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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IO. BAPT. BENED.
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            <div xml:id="echoid-div203" type="math:theorem" level="3" n="108">
              <head xml:id="echoid-head125" xml:space="preserve">THEOREMA
                <num value="108">CVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s947" xml:space="preserve">AT ſi numerus milliarium primi viatoris diſpar fuerit, ſecundum numero pari
                  <lb/>
                ſemper ſuperabit, vt facile erit ſequentem figuram conſideranti intelligere,
                  <lb/>
                ex quo illud ſequetur, futuram quandam diem, qua paria milliaria conficient. </s>
                <s xml:id="echoid-s948" xml:space="preserve">Sit-
                  <lb/>
                q́ue illa dies
                  <var>.u.n.</var>
                ſequitur
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                tranſacta ea die, tot diebus vtrique ambulandum eſſe
                  <lb/>
                quot iter egere anteaquam ad diem
                  <var>.u.n.</var>
                  <lb/>
                peruenirent, vt tanto numero primus à ſe-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/>
                cundo ſuperetur,
                  <reg norm="quanto" type="context">quãto</reg>
                ſecundum primus
                  <lb/>
                ſuperauerat, vnde totalis numerus
                  <var>.o.f.</var>
                mi
                  <lb/>
                nor erit duplo
                  <var>.o.n.</var>
                vnitate ex communi
                  <lb/>
                conceptione, ſed ita etiam ſe habet termi-
                  <lb/>
                nus
                  <var>.u.n.</var>
                hoc eſt minor duplo
                  <var>.o.n.</var>
                per
                  <var>.o.</var>
                vt
                  <lb/>
                94. theoremate dictum fuit, itaque
                  <var>.o.f.</var>
                æ-
                  <lb/>
                qualis erit
                  <var>.u.n.</var>
                quoderat propoſitum.</s>
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                <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a">
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            <div xml:id="echoid-div205" type="math:theorem" level="3" n="109">
              <head xml:id="echoid-head126" xml:space="preserve">THEOREMA
                <num value="109">CIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s949" xml:space="preserve">SIN verò progreſſio ſecundi viatoris, non ab vnitate ſed à binario inchoata,
                  <lb/>
                per binarium quoque aſcenderet,
                  <reg norm="numerusque" type="simple">numerusq́;</reg>
                milliarium diurnorum primi via
                  <lb/>
                toris par eſſet, abſque dubio quadam die paria milliaria
                  <reg norm="vterque" type="simple">vterq;</reg>
                conficeret, quę ſigni
                  <lb/>
                ficetur
                  <var>.u.n.</var>
                qua tranſacta, tot diebus vtrique
                  <reg norm="ambulandum" type="context">ambulãdum</reg>
                erit, quot
                  <reg norm="fuerunt" type="context">fuerũt</reg>
                  <reg norm="dum" type="context">dũ</reg>
                primus
                  <lb/>
                ſecundum ſuperaret, vt totidem alijs pri-
                  <lb/>
                mus à ſecundo ſuperetur, in qua tamen
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0082-02a" xlink:href="fig-0082-02"/>
                progreſſione terminus
                  <var>.u.n.</var>
                ſemper duplus
                  <lb/>
                eſt numero terminorum
                  <var>.o.n.</var>
                ex .95. theo-
                  <lb/>
                remate,
                  <reg norm="totque" type="simple">totq́;</reg>
                ſunt infra
                  <var>.u.n.</var>
                termini vſque
                  <lb/>
                ad
                  <var>.f.</var>
                quot ſupra. </s>
                <s xml:id="echoid-s950" xml:space="preserve">ex quo illud ſequitur om
                  <lb/>
                nesterminosaut dies
                  <var>.o.n.f.</var>
                pauciores eſſe
                  <lb/>
                  <var>u.n.</var>
                vnitate, atque ita præcipit, regula de-
                  <lb/>
                trahendam eſſe vnitatem ex numero mil-
                  <lb/>
                liarium diurnorum primi viatoris, ſi dierum numerum habere voluerimus.</s>
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                <figure xlink:label="fig-0082-02" xlink:href="fig-0082-02a">
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            <div xml:id="echoid-div207" type="math:theorem" level="3" n="110">
              <head xml:id="echoid-head127" xml:space="preserve">THEOREMA
                <num value="110">CX</num>
              .</head>
              <p>
                <s xml:id="echoid-s951" xml:space="preserve">SED ſi in eiuſmodi progreſſione numerus milliarium diurnorum primi viato-
                  <lb/>
                ris diſpar fuerit, patet quòd primus ſecundum numero diſpari ſuperabit, do-
                  <lb/>
                necad vnitatem perueniatur
                  <reg norm="vi- ciſſimque" type="simple">vi-
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0082-03a" xlink:href="fig-0082-03"/>
                  ciſſimq́;</reg>
                primum ſecundus, in-
                  <lb/>
                choando ab vnitate, </s>
                <s xml:id="echoid-s952" xml:space="preserve">quare nul-
                  <lb/>
                la
                  <reg norm="vnquam" type="context">vnquã</reg>
                die paria milliaria vter-
                  <lb/>
                que conficiet, ſit itaque vltima
                  <lb/>
                dies, qua primus ſecundum vnita
                  <lb/>
                tem antecedit
                  <var>.u.n.</var>
                qui terminus
                  <lb/>
                duplus eſt numero terminorum
                  <lb/>
                  <var>o.n.</var>
                & cum illa die primus ſecun-
                  <lb/>
                dum milliario antecedat, ſequen­ </s>
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