Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]

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            <div xml:id="echoid-div217" type="math:theorem" level="3" n="114">
              <p>
                <s xml:id="echoid-s1007" xml:space="preserve">
                  <pb o="76" rhead="IO. BAPT. BENED." n="88" file="0088" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0088"/>
                Taurino Patauium .220. quæ quiſque confecerit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1008" xml:space="preserve">Dum autem hæc ſpecularer attentius, occurrit alius ſoluendi modus, quamuis pro
                  <lb/>
                lixior. </s>
                <s xml:id="echoid-s1009" xml:space="preserve">Is
                  <reg norm="autem" type="wordlist">aũt</reg>
                eſt eiuſmodi. </s>
                <s xml:id="echoid-s1010" xml:space="preserve">
                  <reg norm="Accipiatur" type="simple">Accipiat̃</reg>
                medietas minoris numeri
                  <reg norm="dierum" type="context">dierũ</reg>
                ,
                  <reg norm="nempe" type="context">nẽpe</reg>
                .4.
                  <reg norm="cum" type="context">cũ</reg>
                dimi
                  <lb/>
                dio, & per .400. multiplicetur,
                  <reg norm="productumque" type="context simple">productũq́;</reg>
                per
                  <reg norm="maiorem" type="context">maiorẽ</reg>
                numerum diuidemus ſcilicet
                  <lb/>
                11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio
                  <reg norm="millia- riorum" type="context">millia-
                    <lb/>
                  riorũ</reg>
                itineris .200. detracto, &
                  <reg norm="preſiduum" type="context">preſiduũ</reg>
                  <reg norm="nempe" type="context">nẽpe</reg>
                .36.
                  <reg norm="cum" type="context">cũ</reg>
                .4. vndecimis multiplicato pro
                  <lb/>
                  <reg norm="ductoque" type="simple">ductoq́;</reg>
                diuiſo
                  <reg norm="per" type="simple">ꝑ</reg>
                  <reg norm="ſummam" type="context">ſummã</reg>
                dimidij itineris .200.
                  <reg norm="cum" type="context">cũ</reg>
                primo
                  <reg norm="prouentu" type="context">prouẽtu</reg>
                .163. et .7. vndecimis
                  <lb/>
                  <reg norm="nempe" type="context">nẽpe</reg>
                  <reg norm="per" type="simple">ꝑ</reg>
                .363. ct .7. vndecimas partes
                  <reg norm="proueniet" type="simple">ꝓueniet</reg>
                .16.
                  <reg norm="cum" type="context">cũ</reg>
                .4. vndecimis, quo
                  <reg norm="coniuncto" type="context">cõiuncto</reg>
                pri
                  <lb/>
                mo
                  <reg norm="prouenienti" type="simple context">ꝓueniẽti</reg>
                , primus .180. milliaria
                  <reg norm="confecerit" type="context">cõfecerit</reg>
                , quæ è .400. detracta ſupererunt .220.
                  <lb/>
                pro itinere ſecundi, qui .9. diebus iter abſoluit. </s>
                <s xml:id="echoid-s1011" xml:space="preserve">Ad hæc ſi tempus ſcire velimus
                  <lb/>
                eius, qui .11. diebus appellit, multiplicabimus .11. cum .180.
                  <reg norm="productumque" type="simple">productumq́;</reg>
                per .400.
                  <lb/>
                partiemur,
                  <reg norm="prouenientque" type="simple">prouenientq́;</reg>
                paulominus, quam quinque dies, nempe .4. cum .22. horis
                  <lb/>
                et .48. minutis, quod tempus vtrique viatori inſeruiet, quandoquidem idipſum pro
                  <lb/>
                uenit multiplicato .220. per .9.
                  <reg norm="productoque" type="simple">productoq́;</reg>
                per .400. diuiſo.</s>
              </p>
              <p>
                <s xml:id="echoid-s1012" xml:space="preserve">Huius autem, qui à me pręſcribitur modi, ſpeculatio talis eſt. </s>
                <s xml:id="echoid-s1013" xml:space="preserve">Duo termini duabus
                  <lb/>
                rectis lineis æqualibus, & parallelis inter ſe
                  <var>.b.p.</var>
                et
                  <var>.d.q.</var>
                ſignificentur, quæ alijs dua-
                  <lb/>
                bus
                  <var>.b.d.</var>
                et
                  <var>.q.p.</var>
                  <reg norm="coniungantur" type="simple">coniungant̃</reg>
                , quę parallelæ & æquales erunt ex .33. primi, quibus ſigni
                  <lb/>
                ficentur duo itinera. </s>
                <s xml:id="echoid-s1014" xml:space="preserve">Viator primus quidem lentior à. b in
                  <var>.d.</var>
                velocior à
                  <var>.q.</var>
                in
                  <var>.p</var>
                . </s>
                <s xml:id="echoid-s1015" xml:space="preserve">Iam
                  <lb/>
                ſumatur
                  <reg norm="punctum" type="context">punctũ</reg>
                medium
                  <var>.q.p.</var>
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                  <var>.k.</var>
                & ab ipſo ad
                  <var>.b.d.</var>
                ducatur
                  <var>.k.i.</var>
                parallela
                  <var>.d.q.</var>
                aut
                  <lb/>
                  <var>b.p.</var>
                quod idem eſt, ex quo
                  <var>.b.i.</var>
                æqualis erit
                  <var>.p.k.</var>
                ex .34. primi, hoc eſt
                  <var>.q.k.</var>
                  <reg norm="certique" type="simple">certiq́;</reg>
                eri-
                  <lb/>
                mus primum viatorem
                  <var>.q.p.</var>
                in dimidio itineris
                  <var>.q.k.</var>
                occurrere non potuiſſe viatori ip
                  <lb/>
                ſius
                  <var>.b.i.</var>
                quandoquidem eo tempore, quo is, qui ipſius
                  <var>.q.p.</var>
                mouetur per
                  <var>.q.k.</var>
                (cum ſit
                  <lb/>
                altero velocior) qui per
                  <var>.b.d.</var>
                nondum peruenerit ad .i: Sit itaque punctum
                  <var>.c.</var>
                in quo
                  <lb/>
                lentior reperitur, dum velocior eſt in
                  <var>.k.</var>
                ex quo certi erimus eos inter
                  <var>.c.</var>
                et
                  <var>.i.</var>
                ſibi in-
                  <lb/>
                uicem obuiaturos eſſe. </s>
                <s xml:id="echoid-s1016" xml:space="preserve">Cogito deinde rectam lineam ductam
                  <var>.k.c.</var>
                & ut ſe habet
                  <var>.i.
                    <lb/>
                  c.</var>
                ad
                  <var>.c.b.</var>
                ita cogito ſe habere. u
                  <unsure/>
                  <var>.k.</var>
                ad
                  <var>.k.q.</var>
                & à puncto
                  <var>.u.</var>
                ad
                  <var>.i.</var>
                duco
                  <var>.u.i.</var>
                quæ, vt manife
                  <lb/>
                ſtum eſt, lineam
                  <var>.k.c.</var>
                in puncto
                  <var>.e.</var>
                interſecabit, à quo cum fuerit ducta
                  <var>.e.o.n.</var>
                parallela
                  <lb/>
                  <var>k.i.</var>
                habebimus
                  <var>.o.n.</var>
                ea ſcilicet puncta, quibus occurrunt ſibijpſis, nam cum ſic ſe ha
                  <lb/>
                beat
                  <var>.q.k.</var>
                ad
                  <var>.k.u.</var>
                vt
                  <var>.b.c.</var>
                ad
                  <var>.c.i.</var>
                et
                  <var>.k.u.</var>
                ad
                  <var>.k.n.</var>
                vt
                  <var>.c.i.</var>
                ad
                  <var>.c.o.</var>
                ex ſimilitudine manifeſta
                  <lb/>
                triangulorum, ex æqualitate proportionum ſic ſe habebit
                  <var>.q.k.</var>
                ad
                  <var>.k.n.</var>
                vt
                  <var>.b.c.</var>
                ad
                  <var>.c.o.</var>
                  <lb/>
                & permutando ita
                  <var>.k.q.</var>
                ad
                  <var>.b.c.</var>
                vt
                  <var>.k.n.</var>
                ad
                  <var>.c.o.</var>
                & cum
                  <var>.q.k.</var>
                et
                  <var>.b.c.</var>
                ſpatia ſint tempori-
                  <lb/>
                bus æqualibus confecta, itaque ſpatia
                  <var>.k.n.</var>
                et
                  <var>.c.o.</var>
                ex communi ſcientia temporibus
                  <lb/>
                æqualibus conficientur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1017" xml:space="preserve">Quare rectè dicimus, ſi tot diebus à
                  <var>.b.</var>
                in
                  <var>.d.</var>
                aliquis peruenit, quot milliaria in di
                  <lb/>
                midio temporis alterius viatoris idem conficiet? </s>
                <s xml:id="echoid-s1018" xml:space="preserve">ex quo ex regula de tribus quam
                  <lb/>
                primum iter
                  <var>.b.c.</var>
                cognoſcitur, quo ex dimidio itineris detracto, remanet
                  <var>.c.i.</var>
                cogni
                  <lb/>
                tus, ſed cum probauerimus
                  <var>.q.k.</var>
                ad
                  <var>.k.n.</var>
                hoc eſt
                  <var>.i.o.</var>
                (cum ſint æquales inter ſe, ex .34
                  <lb/>
                primi) ita ſe habere. vt
                  <var>.b.c.</var>
                ad
                  <var>.c.o.</var>
                permutando ſic ſe habebit
                  <var>.q.k.</var>
                ad
                  <var>.b.c.</var>
                vt
                  <var>.i.o.</var>
                ad
                  <var>.
                    <lb/>
                  o.c.</var>
                &
                  <reg norm="componendo" type="context">cõponendo</reg>
                  <var>.q.k.</var>
                et
                  <var>.b.c.</var>
                ad
                  <var>.b.c.</var>
                vt
                  <var>.i.c.</var>
                ad
                  <var>.c.o.</var>
                </s>
                <s xml:id="echoid-s1019" xml:space="preserve">quare rectè dicimus ſi ſumma
                  <var>.q.
                    <lb/>
                  k.</var>
                cum
                  <var>.b.c.</var>
                dat
                  <var>.b.c.</var>
                quid dabit
                  <var>.i.c</var>
                ? </s>
                <s xml:id="echoid-s1020" xml:space="preserve">nempe dabit
                  <var>.c.o.</var>
                quo coniuncto cum
                  <var>.b.c.</var>
                cogno-
                  <lb/>
                ſcitur
                  <var>.b.o.</var>
                quo
                  <var>.b.o.</var>
                detracto ex
                  <var>.b.d.</var>
                remanet cognitus
                  <var>.o.d.</var>
                nempe
                  <var>.q.n.</var>
                illi æqualis
                  <lb/>
                ex .34. prædicta. </s>
                <s xml:id="echoid-s1021" xml:space="preserve">Gratia verò
                  <reg norm="temporis" type="context">tẽporis</reg>
                patet nos rectè dicere ſi
                  <var>.b.d.</var>
                tot diebus abſolui
                  <lb/>
                tur, aut etiam
                  <var>.q.p</var>
                : quo
                  <var>.b.o.</var>
                aut
                  <var>.q.n.</var>
                abſoluetur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1022" xml:space="preserve">Vt autem ad ſpeculationem regulæ antiquorum deueniamus, cogitemus pri-
                  <lb/>
                mum viatorem ipſius
                  <var>.q.p.</var>
                velociorem eo, qui per
                  <var>.b.d.</var>
                iter agit, tanto tempore præ
                  <lb/>
                tergredi
                  <var>.p.</var>
                quanto alter
                  <var>.b.d.</var>
                abſoluit. </s>
                <s xml:id="echoid-s1023" xml:space="preserve">Is autem ad
                  <var>.g.</var>
                pertingat, ex quo eadem pro-
                  <lb/>
                portio ſpacij
                  <var>.q.g.</var>
                ad
                  <var>.q.p.</var>
                hoc eſt
                  <var>.b.d.</var>
                dabitur, quæ temporis quo
                  <var>.b.d.</var>
                abſoluitur ab </s>
              </p>
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