [ut
              <lb/>
            demonſtratum eſt 66 n 5] & duæ lineæ d t, d k
              <lb/>
            ſunt æquales lineæ d q [per 15 d 1.</s>
            <s xml:id="echoid-s15973" xml:space="preserve">] Ergo pun-
              <lb/>
            ctum t eſt imago z, & k eſt imago e.</s>
            <s xml:id="echoid-s15974" xml:space="preserve"> Et quia li-
              <lb/>
            neæ d t, d q, d k ſunt æquales:</s>
            <s xml:id="echoid-s15975" xml:space="preserve"> & lineæ d z, d o,
              <lb/>
            d e ſunt æquales:</s>
            <s xml:id="echoid-s15976" xml:space="preserve"> erit [per 7 p 5] proportio d t ad
              <lb/>
            d z, ſicut proportio q d ad d o, & ſicut proportio
              <lb/>
            k d ad d e.</s>
            <s xml:id="echoid-s15977" xml:space="preserve"> Sed proportio q d ad d o, ut in prima
              <lb/>
            figura [præcedentis numeri] præoſtendimus,
              <lb/>
            eſt maior proportione n d ad d u.</s>
            <s xml:id="echoid-s15978" xml:space="preserve"> Ergo propor-
              <lb/>
            tio d t ad d z eſt maior proportione n d ad d u:</s>
            <s xml:id="echoid-s15979" xml:space="preserve"> &
              <lb/>
            ſimiliter k d ad d e.</s>
            <s xml:id="echoid-s15980" xml:space="preserve"> Et quia duæ lineę z d, d e ſunt
              <lb/>
            æquales, & duę lineæ d t, d k ſunt æquales:</s>
            <s xml:id="echoid-s15981" xml:space="preserve"> erit li
              <lb/>
            nea t k æquidiſtans z e [per 2 p 6:</s>
            <s xml:id="echoid-s15982" xml:space="preserve"> eſt enim per 7
              <lb/>
            p 5 d t ad d z, ſicut d k ad d e:</s>
            <s xml:id="echoid-s15983" xml:space="preserve"> & per 17 p 5, ut t z ad
              <lb/>
            z d, ſic k e ad e d.</s>
            <s xml:id="echoid-s15984" xml:space="preserve">] Ergo [per 2 p 6.</s>
            <s xml:id="echoid-s15985" xml:space="preserve"> 18 p 5] utraq;</s>
            <s xml:id="echoid-s15986" xml:space="preserve">
              <lb/>
            proportio d t ad d z, & k d ad d e erit, ſicut pro-
              <lb/>
            portio l d ad d u.</s>
            <s xml:id="echoid-s15987" xml:space="preserve"> Ergo proportio l d ad d u eſt maior proportione n d ad d u:</s>
            <s xml:id="echoid-s15988" xml:space="preserve"> ergo linea l d eſt maior
              <lb/>
            linea n d [per 10 p 5.</s>
            <s xml:id="echoid-s15989" xml:space="preserve">] Ergo n eſt inter l, u.</s>
            <s xml:id="echoid-s15990" xml:space="preserve"> Sed n eſt imago u:</s>
            <s xml:id="echoid-s15991" xml:space="preserve"> & duo puncta t, k ſunt imagines z, e.</s>
            <s xml:id="echoid-s15992" xml:space="preserve"> Er
              <lb/>
            go imago lineæ z u e rectæ, eſt linea tranſiens per puncta t n k:</s>
            <s xml:id="echoid-s15993" xml:space="preserve"> & linea, quæ tranſit per hæc puncta,
              <lb/>
            eſt conuexa.</s>
            <s xml:id="echoid-s15994" xml:space="preserve"> Ex quibus patet, quòd linea in ſpeculis concauis quandoque uidetur conuexa in
              <lb/>
            quibuſdam ſitibus.</s>
            <s xml:id="echoid-s15995" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div534" type="section" level="0" n="0">
          <head xml:id="echoid-head466" xml:space="preserve" style="it">50. In ſpeculo ſphærico cauo imagines linearum: cauæ, conuexæ, aliquando uiden-
            <lb/>
          tur cauæ. 58 p 8.</head>
          <p>
            <s xml:id="echoid-s15996" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15997" xml:space="preserve"> ponamus in linea z u punctum m, quocun que modo ſit:</s>
            <s xml:id="echoid-s15998" xml:space="preserve"> & circa centrum m, & in longitu-
              <lb/>
            dine m u faciamus arcum r u f.</s>
            <s xml:id="echoid-s15999" xml:space="preserve"> Iſte ergo arcus ſecabit arcum u o e in duobus punctis:</s>
            <s xml:id="echoid-s16000" xml:space="preserve"> [per 10 p
              <lb/>
            3] ſecet in r, f:</s>
            <s xml:id="echoid-s16001" xml:space="preserve"> & continuemus lineas d r, d f:</s>
            <s xml:id="echoid-s16002" xml:space="preserve"> & tranſeant rectè, quouſque concurrant in arcu
              <lb/>
            t q k, in p, i.</s>
            <s xml:id="echoid-s16003" xml:space="preserve"> Superficies ergo duarum linearum h d, d p faciet in ſpeculo circulum, à cuius circum-
              <lb/>
            ferentia reflectentur lineę ad r:</s>
            <s xml:id="echoid-s16004" xml:space="preserve"> & ſimiliter ſuperficies duarum linearum h d, d i faciet in ſpeculo cir-
              <lb/>
            culum, à cuius circumferentia reflectentur lineæ ad f.</s>
            <s xml:id="echoid-s16005" xml:space="preserve"> p ergo eſt imago r, & i eſt imago f:</s>
            <s xml:id="echoid-s16006" xml:space="preserve"> & n eſt ima
              <lb/>
            go u.</s>
            <s xml:id="echoid-s16007" xml:space="preserve"> Imago ergo arcus r u f, eſt linea tranſiens per i, p, n.</s>
            <s xml:id="echoid-s16008" xml:space="preserve"> Sed hęc linea erit concaua ex parte uiſus,
              <lb/>
            & arcus r u f eſt concauus ex parte ſuperficiei ſpeculi.</s>
            <s xml:id="echoid-s16009" xml:space="preserve"> Cum ergo uiſus fuerit in h, & unaquęque li-
              <lb/>
            nearum z u e, z o e, r u f fuerit in aliquo uiſibili:</s>
            <s xml:id="echoid-s16010" xml:space="preserve"> tunc linea z u e recta comprehendetur conuexa:</s>
            <s xml:id="echoid-s16011" xml:space="preserve"> &
              <lb/>
            </s>
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