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-div197" type="math:theorem" level="3" n="105">
              <p>
                <s xml:id="echoid-s919" xml:space="preserve">
                  <pb o="70" rhead="IO. BAPT. BENED." n="80" file="0080" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0080"/>
                g. & permutando
                  <var>.o.r.</var>
                ad
                  <var>.f.o.</var>
                hoc eſt ad
                  <var>.m.s.</var>
                vt
                  <var>.h.</var>
                ad
                  <var>.g.</var>
                hoc eſt
                  <var>.y</var>
                . </s>
                <s xml:id="echoid-s920" xml:space="preserve">Quamobrem ea-
                  <lb/>
                dem erit proportio
                  <var>.o.r.</var>
                ad
                  <var>.m.s.</var>
                quæ
                  <var>.n.u.</var>
                ad
                  <var>.n.x</var>
                . </s>
                <s xml:id="echoid-s921" xml:space="preserve">Abfoluantur itaque duo rectangu
                  <lb/>
                la
                  <var>.x.i.</var>
                et
                  <var>.u.z.</var>
                ita tamen vt
                  <reg norm="rectangulum" type="context">rectangulũ</reg>
                  <var>.u.z.</var>
                cogitetur ęquale rectangulo
                  <var>.x.i.</var>
                cuius
                  <var>.x.i.</var>
                  <lb/>
                ſuperficialis numerus ex communi conceptione lineari
                  <var>.n.i.</var>
                æqualis erit, </s>
                <s xml:id="echoid-s922" xml:space="preserve">quare ex
                  <lb/>
                  <reg norm="eadem" type="context">eadẽ</reg>
                communi conceptione, numerus ſuperficialis
                  <var>.u.z.</var>
                lineari
                  <var>.n.i.</var>
                æqualis erit, qui
                  <lb/>
                quidem numerus in figura rectangu-
                  <lb/>
                la ſuperficialis cognitandus erit, cum
                  <lb/>
                  <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a" number="108">
                    <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0080-01"/>
                  </figure>
                diuidendus ſit per
                  <var>.h.</var>
                hoc eſt per
                  <var>.n.u.</var>
                  <lb/>
                ex quo proueniens ex huiuſmodi di
                  <lb/>
                uiſione erit numerus
                  <var>.n.z.</var>
                ex ijs
                  <lb/>
                quæ .10. theoremate dicta fuerunt.
                  <lb/>
                </s>
                <s xml:id="echoid-s923" xml:space="preserve">Sed ex .15. fexti aut .20. ſepti-
                  <lb/>
                mi eadem eſt proportio
                  <var>.n.i.</var>
                ad
                  <var>.n.z.</var>
                  <lb/>
                quæ
                  <var>.n.u.</var>
                ad
                  <var>.n.x.</var>
                hoc eſt
                  <var>.o.r.</var>
                ad
                  <var>.m.s.</var>
                  <lb/>
                videlicet vt
                  <var>.n.i.</var>
                ad aggregatum reli-
                  <lb/>
                quorum omnium terminorum
                  <var>.c.a</var>
                :
                  <var>f.
                    <lb/>
                  r</var>
                : </s>
                <s xml:id="echoid-s924" xml:space="preserve">m.s. ex præcedenti theoremate, & ex .11. quinti Euclidis. </s>
                <s xml:id="echoid-s925" xml:space="preserve">Itaque ex .9. eiuf-
                  <lb/>
                dem numerus
                  <var>.n.z.</var>
                æqualis erit ſummæ trium terminorum
                  <var>.c.a</var>
                : f. num
                  <var>.s.</var>
                cui coniuncto
                  <lb/>
                quarto termino
                  <var>.b.i.</var>
                propoſitum obtinetur.</s>
              </p>
            </div>
            <div xml:id="echoid-div199" type="math:theorem" level="3" n="106">
              <head xml:id="echoid-head123" xml:space="preserve">THEOREMA
                <num value="106">CVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s926" xml:space="preserve">PRopoſuere veteres quæſita nonnulla de itineribus
                  <reg norm="interque" type="simple">interq́;</reg>
                hoc vnum fuit. </s>
                <s xml:id="echoid-s927" xml:space="preserve">Po-
                  <lb/>
                namus duos iter agere per eandem viam quorum alter quatuor milliaria ſin-
                  <lb/>
                gulis diebus conficiat, alter verò prima die milliare vnum, ſecunda duo, tertia tria,
                  <lb/>
                atque ita ſingulis diebus milliare addit; </s>
                <s xml:id="echoid-s928" xml:space="preserve">quærimus quot dierum ſpacio ſocium con
                  <lb/>
                ſequetur.</s>
              </p>
              <p>
                <s xml:id="echoid-s929" xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, ſic ſunt .8. milliaria.
                  <lb/>
                </s>
                <s xml:id="echoid-s930" xml:space="preserve">ex quo ſemper vnitas detrahicur, quæ in præſenti exemplo erit .7.
                  <reg norm="totque" type="simple">totq́;</reg>
                dies erunt
                  <lb/>
                quibus ſocius ſocium conſequetur, & milliarium numerum æqualem abſoluerit.</s>
              </p>
              <p>
                <s xml:id="echoid-s931" xml:space="preserve">Cuius rei facilis erit ſpeculatio, ſi ſubſcripta figura diligenter conſideretur, in
                  <lb/>
                qua primus viator, die prima, quatuor milliaria linea
                  <var>.q.d.</var>
                ſignificata conficit, at-
                  <lb/>
                que illa ipſa die alter vnum tantum defignatum per
                  <var>.d.</var>
                perficit, ita vt primus via-
                  <lb/>
                tor tribus milliaribus ſocium anteceſſerit, altera verò die fecundus uiator cum duo
                  <lb/>
                milliaria
                  <reg norm="conficiat" type="context">cõficiat</reg>
                , excedètur à primo duobus milliaribus tantummodo, quę cum tri-
                  <lb/>
                bus primæ diei quinque erunt; </s>
                <s xml:id="echoid-s932" xml:space="preserve">tertia die ijſdem de cauſis primus ſex tantum millia-
                  <lb/>
                ribus à ſecundo diſtabit, cum verò quarta die tot ſecundus quot primus milliaria
                  <lb/>
                conficiat, primus à ſecundo amplius quam antea non diſtabit; </s>
                <s xml:id="echoid-s933" xml:space="preserve">quinta verò cum ſe
                  <lb/>
                cundus vnum milliare amplius quam primus conficiat. </s>
                <s xml:id="echoid-s934" xml:space="preserve">propius accedit ad primum
                  <lb/>
                vno ex ſex milliaribus, quibus anteà diſtabat, tum ſexta cum duobus primum ſupe-
                  <lb/>
                ret, detrahet ex ſex milliaribus præteritæ diſtantiæ tria, ſeptima tandem illa ſex
                  <lb/>
                detraxerit. </s>
                <s xml:id="echoid-s935" xml:space="preserve">In quo conſiderandum eſt ſecundum viatorem iter agere progreſſio-
                  <lb/>
                ne arithmetica continua naturali
                  <var>.d.c.f.</var>
                primum autem per rectangulum
                  <var>.q.f.</var>
                  <reg norm="quarum" type="context">quarũ</reg>
                  <lb/>
                duarum figurarum
                  <var>.d.o.p.f.</var>
                pars
                  <reg norm="communis" type="context">cõmunis</reg>
                eſſe reperitur, quæ quantitates ſi inuicem
                  <lb/>
                æquales eſſe debent, neceſſe eſt ſeparatas partes
                  <var>.u.q.n.</var>
                et
                  <var>.t.i.c.</var>
                inter ſe æquales eſſe,
                  <lb/>
                & quoniam quarta die (hoc eſt die ſic diſtante à primo, nempè numero milliarium </s>
              </p>
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