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

Page concordance

< >
Scan Original
191 185
192 186
193 187
194 188
195 189
196 190
197 191
198 192
199 193
200 194
201 195
202 196
203 197
204 198
205 199
206 200
207 201
208 202
209 203
210 204
211 205
212 206
213 207
214 208
215 209
216 210
217 211
218 212
219 213
220 214
< >
page |< < (194) of 778 > >|
    <echo version="1.0RC">
      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div461" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s13287" xml:space="preserve">
              <pb o="194" file="0200" n="200" rhead="ALHAZEN"/>
            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>
          </p>
        </div>
      </text>
    </echo>
Impressum