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-div187" type="math:theorem" level="3" n="96">
              <p>
                <s xml:id="echoid-s851" xml:space="preserve">
                  <pb o="64" rhead="IO. BAPT. BENED." n="76" file="0076" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0076"/>
                numerus quæſitus erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s852" xml:space="preserve">Quod intelligendum eſttamen quoties primus terminus differentia
                  <reg norm="terminorum" type="context">terminorũ</reg>
                  <lb/>
                eſt, nempe aſcendens ipſorum ter minorum.</s>
              </p>
              <p>
                <s xml:id="echoid-s853" xml:space="preserve">Cuius ratio manifeſtè ſpeculari poteſt in figura præcedentis theorematis. </s>
                <s xml:id="echoid-s854" xml:space="preserve">Nam
                  <lb/>
                diuiſa
                  <var>.a.o.</var>
                per
                  <var>.n.n.n.n.</var>
                eadem proportio erit
                  <var>.a.o.</var>
                ad proueniens, quæ. n
                  <var>.n.n.
                    <lb/>
                  n.</var>
                ad vnitatem
                  <var>.n.</var>
                ex definitione diuiſionis. </s>
                <s xml:id="echoid-s855" xml:space="preserve">At ſuperius dictum fuit ita ſe ha bere
                  <var>.a.
                    <lb/>
                  o.</var>
                ad
                  <var>.o.n.</var>
                vt
                  <var>.n.n.n.n.</var>
                ad
                  <var>.n.</var>
                ex quo ſequitur ex .11. et .9. quinti pr oueniens eſſe nume-
                  <lb/>
                rum quæſitum
                  <var>.o.n</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div188" type="math:theorem" level="3" n="97">
              <head xml:id="echoid-head114" xml:space="preserve">THEOREMA
                <num value="97">XCVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s856" xml:space="preserve">VBI verò primus terminus, reliquorum non erit differentia. </s>
                <s xml:id="echoid-s857" xml:space="preserve">Hac de caufa ne-
                  <lb/>
                ceſſe eſt detrahere primum ex vltimo,
                  <reg norm="reſiduumque" type="simple">reſiduumq́;</reg>
                per numerum aſcenden-
                  <lb/>
                tem differentiam ſcilicet, partiri,
                  <reg norm="proueniensque" type="simple">proueniensq́;</reg>
                vnitati coniungere, quò numerum
                  <lb/>
                terminorum habere poſſimus. </s>
                <s xml:id="echoid-s858" xml:space="preserve">Scimus etenim tam multas vnitates eſſe in vltimo
                  <lb/>
                terminorum quot in omnibus interuallis aut differentijs in ſummam collectis ſimul
                  <lb/>
                cum vnitatibus primi termini,
                  <reg norm="totque" type="simple">totq́;</reg>
                funt termini, quot interualla ſimul cum pri-
                  <lb/>
                motermino. </s>
                <s xml:id="echoid-s859" xml:space="preserve">Quare fi minimus terminus interuallo æqualis fuerit. </s>
                <s xml:id="echoid-s860" xml:space="preserve">Vltimo per pri-
                  <lb/>
                mum diuiſo, ex a dductis præcedenti theoremate propofitum confequemur. </s>
                <s xml:id="echoid-s861" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <lb/>
                primo termino ex vltimo detracto
                  <reg norm="refiduoque" type="simple">refiduoq́;</reg>
                per interuallum, hoc eft numerum dif-
                  <lb/>
                ferentiæ diuifo, proueniens erit numerus terminorum abſque primo quod vnus eft,
                  <lb/>
                coni uncto quoque dicto prouenienti propoſitum conſequemur.</s>
              </p>
            </div>
            <div xml:id="echoid-div189" type="math:theorem" level="3" n="98">
              <head xml:id="echoid-head115" xml:space="preserve">THEOREMA
                <num value="98">XCVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s862" xml:space="preserve">CVR fi quis arithmeticæ progreſſionis dato primo & vltimo fimul cum nume
                  <lb/>
                ro terminorum, afcendentem numerum cognofcere voluerit. </s>
                <s xml:id="echoid-s863" xml:space="preserve">Rectè primuin
                  <lb/>
                ex vltimo detrahet,
                  <reg norm="refiduumque" type="simple">refiduumq́;</reg>
                per numerum terminorum excepto vno diuidet.</s>
              </p>
              <p>
                <s xml:id="echoid-s864" xml:space="preserve">Huius theorematis ſpeculatio ex .13. theoremate manifeſta crit, nam in præce-
                  <lb/>
                denti cap. numerus terminorum erat proueniens diuiſionis reſidui ſubtractionis pri-
                  <lb/>
                mi termini ex vltimo.</s>
              </p>
            </div>
            <div xml:id="echoid-div190" type="math:theorem" level="3" n="99">
              <head xml:id="echoid-head116" xml:space="preserve">THEOREMA
                <num value="99">XCIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s865" xml:space="preserve">CVR ſi quis maximum omnium terminorum dictæ progreffionis cognofcere
                  <lb/>
                voluerit, dato primo vnà cum numero aſcendenti,
                  <reg norm="numeroque" type="simple">numeroq́;</reg>
                terminorum. </s>
                <s xml:id="echoid-s866" xml:space="preserve">Re-
                  <lb/>
                ctè numerum afcendentem cum numero terminorum excepto vno multiplicabit,
                  <lb/>
                  <reg norm="productoque" type="simple">productoq́;</reg>
                primum terminum coniunget.</s>
              </p>
              <p>
                <s xml:id="echoid-s867" xml:space="preserve">Cuius quidem theorematis tum ex vndecimo, tum ex ijs quæ præcedentibus ca-
                  <lb/>
                pitibus dicta fuerunt, aperta eſt ratio.</s>
              </p>
            </div>
            <div xml:id="echoid-div191" type="math:theorem" level="3" n="100">
              <head xml:id="echoid-head117" xml:space="preserve">THEOREMA
                <num value="100">C</num>
              .</head>
              <p>
                <s xml:id="echoid-s868" xml:space="preserve">CVR veteres cupientes obtinere ſummam pr
                  <unsure/>
                ogreffionis continuæ naturalis,
                  <lb/>
                quæab vnitate initium ducit, dato vltimo termino tantummodo. </s>
                <s xml:id="echoid-s869" xml:space="preserve">Dimidium
                  <lb/>
                vltimi-termini
                  <reg norm="cum" type="context">cũ</reg>
                toto fequente multiplicabant,
                  <reg norm="productumque" type="simple">productumq́;</reg>
                ſumma quæſita erat.</s>
              </p>
              <p>
                <s xml:id="echoid-s870" xml:space="preserve">Exempli gratia, ſi vltimus terminus eiuſmodi progreſſionis fuerit .7. multiplica- </s>
              </p>
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