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-div148" type="math:theorem" level="3" n="75">
              <pb o="50" rhead="IO. BAPT. BENED." n="62" file="0062" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0062"/>
              <p>
                <s xml:id="echoid-s656" xml:space="preserve">Progredi nihilominus etiam hac in re poſſemus per differentiam primi & ſecun-
                  <lb/>
                di termini, eam detrahendo aut in ſummam cum ſecunda colligendo, attamen prior
                  <lb/>
                ratio magis latè patet, ideſt vniuerſalior eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div150" type="math:theorem" level="3" n="76">
              <head xml:id="echoid-head93" xml:space="preserve">THEOREMA
                <num value="76">LXXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s657" xml:space="preserve">CVR ſi quis cupiat ſecundum terminum inuenire, quatuor terminorum arith-
                  <lb/>
                meticè proportionalis continuæ, quorum nobis duo extrema proponantur.
                  <lb/>
                </s>
                <s xml:id="echoid-s658" xml:space="preserve">Rectè primum duplicabit
                  <reg norm="coniungetque" type="simple">coniungetq́;</reg>
                vltimo termino, nempe quarto, ex qua ſum-
                  <lb/>
                ma tertiam partem deſumet, quæ erit ſecundus terminus quęſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s659" xml:space="preserve">Exempli gratia, ſi horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema
                  <lb/>
                proponantur. </s>
                <s xml:id="echoid-s660" xml:space="preserve">nempe .12. et .3. quorum ſecundus inueniendus ſit, ſumpto quolibet
                  <lb/>
                pro primo, ſit autem .3. primus numerus, quartus verò .12. </s>
                <s xml:id="echoid-s661" xml:space="preserve">quare duplicato 3. vtpo
                  <lb/>
                tè primo, & coniuncto .12. quarto, ſumma erit .18. cuius eſt tertia pars .6. ſecundus
                  <lb/>
                numerus ſcilicet ſumpto principio à minimo. </s>
                <s xml:id="echoid-s662" xml:space="preserve">Idipſum euenit ſumpto principio à
                  <lb/>
                maximo. </s>
                <s xml:id="echoid-s663" xml:space="preserve">Nam ſi datur ſecundus à minimo aut à maximo, illico tertius datur diffe-
                  <lb/>
                rentia inter hunc & primum, ſecundo coniuncta, aut ex eodem detracta.</s>
              </p>
              <p>
                <s xml:id="echoid-s664" xml:space="preserve">Cuius ratio ſic demonſtratur, quatuor termini quatuor lineis
                  <var>.m.g</var>
                :
                  <var>q.p</var>
                :
                  <var>u.n</var>
                :
                  <var>c.t.</var>
                  <lb/>
                ſignificentur, quorum
                  <var>.m.g.</var>
                et
                  <var>.c.t.</var>
                tantummodo cognoſcantur. </s>
                <s xml:id="echoid-s665" xml:space="preserve">
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                  <var>.m.g.</var>
                primus ac
                  <lb/>
                maior terminus: </s>
                <s xml:id="echoid-s666" xml:space="preserve">k.g. verò ſit duplum primi
                  <var>.m.g</var>
                : cui coniungatur
                  <var>.b.k.</var>
                æqualis
                  <var>.c.t.</var>
                  <lb/>
                Dico tertiam partem
                  <var>.b.g.</var>
                quæ ſumma totalis eſt, æqualem eſſe
                  <var>.q.p</var>
                . </s>
                <s xml:id="echoid-s667" xml:space="preserve">In primis enim
                  <lb/>
                certi ſumus
                  <var>.m.f.</var>
                in
                  <var>.m.g.</var>
                reperiri æqualem
                  <var>.q.p.</var>
                  <reg norm="ſupereſtque" type="simple">ſupereſtq́;</reg>
                  <var>.f.g.</var>
                differentia inter
                  <var>.m.g.</var>
                  <lb/>
                et
                  <var>.q.p.</var>
                æqualis
                  <var>.e.p.</var>
                differentiæ inter
                  <var>.q.p.</var>
                et
                  <var>.u.n.</var>
                & æqualis
                  <var>.o.n.</var>
                differen-
                  <lb/>
                tiæ inter
                  <var>.u.n.</var>
                et
                  <var>.c.t</var>
                : ſimul etiam in
                  <var>.k.m.</var>
                habemus
                  <var>.d.m.</var>
                æqualem
                  <var>.m.f.</var>
                </s>
                <s xml:id="echoid-s668" xml:space="preserve">quare etiam
                  <var>.q.
                    <lb/>
                  p.</var>
                et
                  <var>.k.d.</var>
                æqualem
                  <var>.f.g.</var>
                nempe
                  <var>.e.p.</var>
                aut
                  <var>.o.n</var>
                : Hactenus in
                  <var>.k.g.</var>
                reperimus duplum
                  <var>.q.
                    <lb/>
                  p.</var>
                ſimul cum
                  <var>.f.g.</var>
                et
                  <var>.k.d.</var>
                æqualibus
                  <var>.e.p.</var>
                et
                  <var>.o.n.</var>
                & quia
                  <var>.b.K.</var>
                æqualis
                  <var>.c.t.</var>
                fuit coniuncta.
                  <lb/>
                </s>
                <s xml:id="echoid-s669" xml:space="preserve">conſiderandum eſt an hætres quantitates
                  <var>.f.g</var>
                :
                  <var>K.d.</var>
                et
                  <var>.b.K.</var>
                ſimul æquales ſint
                  <var>.q.p.</var>
                  <lb/>
                quod tamen per ſe manifeſtum eſt. </s>
                <s xml:id="echoid-s670" xml:space="preserve">nam
                  <var>.q.p.</var>
                ſuperat
                  <var>.u.n.</var>
                per
                  <var>.e.p.</var>
                et
                  <var>.u.n.</var>
                ex-
                  <lb/>
                cedit
                  <var>.c.t.</var>
                per
                  <var>.o.n.</var>
                æqualem
                  <var>.e.p.</var>
                </s>
                <s xml:id="echoid-s671" xml:space="preserve">quare
                  <var>.q.p.</var>
                per duplum differentię
                  <var>.f.g.</var>
                ſuperat
                  <var>.c.t.</var>
                ita
                  <lb/>
                que
                  <var>.f.g</var>
                :
                  <var>k.d.</var>
                et
                  <var>.K.b.</var>
                ipſi
                  <var>.q.p.</var>
                ſunt
                  <reg norm="ae- quales" type="simple">ę-
                    <lb/>
                  quales</reg>
                , ex quo ſequitur
                  <var>.q.p.</var>
                  <reg norm="tertiam" type="context">tertiã</reg>
                  <lb/>
                  <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a" number="85">
                    <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0062-01"/>
                  </figure>
                partem eſſe
                  <var>.b.g.</var>
                Hæc quæ hacte-
                  <lb/>
                nus dicta fuerunt, in genere maio-
                  <lb/>
                ris inæqualitatis probata fuerunt.
                  <lb/>
                </s>
                <s xml:id="echoid-s672" xml:space="preserve">At in genere minoris, ſumpto or-
                  <lb/>
                dinis principio à minimo termino
                  <lb/>
                rum, duplicetur
                  <var>.c.t.</var>
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                duplum
                  <lb/>
                hoc
                  <var>.K.t.</var>
                cui
                  <var>.k.b.</var>
                æqualis
                  <var>.m.g.</var>
                con-
                  <lb/>
                iungatur, quæſumma ſit
                  <var>.b.t</var>
                . </s>
                <s xml:id="echoid-s673" xml:space="preserve">Di-
                  <lb/>
                co
                  <var>.u.n.</var>
                tertiam eſſe partem ipſius.
                  <lb/>
                </s>
                <s xml:id="echoid-s674" xml:space="preserve">Nam in primis in
                  <var>.b.t.</var>
                datur termi
                  <lb/>
                nus
                  <var>.b.K.</var>
                æqualis vltimo
                  <var>.m.g.</var>
                in
                  <lb/>
                quo ſemel reperitur
                  <var>.u.n.</var>
                vnà cum
                  <lb/>
                duabus differentijs, nempe
                  <var>.i.g.</var>
                in
                  <lb/>
                ipſa autem
                  <var>.b.t</var>
                :
                  <var>u.n.</var>
                ſignificetur pri
                  <lb/>
                mo loco per
                  <var>.r.K.</var>
                ex quo ſupererit
                  <var>.b.r.</var>
                duabus differentijs prædictis æqualis, ſed ex
                  <lb/>
                præſuppoſito
                  <var>.u.n.</var>
                componitur ex
                  <var>.o.u.</var>
                æquali
                  <var>.c.t.</var>
                et
                  <var>.o.n.</var>
                ęquali vni differentiæ. </s>
                <s xml:id="echoid-s675" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                </s>
              </p>
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