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-div102" type="math:theorem" level="3" n="48">
              <p>
                <s xml:id="echoid-s425" xml:space="preserve">
                  <pb o="32" rhead="I.O. BAPT. BENED." n="44" file="0044" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0044"/>
                licet quanta ſumma eſt maioris cum proueniente.</s>
              </p>
              <p>
                <s xml:id="echoid-s426" xml:space="preserve">Cuius ſpeculationis cauſa, maior numerus ſignificetur
                  <var>.a.i.</var>
                et minor linea
                  <var>.a.o.</var>
                ex
                  <lb/>
                quo ex præſupoſito
                  <var>.o.i.</var>
                vnitas erit. </s>
                <s xml:id="echoid-s427" xml:space="preserve">Sit autem proueniens ex diuiſione
                  <var>.a.i.</var>
                per
                  <var>.a.o.
                    <lb/>
                  a.e</var>
                : </s>
                <s xml:id="echoid-s428" xml:space="preserve">quod
                  <var>.e.a.</var>
                directè coniungatur ipſi
                  <var>.a.i.</var>
                et productum
                  <var>.a.i.</var>
                in
                  <var>.a.e.</var>
                ſit
                  <var>.u.i</var>
                . </s>
                <s xml:id="echoid-s429" xml:space="preserve">Probabo
                  <lb/>
                numerum ſuperficialem
                  <var>.u.i.</var>
                æqualem eſſe lineari
                  <var>.i.a.e</var>
                . </s>
                <s xml:id="echoid-s430" xml:space="preserve">quare meminiſſe oportet,
                  <lb/>
                decimotertio theoremate probatum fuiſſe, quod ſi numerus diuiſibilis per pro-
                  <lb/>
                ueniens diuidatur, proueniens futurus ſit numerus diuidens, </s>
                <s xml:id="echoid-s431" xml:space="preserve">quare
                  <var>.a.o.</var>
                erit pro-
                  <lb/>
                ueniens ex diuiſione
                  <var>.a.i.</var>
                per
                  <var>.a.e.</var>
                & ex deſinitione diuiſionis ita ſe habebit
                  <var>.e.a.</var>
                ad
                  <var>.
                    <lb/>
                  a.i.</var>
                ſicut
                  <var>.o.i.</var>
                ad
                  <var>.o.a.</var>
                & componondo ita
                  <var>.e.i.</var>
                ad
                  <var>.a.i.</var>
                ſicut
                  <var>.i.a.</var>
                ad
                  <var>.o.a.</var>
                </s>
                <s xml:id="echoid-s432" xml:space="preserve">quare
                  <var>.a.i.</var>
                erit me-
                  <lb/>
                dia pportionalis inter
                  <var>.e.i.</var>
                et
                  <var>.a.o.</var>
                ſed
                  <var>.a.i.</var>
                non modò diuiſa
                  <reg norm="nunc" type="context">nũc</reg>
                cogitatur ab
                  <var>.e.a.</var>
                ex
                  <lb/>
                quo ſit proueniens
                  <var>.a.o.</var>
                ſed etiam per eandem
                  <var>.e.a.</var>
                multiplicata, ex quo produ-
                  <lb/>
                ctum oriatur
                  <var>.u.i</var>
                . </s>
                <s xml:id="echoid-s433" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex .25. theobema-
                  <lb/>
                te
                  <var>.a.i.</var>
                media eſt proportionalis inter
                  <var>.u.</var>
                  <lb/>
                  <figure xlink:label="fig-0044-01" xlink:href="fig-0044-01a" number="60">
                    <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-01"/>
                  </figure>
                i. et
                  <var>.a.o</var>
                . </s>
                <s xml:id="echoid-s434" xml:space="preserve">Quare. ex .11. quinti. eadem erit
                  <lb/>
                proportio
                  <var>.u.i.</var>
                ad
                  <var>.a.i.</var>
                ſicut
                  <var>.e.i.</var>
                ad eandem
                  <var>.
                    <lb/>
                  a.i</var>
                . </s>
                <s xml:id="echoid-s435" xml:space="preserve">Igitur ex .9. prædicti numerus
                  <var>.u.i.</var>
                  <lb/>
                æqualis erit numero
                  <var>.e.i.</var>
                quod erat propoſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div104" type="math:theorem" level="3" n="49">
              <head xml:id="echoid-head65" xml:space="preserve">THEOREMA
                <num value="49">XLIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s436" xml:space="preserve">IDipſtim etiam alia ratione conſiderari poteſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s437" xml:space="preserve">Linea
                  <var>.u.a.</var>
                ſecetur in puncto
                  <var>.t.</var>
                ita vt
                  <var>.a.t.</var>
                æqualis ſit vnitati
                  <var>.o.i.</var>
                & media paral
                  <lb/>
                lela
                  <var>.t.n.</var>
                terminetur productum
                  <var>.t.i.</var>
                quod conſtabit æquali numero, quamuis ſuperfi-
                  <lb/>
                ciali, numero
                  <var>.a.i.</var>
                tametſi lineari. </s>
                <s xml:id="echoid-s438" xml:space="preserve">Tumparallela ducatur à puncto
                  <var>.o.</var>
                ipſi
                  <var>.a.u.</var>
                termi
                  <lb/>
                  <reg norm="neturque" type="simple">neturq́;</reg>
                productum
                  <var>.o.u.</var>
                ex quo bina producta dabuntur
                  <var>.u.o.</var>
                et
                  <var>.t.i.</var>
                inter ſe æqualia
                  <lb/>
                ex .15. ſexti aut .20. ſeptimi cum ita ſe habeat
                  <var>.a.i.</var>
                ad
                  <var>.a.u.</var>
                ſicut
                  <var>.a.o.</var>
                ad
                  <var>.a.t.</var>
                ſed
                  <var>.a.i.</var>
                ad
                  <var>.
                    <lb/>
                  a.o.</var>
                permutando ſic ſe habet ſicut
                  <var>.a.u.</var>
                ad
                  <var>.a.t.</var>
                & ex prima ſexti aut .18. vel .19. ſepti-
                  <lb/>
                mi ſic ſe habet
                  <var>.u.i.</var>
                ad
                  <var>.u.o.</var>
                ſicut
                  <var>.a.i.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                  <figure xlink:label="fig-0044-02" xlink:href="fig-0044-02a" number="61">
                    <image file="0044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-02"/>
                  </figure>
                o. hoc eſt
                  <var>.u.i.</var>
                ad
                  <var>.t.i.</var>
                ope .11. quinti. </s>
                <s xml:id="echoid-s439" xml:space="preserve">Iam
                  <lb/>
                ex definitione diuiſionis ita ſe habet
                  <var>.a.e.</var>
                  <lb/>
                ad
                  <var>.a.i.</var>
                ſicut
                  <var>.o.i.</var>
                ad
                  <var>.o.a.</var>
                & componendo
                  <var>.
                    <lb/>
                  e.i.</var>
                ad
                  <var>.a.i.</var>
                ſicut
                  <var>.i.a.</var>
                ad
                  <var>.o.a</var>
                . </s>
                <s xml:id="echoid-s440" xml:space="preserve">Itaque ex præ-
                  <lb/>
                dicta .11. ſic ſe habebit
                  <var>.e.i.</var>
                ad
                  <var>.i.a.</var>
                ſicut
                  <var>.u.
                    <lb/>
                  i.</var>
                ad
                  <var>.t.i.</var>
                ſed
                  <var>.t.i.</var>
                numero conſtat æquali
                  <var>.a.
                    <lb/>
                  i</var>
                . </s>
                <s xml:id="echoid-s441" xml:space="preserve">quare ex .9. quinti numerus
                  <var>.u.i.</var>
                numero
                  <var>.e.i.</var>
                æqualis erit.</s>
              </p>
            </div>
            <div xml:id="echoid-div106" type="math:theorem" level="3" n="50">
              <head xml:id="echoid-head66" xml:space="preserve">THEOREMA
                <num value="50">L</num>
              .</head>
              <p>
                <s xml:id="echoid-s442" xml:space="preserve">CVR diuidentes numerum propoſitum in duas eiuſmodi partes, vt
                  <reg norm="productum" type="context">productũ</reg>
                  <lb/>
                vnius in alteram cum i pſarum differentia in ſummam collectum, æquale ſit
                  <lb/>
                alicui alteri numero maiori primo. </s>
                <s xml:id="echoid-s443" xml:space="preserve">Rectè primum ex ſecundo detrahunt, reſiduum
                  <lb/>
                verò conſeruant, tum ex primo ſemper binarium deſumunt,
                  <reg norm="dimidiumque" type="simple">dimidiumq́;</reg>
                conſer-
                  <lb/>
                uant, alterum verò dimidium in ſeipſo multiplicant, & ex quadrato numerum con
                  <lb/>
                ſeruatum eruunt,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                radicem ex dimidio conſeruato, quod vltimum reſi-
                  <lb/>
                duum propoſiti numeri quæſita pars minor eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s444" xml:space="preserve">Exempli gratia, ſi proponatur numerus .20. ita
                  <reg norm="diuidendus" type="context">diuidẽdus</reg>
                , vt
                  <reg norm="productum" type="context">productũ</reg>
                vnius partis
                  <lb/>
                in alteram, cum partium differentia collectum in ſummam, æquale ſit propoſito </s>
              </p>
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