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. ARIT.
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            <div xml:id="echoid-div150" type="math:theorem" level="3" n="76">
              <p>
                <s xml:id="echoid-s675" xml:space="preserve">
                  <pb o="51" rhead="THEOREM. ARIT." n="63" file="0063" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0063"/>
                cum in
                  <var>.b.t.</var>
                præter
                  <var>.r.K.</var>
                bis detur
                  <var>.c.t.K.t.</var>
                et
                  <var>.b.r.</var>
                duabus differentijs æquipol-
                  <lb/>
                lens, illud efficitur
                  <var>.u.n.</var>
                pariter ipſius
                  <var>.b.t.</var>
                eſſe tertiam partem, quod erat
                  <reg norm="propoſitum" type="context">propoſitũ</reg>
                .</s>
              </p>
              <div xml:id="echoid-div150" type="float" level="4" n="1">
                <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a">
                  <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0062-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div152" type="math:theorem" level="3" n="77">
              <head xml:id="echoid-head94" xml:space="preserve">THEOREMA
                <num value="77">LXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s676" xml:space="preserve">CVR ſi quis velit ſecundum quinque continuorum proportionalium termi-
                  <lb/>
                num inuenire, ſolis extremis cognitis. </s>
                <s xml:id="echoid-s677" xml:space="preserve">Rectè
                  <reg norm="vltimum" type="context">vltimũ</reg>
                triplo primi coniunget,
                  <lb/>
                ex qua ſumma quartam partem detraher, quæ erit ſecundus terminus quæſitus.
                  <lb/>
                </s>
                <s xml:id="echoid-s678" xml:space="preserve">Quod ipſum faciet qui inuenire vult ſecundum terminum ſenarij ſeptenarij, octo-
                  <lb/>
                narij aut alterius cuiuſcunque, creſcente tamen multiplicatione primi,
                  <reg norm="vltimoque" type="simple">vltimoq́;</reg>
                  <lb/>
                coniuncto.</s>
              </p>
              <p>
                <s xml:id="echoid-s679" xml:space="preserve">Exempli gratia, dantur duo extremi termini, horum quinque numerorum .18.
                  <lb/>
                16. 14. 12. 10. nempe .18. et .10. ſi .18. primus erit, hoc eſt, ſi à genere maioris inæ-
                  <lb/>
                qualitatis progrediemur, triplicabimus terminum .18.
                  <reg norm="dabunturque" type="simple">dabunturq́;</reg>
                .54. cui numero
                  <lb/>
                coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo
                  <lb/>
                tè ſecundus terminus gratia, aut ſecundi ſex terminorum, quadruplicandus eſſet pri
                  <lb/>
                mus .18. deinde adiuncto vltimo, quinta pars ſummæ eſſet ſecundus terminus,
                  <reg norm="atque" type="simple">atq;</reg>
                  <lb/>
                ita deinceps.</s>
              </p>
              <p>
                <s xml:id="echoid-s680" xml:space="preserve">Cuius ſpeculationis gratia, dicti termini lineis
                  <var>.z.h</var>
                :
                  <var>f.s</var>
                :
                  <var>u.p</var>
                :
                  <var>e.g.</var>
                et
                  <var>.r.x.</var>
                  <reg norm="ſigniſicentur" type="context">ſigniſicẽtur</reg>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s681" xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus
                  <var>.z.h.</var>
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                triplum hoc
                  <var>.k.
                    <lb/>
                  h.</var>
                  <reg norm="cuiconiungatur" type="context">cuicõiungatur</reg>
                  <var>.b.k.</var>
                ęqualis vltimo termino
                  <var>.r.x</var>
                . </s>
                <s xml:id="echoid-s682" xml:space="preserve">Dico
                  <var>.f.s.</var>
                  <reg norm="quartam" type="context">quartã</reg>
                partem eſſe ſum-
                  <lb/>
                  <var>.b.h</var>
                . </s>
                <s xml:id="echoid-s683" xml:space="preserve">Nam in
                  <var>.k.h.</var>
                ſecundus terminus
                  <var>.f.s.</var>
                ter cum tribus differentijs æqualibus
                  <var>.n.h.</var>
                  <lb/>
                reperitur. </s>
                <s xml:id="echoid-s684" xml:space="preserve">Probandum nunc eſt tres has differentias
                  <var>.n.h</var>
                :
                  <var>a.c.</var>
                et
                  <var>.d.k.</var>
                ſimul cum
                  <var>.b.
                    <lb/>
                  K.</var>
                ęquales eſſe
                  <var>.f.s.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0063-01a" xlink:href="fig-0063-01"/>
                quod in
                  <reg norm="dubium" type="context">dubiũ</reg>
                re
                  <lb/>
                uocari
                  <reg norm="non" type="context">nõ</reg>
                poteſt,
                  <lb/>
                cum
                  <var>.f.s.</var>
                ſuperet
                  <var>.
                    <lb/>
                  r.x.</var>
                per
                  <var>.o.s</var>
                :
                  <var>t.p.</var>
                et
                  <var>.
                    <lb/>
                  i.g</var>
                . </s>
                <s xml:id="echoid-s685" xml:space="preserve">At in genere
                  <lb/>
                minoris inæquali
                  <lb/>
                tatis, triplum
                  <var>.r.x.</var>
                  <lb/>
                ſit
                  <var>.x.a.</var>
                et
                  <var>.a.b.</var>
                ſit
                  <lb/>
                æqualis
                  <var>.z.h.</var>
                &
                  <reg norm="cum" type="context">cũ</reg>
                  <lb/>
                  <var>z.h.</var>
                tribus
                  <reg norm="differem" type="context">differẽ</reg>
                  <lb/>
                tijs
                  <var>.n.h</var>
                :
                  <var>o.s</var>
                :
                  <var>t.p.</var>
                ſu-
                  <lb/>
                peret
                  <var>.e.g.</var>
                quæ in
                  <var>.
                    <lb/>
                  a.b.</var>
                ſint
                  <var>.b.K</var>
                :
                  <var>K.d</var>
                :
                  <lb/>
                  <var>d.c.</var>
                ex quo
                  <var>.a.c.</var>
                  <lb/>
                æqualis erit
                  <var>.e.g.</var>
                  <lb/>
                et
                  <var>.a.x.</var>
                cum
                  <var>.b.c.</var>
                tripla
                  <var>.e.g</var>
                . </s>
                <s xml:id="echoid-s686" xml:space="preserve">Itaque tota ſumma
                  <var>.b.x.</var>
                qua drupla erit
                  <var>.e.g</var>
                .</s>
              </p>
              <div xml:id="echoid-div152" type="float" level="4" n="1">
                <figure xlink:label="fig-0063-01" xlink:href="fig-0063-01a">
                  <image file="0063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0063-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div154" type="math:theorem" level="3" n="78">
              <head xml:id="echoid-head95" xml:space="preserve">THEOREMA
                <num value="78">LXXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s687" xml:space="preserve">QVantitates quæ fuerint inuicem in proportionalitate arithmetica proportio-
                  <lb/>
                nales, permutan do quoque proportionales erunt.</s>
              </p>
            </div>
          </div>
        </div>
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