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-div199" type="math:theorem" level="3" n="106">
              <p>
                <s xml:id="echoid-s935" xml:space="preserve">
                  <pb o="71" rhead="THEOREM. ARITH." n="81" file="0081" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0081"/>
                primi viatoris) tot milliaria abſoluat vnus
                  <lb/>
                  <figure xlink:label="fig-0081-01" xlink:href="fig-0081-01a" number="109">
                    <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-01"/>
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                quot alter abſque vlla differentia, quæ ſigni-
                  <lb/>
                ficetur per
                  <var>.o.s.</var>
                neceſſe eſtitaque ex communi
                  <lb/>
                conceptione tot dies eſſe poſt
                  <var>.o.s.</var>
                quotante-
                  <lb/>
                ceſſerant, vt exceſſus æqualis ſit defectui, qui
                  <lb/>
                ſimul collecti, iuncta etiam
                  <var>.o.s.</var>
                duplum erunt
                  <lb/>
                  <var>d.s.</var>
                dempta vnitate, prout facilè in ſubſcripta
                  <lb/>
                figura qui ſque per ſe ſcientificè poterit ſpecu
                  <lb/>
                lari. </s>
                <s xml:id="echoid-s936" xml:space="preserve">Quamobrem conſultum erit duplicare
                  <lb/>
                numerum
                  <var>.o.s.</var>
                & exduplo vnitatem detrahe-
                  <lb/>
                re, quandoquidem dies ſupra
                  <reg norm="infraque" type="simple">infraq́;</reg>
                  <var>.o.s.</var>
                cum
                  <lb/>
                die
                  <var>.o.s.</var>
                minores ſunt duplo numeri
                  <var>.d.s.</var>
                aut
                  <var>.o.
                    <lb/>
                  s.</var>
                (quodidem eſt) vnitate.</s>
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            <div xml:id="echoid-div201" type="math:theorem" level="3" n="107">
              <head xml:id="echoid-head124" xml:space="preserve">THEOREMA
                <num value="107">CVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s937" xml:space="preserve">
                  <emph style="sc">QVod</emph>
                  <reg norm="ſiſecundus" type="context">ſiſecũdus</reg>
                viator
                  <reg norm="ordinem" type="context">ordinẽ</reg>
                  <reg norm="ſecundæ" type="context">ſecũdæ</reg>
                progreſſionis arithmeticæ
                  <reg norm="ſeruansiter" type="context">ſeruãsiter</reg>
                  <lb/>
                agat, nempe ea quæ ab vno per binarium aſcendit, ſemper numerus dierum
                  <lb/>
                æqualis erit numero milliarium diurnorum primi viatoris.</s>
              </p>
              <p>
                <s xml:id="echoid-s938" xml:space="preserve">In cuius gratiam animaduertendum eſt numerus ne milliarium diurnorum pri-
                  <lb/>
                mi viatoris par an impar ſit. </s>
                <s xml:id="echoid-s939" xml:space="preserve">Etenim ſi par eſt, primus viator in fine ſingulorum die-
                  <lb/>
                rum primæ medietatis numeri omnium dierum ſecundum antecedet numero diſpa
                  <lb/>
                ri milliarium; </s>
                <s xml:id="echoid-s940" xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diſpa
                  <lb/>
                ri præteribitur, vt in ſequenti figura patet. </s>
                <s xml:id="echoid-s941" xml:space="preserve">Nam prima die, ſecundus ex primo
                  <lb/>
                milliare vnum ex numero pari, qui à primo conficitur detrahit; </s>
                <s xml:id="echoid-s942" xml:space="preserve">ſecunda verò die
                  <lb/>
                idem ſecundus, duo ſubtrahit milliaria exdiſpari, qui primo reliquus fuerat,
                  <reg norm="ſicque" type="simple">ſicq́;</reg>
                  <lb/>
                perpetuò diſpar remanet vſque ad vnitatem, ad quam cum peruenerint, nempe ad
                  <lb/>
                illius diei exitum, quo primus ſecundum vnitate tantummodò ſuperat, manifeſtè
                  <lb/>
                depræhendetur ſubſequente dieſecundum vnitate primum ſuperaturum, altera ve
                  <lb/>
                rò tribus vnitatibus, prout penultima die ſecundus à primo tribus vnitatibus ſupera
                  <lb/>
                batur. </s>
                <s xml:id="echoid-s943" xml:space="preserve">Quare neceſſe erit, tot diebus ſecundum cum primo iter agere, inchoan-
                  <lb/>
                do ab ea die, qua fecundus primam ſuperabit, quot egerat dum à primo ſuperare-
                  <lb/>
                tur, vt ex communi conceptione, media figura
                  <var>.A.</var>
                depræhendi poteſt. </s>
                <s xml:id="echoid-s944" xml:space="preserve">Quod au-
                  <lb/>
                tem ſingula dimidia dierum, dimi-
                  <lb/>
                  <figure xlink:label="fig-0081-02" xlink:href="fig-0081-02a" number="110">
                    <image file="0081-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-02"/>
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                dia ſint numeri milliarium diurno-
                  <lb/>
                rum primi; </s>
                <s xml:id="echoid-s945" xml:space="preserve">patebit exſequenti fi-
                  <lb/>
                gura, cogitato termino
                  <var>.u.n.</var>
                vlti-
                  <lb/>
                mo progreſſionis ſuperatę à primo
                  <lb/>
                vſque ad vnitatem
                  <var>.e.</var>
                quiterminus
                  <lb/>
                  <var>u.n.</var>
                coniunctus primo
                  <var>.o.</var>
                nempe
                  <var>.e.</var>
                  <lb/>
                ſemper
                  <reg norm="duplum" type="context">duplũ</reg>
                  <unsure/>
                eſt numeritermino-
                  <lb/>
                rum
                  <var>.o.n.</var>
                vt .94. theoremate circa
                  <lb/>
                finem dictum fuit. </s>
                <s xml:id="echoid-s946" xml:space="preserve">Sed
                  <var>.u.n.</var>
                cum
                  <var>.e.</var>
                  <lb/>
                numero æquali conſtat numero
                  <lb/>
                milliarium diurnorum primi viatoris, ex quo ſequitur totum numerum dierum, quo
                  <lb/>
                rum
                  <var>.o.n.</var>
                dimidium eſt, æqualem eſſe numero milliarium diuruorum primi via-
                  <lb/>
                toris.</s>
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