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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THEOREM. ARITH.
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            <div xml:id="echoid-div244" type="math:theorem" level="3" n="128">
              <p>
                <s xml:id="echoid-s1134" xml:space="preserve">
                  <pb o="87" rhead="THEOREM. ARITH." n="99" file="0099" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0099"/>
                tionatus .216. ad .156. vt .18. ad .13. maniteſtum eſt exijſdem, nam tam .18. quam
                  <num value="13">.
                    <lb/>
                  13.</num>
                multiplicatus fuit per .12.</s>
              </p>
            </div>
            <div xml:id="echoid-div245" type="math:theorem" level="3" n="129">
              <head xml:id="echoid-head147" xml:space="preserve">THEOREMA
                <num value="129">CXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s1135" xml:space="preserve">ALIVD proponitur problema hoc modo: </s>
                <s xml:id="echoid-s1136" xml:space="preserve">ſupponitur obſidio alicuius loci, vbi
                  <lb/>
                alimento ad nutriendos .10000. homines ſufficiunt pro quinque menſibus tan-
                  <lb/>
                tum, ſed quia eum locum obſidione non liberari putatur niſi .18. menſibus exactis,
                  <lb/>
                quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eſt .18.
                  <lb/>
                menſibus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1137" xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eſt hominum .10000. cum
                  <lb/>
                ſecundo, hoc eſt menſium quinque, productum verò diuidatur per .18. hoc eſt men-
                  <lb/>
                ſium, </s>
                <s xml:id="echoid-s1138" xml:space="preserve">tunc proueniet .2777. cum .7. nonis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1139" xml:space="preserve">Cuius operationis ratio eſt hæc, ſint exempli gratia duo hic ſubſcripta producta
                  <lb/>
                ſuperficialia
                  <var>.a.n.</var>
                et
                  <var>.o.u.</var>
                inuicem æqualia, ſed tal@ figura delineata, vt proportio
                  <var>.u.
                    <lb/>
                  x.</var>
                ad
                  <var>.x.o.</var>
                ſit, vt .10000. ad quinque, & proportio
                  <var>a.x.</var>
                ad
                  <var>.x.o.</var>
                ſit vt .18. ad quinque,
                  <lb/>
                ct
                  <var>.x.n.</var>
                ſit nobis ignota, quæ quidem eſt illa, quæ indagatur, ita
                  <reg norm="quod" type="simple">ꝙ</reg>
                vnumquodque
                  <lb/>
                iſtorum productorum ſignificabit alimentum, et
                  <var>.u.x.</var>
                ſignificabit numerum homi-
                  <lb/>
                num .10000. qui quidem homines comederent totum alimentum
                  <var>.u.o.</var>
                ſpacio tem-
                  <lb/>
                poris
                  <var>.x.o.</var>
                quinque menſium, proptereà quòd
                  <var>u.o.</var>
                ſupponitur productum eſſe ab
                  <var>.
                    <lb/>
                  u.x.</var>
                in
                  <var>.x.o</var>
                . </s>
                <s xml:id="echoid-s1140" xml:space="preserve">Deinde
                  <reg norm="ſupponendo" type="context">ſupponẽdo</reg>
                  <var>.a.x.</var>
                tem
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/>
                pus eſſe .18. menſium, ergo
                  <var>.x.n.</var>
                ſignifi-
                  <lb/>
                cabit numerum hominum, qui eo tem-
                  <lb/>
                poris ſpacio ali poſſunt, hoc eſt
                  <var>.x.a.</var>
                ali-
                  <lb/>
                mento
                  <var>.n.a.</var>
                eo quòd
                  <var>.a.n.</var>
                producitur ex
                  <var>.
                    <lb/>
                  n.x.</var>
                in
                  <var>.a.x.</var>
                vnde ex .15. ſexti, ſeu ex, 20.
                  <lb/>
                ſeptimi proportio
                  <var>.x.u.</var>
                ad
                  <var>.x.n.</var>
                  <reg norm="eadem" type="context">eadẽ</reg>
                erit,
                  <lb/>
                quę
                  <var>.a.x.</var>
                ad
                  <var>.x.o.</var>
                quapropter rectè factum
                  <lb/>
                erit accipere
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.u.o.</var>
                quodidem
                  <lb/>
                eſt in quantitate, quod productum .2. n. & ipſum diuidere per
                  <var>.a.x.</var>
                vnde nobis
                  <lb/>
                proueniat
                  <var>.n.x</var>
                .</s>
              </p>
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                <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a">
                  <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0099-01"/>
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              </div>
            </div>
            <div xml:id="echoid-div247" type="math:theorem" level="3" n="130">
              <head xml:id="echoid-head148" xml:space="preserve">THEOREMA
                <num value="130">CXXX</num>
              .</head>
              <p>
                <s xml:id="echoid-s1141" xml:space="preserve">QVotieſcunque nobis propoſitum fuerit inuenire tertium terminum, trium ter
                  <lb/>
                minorum continuè proportionalium armonicæ proportionalitatis, quo-
                  <lb/>
                tum duo nobis cogniti ſint, ita agemus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1142" xml:space="preserve">Sint, exempli gratia, tres termini
                  <var>.q.p</var>
                :
                  <var>a.g.</var>
                et
                  <var>.e.c.</var>
                continuæ proportionalium at
                  <lb/>
                monicæ proportionalitatis, quorum
                  <var>.q.p.</var>
                maior et
                  <var>.a.g.</var>
                medius ſint nobis cogniti,
                  <lb/>
                cum ergo voluerimus tertium
                  <var>.e.
                    <lb/>
                  c.</var>
                cognitum nobis eſſe: </s>
                <s xml:id="echoid-s1143" xml:space="preserve">a.g. detra-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0099-02a" xlink:href="fig-0099-02"/>
                hatur ex
                  <var>.q.p.</var>
                differentia verò
                  <var>.d.
                    <lb/>
                  p.</var>
                addatur
                  <var>.q.p.</var>
                quorum ſumma
                  <lb/>
                erit
                  <var>.q.o.</var>
                cognita, qua mediante
                  <lb/>
                diuidatur productum, quod ex
                  <var>.a.
                    <lb/>
                  g.</var>
                in
                  <var>.d.p.</var>
                exurgit, & proueniet no
                  <lb/>
                bis
                  <var>.n.g.</var>
                hoc e@t minor differentia, eo quòd productum
                  <var>.q.o.</var>
                in
                  <var>.n.g.</var>
                æquale eſt pro- </s>
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