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
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            illud, quod fit ex ductu a h in h f bis, ſecundũ quod dicit Euclides [13 p 2.</s>
            <s xml:id="echoid-s13288" xml:space="preserve">] Igitur quadratũ a h cũ qua
              <lb/>
            drato h b, ſuperat quadratum a d quæ eſt æqualis a b) in ductu a h in h f bis:</s>
            <s xml:id="echoid-s13289" xml:space="preserve"> & ita [per 1 p 2] in ductu
              <lb/>
            a h in h d bis, & a h in d f bis:</s>
            <s xml:id="echoid-s13290" xml:space="preserve"> Sed [per 7 p 2] multiplicatio a h in h d bis, cum quadrato a d, eſt æqua-
              <lb/>
            lis quadrato a h cum quadrato h d:</s>
            <s xml:id="echoid-s13291" xml:space="preserve"> & ita ablato cõmuni quadrato a d, cũ ductu a h in h d bis:</s>
            <s xml:id="echoid-s13292" xml:space="preserve"> reſtabit
              <lb/>
            quadratũ h d cũ ductu a h in f d bis, æquale quadrato h b.</s>
            <s xml:id="echoid-s13293" xml:space="preserve"> Sed [per fabricationẽ] multiplicatio a h in
              <lb/>
            h t æqualis eſt quadrato h d:</s>
            <s xml:id="echoid-s13294" xml:space="preserve"> & multiplicatio a h in h u, æqualis quadrato h b:</s>
            <s xml:id="echoid-s13295" xml:space="preserve"> erit ergo multiplicatio
              <lb/>
            a h in h u, æqualis multiplicationi a h in h t, & multiplicationi a h in d f bis, ſubtractoq́;</s>
            <s xml:id="echoid-s13296" xml:space="preserve"> ductu a h in
              <gap/>
              <lb/>
            t(quẽ communẽ ponimus utriq;</s>
            <s xml:id="echoid-s13297" xml:space="preserve"> multiplicationi.</s>
            <s xml:id="echoid-s13298" xml:space="preserve">) [Quia enim oblonga cõprehenſa ſub a h t & ſub
              <lb/>
            a h & t u, æquãtur oblongo cõprehenſo ſub a h u perip 2:</s>
            <s xml:id="echoid-s13299" xml:space="preserve"> ergo æquãtur oblõgis cõprehenſis ſub a h
              <lb/>
            t & ſub a h & d f bis:</s>
            <s xml:id="echoid-s13300" xml:space="preserve"> cõmune igitur eſt oblongũ cõprehenſum ſub a h t] reſtabit multiplicatio a h in
              <lb/>
            t u ęqualis multiplicationi a h in d f bis.</s>
            <s xml:id="echoid-s13301" xml:space="preserve"> Igitur t u eſt dupla d f:</s>
            <s xml:id="echoid-s13302" xml:space="preserve"> [Quia enim oblongũ comprehenſum
              <lb/>
            ſub altitudine a h & baſi t u, æquatur duplici oblongo, comprehenſo ſub eadem altitudine & baſi d
              <lb/>
            f:</s>
            <s xml:id="echoid-s13303" xml:space="preserve">erit per 1 p 6 baſis t u dupla baſis d f.</s>
            <s xml:id="echoid-s13304" xml:space="preserve">] Amplius:</s>
            <s xml:id="echoid-s13305" xml:space="preserve"> cũ angulus a t g ſit acutus [ut oſtenſũ eſt 60 n 5] erit
              <lb/>
            ſecundũ prædictũ modũ, quadratũ a t cum quadrato t g, æquale quadrato a d, cũ ductu a t in t k bis:</s>
            <s xml:id="echoid-s13306" xml:space="preserve">
              <lb/>
            & ita [per 1 p 2] cũ ductu a t in t d bis, & in d k bis.</s>
            <s xml:id="echoid-s13307" xml:space="preserve"> Et probabitur modo prædicto, quòd quadratũ t g
              <lb/>
            æquale eſt quadrato t d, cũ ductu a t in d k bis:</s>
            <s xml:id="echoid-s13308" xml:space="preserve"> ſed ductus a t in t u, æqualis eſt quadrato t g [excõclu
              <lb/>
            ſo] & ita æqualis quadrato t d, cũ ductu a t in d k bis.</s>
            <s xml:id="echoid-s13309" xml:space="preserve"> Sit aũt ductus a t in t æ æqualis quadrato t d [ut
              <lb/>
            oſtenſũ eſt in principio huius numeri] reſtat ergo, ut ductus a t in æ u, ſit ęqualis ductui a t in d k bis,
              <lb/>
            per ablationẽ cõmunis, qui eſt ductus a t in t æ [nam oblonga cõprehenſa ſub a t æ, itẽ ſub a t & æ u,
              <lb/>
            æquãtur oblongo cõprehenſo ſub a t u per 1 p 2:</s>
            <s xml:id="echoid-s13310" xml:space="preserve"> ergo æquãtur oblongis cõprehenſis ſub a t æ ſemel,
              <lb/>
            & ſub a t & d k bis.</s>
            <s xml:id="echoid-s13311" xml:space="preserve"> Cõmune igitur eſt a t æ, quo ſublato:</s>
            <s xml:id="echoid-s13312" xml:space="preserve"> reliquũ oblongũ coprehenſum ſub at & æ u
              <lb/>
            æquatur oblongo ſub a t & d k bis cõprehenſo.</s>
            <s xml:id="echoid-s13313" xml:space="preserve">] Igitur æ u eſt dupla k d [per 1 p 6] ſed iam dictũ eſt,
              <lb/>
            quòd t u eſt dupla d f:</s>
            <s xml:id="echoid-s13314" xml:space="preserve"> reſtat ergo t æ dupla k f.</s>
            <s xml:id="echoid-s13315" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13316" xml:space="preserve"> proportio a h ad h t eſt, ſicut a h ad h d dupli
              <lb/>
            cata [per 10 d 5] h d enim media eſt in proportione interillas:</s>
            <s xml:id="echoid-s13317" xml:space="preserve"> cũ eius quadratũ ſit æquale ductui a h
              <lb/>
            in h t [per fabricationẽ.</s>
            <s xml:id="echoid-s13318" xml:space="preserve">] Et ſimiliter proportio a t ad t æ, ſicut a t ad t d duplicata [eſt enim ex ſabri-
              <lb/>
            catione & 17 p 6 a t ad t d, ſicut t d ad t æ.</s>
            <s xml:id="echoid-s13319" xml:space="preserve">] Sed maior eſt proportio a t ad t d, quàm a h ad h d.</s>
            <s xml:id="echoid-s13320" xml:space="preserve"> [Quia
              <lb/>
            enim h t minor eſt quinta parte h d, ut patuit:</s>
            <s xml:id="echoid-s13321" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s13322" xml:space="preserve"> ſi a t, uerbi gratia, ipſam t d quater contineat:</s>
            <s xml:id="echoid-s13323" xml:space="preserve"> a h
              <lb/>
            eandem t d quater continebit, & h d ſemel.</s>
            <s xml:id="echoid-s13324" xml:space="preserve"> Quare a h nõ continebit h d quater.</s>
            <s xml:id="echoid-s13325" xml:space="preserve"> Ratio igitur a t ad t d
              <lb/>
            maior eſt, quàm a h ad h d.</s>
            <s xml:id="echoid-s13326" xml:space="preserve">] Et cum a h ſit maior a t:</s>
            <s xml:id="echoid-s13327" xml:space="preserve"> [per 9 ax:</s>
            <s xml:id="echoid-s13328" xml:space="preserve">] erit h t maior t æ [quia enim a h maior
              <lb/>
            eſt a t:</s>
            <s xml:id="echoid-s13329" xml:space="preserve"> erit ք 8 p 5 ratio a h ad t æ maior, quàm a t ad t æ:</s>
            <s xml:id="echoid-s13330" xml:space="preserve"> ſed ratio a t ad t æ maior eſt, quàm a h ad h t.</s>
            <s xml:id="echoid-s13331" xml:space="preserve">
              <lb/>
            Ergo per 11 p 5 ratio a h ad t æ maior eſt, quàm a h ad h t.</s>
            <s xml:id="echoid-s13332" xml:space="preserve"> Quare ք 10 p 5 h t maior eſt t æ.</s>
            <s xml:id="echoid-s13333" xml:space="preserve">] Sed t æ du-
              <lb/>
            pla ad k f:</s>
            <s xml:id="echoid-s13334" xml:space="preserve"> ergo h t maior eſt, quàm dupla ad k f.</s>
            <s xml:id="echoid-s13335" xml:space="preserve"> Item.</s>
            <s xml:id="echoid-s13336" xml:space="preserve"> Vt dictũ eſt, proportio b g ad g s, ſicut o a ad o y,
              <lb/>
            erit [per 16 p 5] b g ad o a, ſicut g s ad o y:</s>
            <s xml:id="echoid-s13337" xml:space="preserve"> ſed o a ęqualis b a [per 15 d 1] & g s ęqualis f k [per 34 p 1] pro-
              <lb/>
            pter ęquidiſtantiã:</s>
            <s xml:id="echoid-s13338" xml:space="preserve"> erit [per 7 p 5] proportio b g ad b a, ſicut f k ad o y.</s>
            <s xml:id="echoid-s13339" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13340" xml:space="preserve"> quia i h minor eſt me-
              <lb/>
            dietate o h [ut patuit] & o h tripla th:</s>
            <s xml:id="echoid-s13341" xml:space="preserve"> eriti h minor h t, & medietate ipſius:</s>
            <s xml:id="echoid-s13342" xml:space="preserve"> ſed h t minor quinta parte
              <lb/>
            h d.</s>
            <s xml:id="echoid-s13343" xml:space="preserve"> Igitur i h minor eſt t d:</s>
            <s xml:id="echoid-s13344" xml:space="preserve"> quare i h multò minor n d:</s>
            <s xml:id="echoid-s13345" xml:space="preserve"> quare m i multò minor n d [quia m i minor eſt
              <lb/>
            i h, quæ minor eſt n d.</s>
            <s xml:id="echoid-s13346" xml:space="preserve">] Et palàm per hoc, quòd i cadit inter h & z.</s>
            <s xml:id="echoid-s13347" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13348" xml:space="preserve"> quod fit ex ductu e z in z
              <lb/>
            d, eſt æquale quadrato a d:</s>
            <s xml:id="echoid-s13349" xml:space="preserve"> [per theſin] igitur quod fit ex ductu e m in m d, eſt minus quadrato a d.</s>
            <s xml:id="echoid-s13350" xml:space="preserve">
              <lb/>
            Sed quoniam m g circulum d b e cõtingit, quod fit ex ductu e m in m d, eſt æquale quadrato m g, ſe-
              <lb/>
            cundũ quod dicit Euclides [36 p 3.</s>
            <s xml:id="echoid-s13351" xml:space="preserve">] Igitur m g eſt minor a d:</s>
            <s xml:id="echoid-s13352" xml:space="preserve"> igitur minor eſt a g.</s>
            <s xml:id="echoid-s13353" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13354" xml:space="preserve"> triangula
              <lb/>
            a g m, m g k habent unum angulũ communem [a d m] & utrunq;</s>
            <s xml:id="echoid-s13355" xml:space="preserve"> eorum habet unũ angulum rectũ
              <lb/>
            [ad g & k.</s>
            <s xml:id="echoid-s13356" xml:space="preserve">] Igitur [per 32 p 1.</s>
            <s xml:id="echoid-s13357" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s13358" xml:space="preserve"> 1 d 6] ſunt ſimilia.</s>
            <s xml:id="echoid-s13359" xml:space="preserve"> Quare proportio m k ad k g, ſicut m g ad g a:</s>
            <s xml:id="echoid-s13360" xml:space="preserve"> & ita
              <lb/>
            m k minor eſt k g [eſt enim m g minor g a ex concluſo.</s>
            <s xml:id="echoid-s13361" xml:space="preserve">] Et cum [per 15 p 3] o y ſit maior g k:</s>
            <s xml:id="echoid-s13362" xml:space="preserve"> erit h d
              <lb/>
            minor o y [quia h d minor eſt m k, & m k minor k g, & k g minor o y.</s>
            <s xml:id="echoid-s13363" xml:space="preserve">] Amplius:</s>
            <s xml:id="echoid-s13364" xml:space="preserve"> quia a h ad h d, ſicut h
              <lb/>
            d ad h t:</s>
            <s xml:id="echoid-s13365" xml:space="preserve"> [per theſin & 17 p 6] erit ſic [per 15 p 5] medietas h d ad medietatem h t:</s>
            <s xml:id="echoid-s13366" xml:space="preserve"> & ita a h ad h d, ſicut
              <lb/>
            q h ad medietatem h t:</s>
            <s xml:id="echoid-s13367" xml:space="preserve"> cum q h ſit medietas h d:</s>
            <s xml:id="echoid-s13368" xml:space="preserve"> [per fabricationem] & ita a h ad q h, ſicut h d ad
              <lb/>
            medietatem h t:</s>
            <s xml:id="echoid-s13369" xml:space="preserve"> & ita [per conſectarium 4 p 5] q h ad a h, ſicut medietas h t ad h d.</s>
            <s xml:id="echoid-s13370" xml:space="preserve"> Sed medietas h t
              <lb/>
            maior eſt f k [demonſtratũ enim eſt ipſam h t maiorẽ eſſe, quàm duplam ipſius k f] & h d minor o y.</s>
            <s xml:id="echoid-s13371" xml:space="preserve">
              <lb/>
            Erit igitur proportio medietatis h t ad h d maior, quàm f k ad o y [ut conſtat ex 8 p 5.</s>
            <s xml:id="echoid-s13372" xml:space="preserve">] Quare [per 11 p
              <lb/>
            5] erit proportio q h ad a h maior, quàm f k ad o y.</s>
            <s xml:id="echoid-s13373" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13374" xml:space="preserve"> linea a q ſecat circulum e b d:</s>
            <s xml:id="echoid-s13375" xml:space="preserve"> ſit punctũ
              <lb/>
            ſectiõis œ:</s>
            <s xml:id="echoid-s13376" xml:space="preserve"> & ducatur linea d œ:</s>
            <s xml:id="echoid-s13377" xml:space="preserve"> quę erit æquidiſtãs q h:</s>
            <s xml:id="echoid-s13378" xml:space="preserve"> [Quia enim tota a h æquatur toti a q, & pars
              <lb/>
            a d parti a œ per 15 d 1:</s>
            <s xml:id="echoid-s13379" xml:space="preserve"> reliqua igitur d h ęquatur reliquę œ q:</s>
            <s xml:id="echoid-s13380" xml:space="preserve"> quare per 7 p 5, ut a d ad d h, ſic a œ ad œ
              <lb/>
            q.</s>
            <s xml:id="echoid-s13381" xml:space="preserve"> Ita q;</s>
            <s xml:id="echoid-s13382" xml:space="preserve"> per 2 p 6 œ d parallela eſt ipſi q h] eritq́;</s>
            <s xml:id="echoid-s13383" xml:space="preserve"> per 29 p 1.</s>
            <s xml:id="echoid-s13384" xml:space="preserve"> 4 p 6 proportio q h ad h a, ſicut œ ad d a:</s>
            <s xml:id="echoid-s13385" xml:space="preserve"> &
              <lb/>
            ita proportio œ ad d a maior, quàm f k ad o y.</s>
            <s xml:id="echoid-s13386" xml:space="preserve"> Sed fk ad o y, ſicut g b ad b a [ex concluſo.</s>
            <s xml:id="echoid-s13387" xml:space="preserve">] Erit igitur
              <lb/>
            maior proportio œ d ad d a, quàm b g ad b a [id eſt ad d a:</s>
            <s xml:id="echoid-s13388" xml:space="preserve"> æquales enim ſunt d a & b a per 15 d 1] & ita
              <lb/>
            œ d maior b g:</s>
            <s xml:id="echoid-s13389" xml:space="preserve"> [per 10 p 5] & arcus œ d maior arcu g b [per 28 p 3.</s>
            <s xml:id="echoid-s13390" xml:space="preserve">] Amplius:</s>
            <s xml:id="echoid-s13391" xml:space="preserve"> producatur a q uſq;</s>
            <s xml:id="echoid-s13392" xml:space="preserve"> ad
              <lb/>
            punctũ s, ut ſit a s æqualis a i:</s>
            <s xml:id="echoid-s13393" xml:space="preserve"> [per 3 p 1] & ducatur linea s i:</s>
            <s xml:id="echoid-s13394" xml:space="preserve"> quę erit æquidiſtãs q h:</s>
            <s xml:id="echoid-s13395" xml:space="preserve"> [eodẽ argumẽto,
              <lb/>
            quo œ d parallela cõcluſa eſt ipſi q h] & erit [per 29 p 1.</s>
            <s xml:id="echoid-s13396" xml:space="preserve"> 4 p 6] si ad q h, ſicut i a ad h a.</s>
            <s xml:id="echoid-s13397" xml:space="preserve"> Sed ſuprà poſi-
              <lb/>
            tũ eſt, quòd i a ad a h, ſicut t q ad q h:</s>
            <s xml:id="echoid-s13398" xml:space="preserve"> erit igitur [per 9 p 5] si æqualis t q.</s>
            <s xml:id="echoid-s13399" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13400" xml:space="preserve"> mutetur figura ad
              <lb/>
            euitandam linearũ intricationẽ multiplicẽ, & propter defectũ literarũ ad diſtinctionẽ linearũ.</s>
            <s xml:id="echoid-s13401" xml:space="preserve"> Cum
              <lb/>
            ergo i a ſit æqualis lineę, quã diximus a s:</s>
            <s xml:id="echoid-s13402" xml:space="preserve"> fiat circulus ſecundũ quantitatẽ ipſarũ, & loco s ponatur li
              <lb/>
            tera n:</s>
            <s xml:id="echoid-s13403" xml:space="preserve"> & producantur a g i ab uſq;</s>
            <s xml:id="echoid-s13404" xml:space="preserve"> ad circulũ hunc:</s>
            <s xml:id="echoid-s13405" xml:space="preserve"> & ſint a b c, a g r:</s>
            <s xml:id="echoid-s13406" xml:space="preserve"> & loco literæ œ ponamus f.</s>
            <s xml:id="echoid-s13407" xml:space="preserve"> Di-
              <lb/>
            ctum eſt, quòd arcus d f maior eſt arcu b g:</s>
            <s xml:id="echoid-s13408" xml:space="preserve"> ſit arcus b m æqualis arcui d f:</s>
            <s xml:id="echoid-s13409" xml:space="preserve"> [fiet uerò æqualis, ſiad re-
              <lb/>
            ctam a b eiusq́;</s>
            <s xml:id="echoid-s13410" xml:space="preserve"> punctũ a cõſtituatur per 23 p 1 angulus b a m æqualis angulo d a f:</s>
            <s xml:id="echoid-s13411" xml:space="preserve"> ſic enim per 33 p 6
              <lb/>
            </s>
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