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-div258" type="math:theorem" level="3" n="135">
              <p>
                <s xml:id="echoid-s1187" xml:space="preserve">
                  <pb o="91" rhead="THEOREM. ARIT." n="103" file="0103" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0103"/>
                propoſiti
                  <var>.a.e.</var>
                maior, et
                  <var>.e.o.</var>
                minor,
                  <reg norm="Sitque" type="simple">Sitq́;</reg>
                  <var>.o.k.</var>
                medius arithmeticus inter dictos, vn-
                  <lb/>
                de clarè patebit
                  <var>.o.k.</var>
                eſſe dimidium ſummæ dictorum terminorum ex .75. theorema
                  <lb/>
                te huius libri. </s>
                <s xml:id="echoid-s1188" xml:space="preserve">Sit ergo productum
                  <var>a.t.</var>
                id quod fit ex
                  <var>.a.e.</var>
                in
                  <var>.o.k.</var>
                et
                  <var>.o.t.</var>
                ſit
                  <reg norm="productum" type="context">productũ</reg>
                  <lb/>
                quod fit ex
                  <var>.e.o.</var>
                in
                  <var>.o.k.</var>
                et
                  <var>.n.m.</var>
                ſit productum quod ſit ex
                  <var>.a.e.</var>
                in
                  <var>.e.o.</var>
                quorum vnum-
                  <lb/>
                quodque erit dimid ium vniuſcuiuſque producti præcedentis theorematis,
                  <lb/>
                ex .18. et .19. ſeptimi Eucli. vnumquodque ſui relatiui. </s>
                <s xml:id="echoid-s1189" xml:space="preserve">Quare argumentando per
                  <lb/>
                mutando à concluſionibus præcedentis theorematis ad has præſentis, habebimus
                  <lb/>
                productum.</s>
              </p>
            </div>
            <div xml:id="echoid-div260" type="math:theorem" level="3" n="136">
              <head xml:id="echoid-head154" xml:space="preserve">THEOREMA
                <num value="136">CXXXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s1190" xml:space="preserve">MEDIVM autem contra
                  <reg norm="harmonicum" type="context">harmonicũ</reg>
                inuenire cum quis voluesit inter duos
                  <lb/>
                propoſitos terminos, ita faciendum erit, hoc eſt per ſummam datorum ex
                  <lb/>
                tremorum diuidatur productum quod fit ex minimo termino in
                  <reg norm="differentiam" type="context">differẽtiam</reg>
                dato-
                  <lb/>
                rum, prouentus poſtea erit differentia inter maximum & med
                  <unsure/>
                um quæſitum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1191" xml:space="preserve">Vt exempli gratia, ſi nobis propoſiti fuerint hi duo termini .3. et .2. ſumma eo-
                  <lb/>
                rum erit quinque, per quam cum diuiſerimus productum, quod naſcitur ex mini-
                  <lb/>
                mo .2. in differentiam eorum, quæ eſt vnum, quod quidem erit .2. </s>
                <s xml:id="echoid-s1192" xml:space="preserve">tunc duæ quintæ
                  <lb/>
                partes prouenient, quæ ſi demptæ fuerint ex maximo termino, reliquum erit .2.
                  <reg norm="cum" type="context">cũ</reg>
                  <lb/>
                3. quintis, hoc eſt medius terminus contta harmonicus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1193" xml:space="preserve">Pro cuius ratione cogitemus
                  <var>.u.d.</var>
                et
                  <var>.x.c.</var>
                eſſe duosterminosnobis propoſitos, in-
                  <lb/>
                ter quos deſideremus inuenire
                  <var>.o.s.</var>
                medium ita illis
                  <reg norm="relatum" type="context">relatũ</reg>
                , vt proportio exceſſus ip-
                  <lb/>
                ſius ſupra
                  <var>.x.c.</var>
                (qui ſit
                  <var>.e.n.</var>
                ) ad exceſ-
                  <lb/>
                ſum
                  <var>.u.d.</var>
                ſupra
                  <var>.o.s.</var>
                (qui ſit
                  <var>.n.d.</var>
                ) ea-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/>
                dem ſit quæ
                  <var>.u.d.</var>
                ad
                  <var>.x.c</var>
                .</s>
              </p>
              <div xml:id="echoid-div260" type="float" level="4" n="1">
                <figure xlink:label="fig-0103-01" xlink:href="fig-0103-01a">
                  <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0103-01"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s1194" xml:space="preserve">Cogitemus igitur
                  <var>.x.c.</var>
                coniunctum
                  <lb/>
                eſſe cum
                  <var>.u.d.</var>
                & hæcſumma vocetur
                  <var>.
                    <lb/>
                  b.d.</var>
                vnde habebimus proportionem
                  <var>.
                    <lb/>
                  u.d.</var>
                ad
                  <var>.u.b.</var>
                vt
                  <var>.e.n.</var>
                ad
                  <var>.n.d</var>
                . </s>
                <s xml:id="echoid-s1195" xml:space="preserve">Quare
                  <reg norm="com- ponendo" type="context">cõ-
                    <lb/>
                  ponendo</reg>
                ita erit
                  <var>.d.b.</var>
                ad
                  <var>.u.b.</var>
                ut
                  <var>.e.d.</var>
                3d.n.d. ſed quia
                  <var>.d.b</var>
                :
                  <var>u.b.</var>
                et
                  <var>.e.d.</var>
                quantitates no-
                  <lb/>
                bis cognitę ſunt, ideò
                  <var>.d.n.</var>
                ex .20. ſeptimi cognita nobis erit.</s>
              </p>
            </div>
            <div xml:id="echoid-div262" type="math:theorem" level="3" n="137">
              <head xml:id="echoid-head155" xml:space="preserve">THEOREMA
                <num value="137">CXXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1196" xml:space="preserve">SVpponunt antiqui aliquot mercatores dantes pecunias lucro in diuerſis vnius
                  <lb/>
                anni temporibus, </s>
                <s xml:id="echoid-s1197" xml:space="preserve">tunc in fine anni ſumma torius lucri datur cognita, ſed quæ-
                  <lb/>
                ritur quantuni
                  <unsure/>
                vnicuique illorum exipſa ſumma debeatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1198" xml:space="preserve">Exempli gratia, primus in principio anni poſuit .100. aurcos, ſecundus verò .100
                  <lb/>
                diebus poſt primum poſuit .50. aureos tertius autem .200. diebus poſt primum po-
                  <lb/>
                ſuit .25. aureos ſumma lucri poſtea in fine anni fuit aureorum .60.</s>
              </p>
              <p>
                <s xml:id="echoid-s1199" xml:space="preserve">Nunc vt ſciamus quantum huius ſummæ vniduique illorum proueniat, præcipit
                  <lb/>
                regula, vt faciamus tria producta, quorum primum ſit ex numero dierum totius an-
                  <lb/>
                ni in numerum aureorum primi, vnde tale productum in præſenti caſu erit .36500.
                  <lb/>
                ſecundum verò ſit ex numero dierum à primo die in quo ipſe ſecundus poſuit uſque
                  <lb/>
                ad finem anni, in numerum ipſorum nummorum, quod erit .13250. tertium autem
                  <lb/>
                productum ex diebus tertij in numerum ſuorum aureorum, quod
                  <reg norm="quidem" type="context">quidẽ</reg>
                erit .4125.
                  <lb/>
                quæ producta ſimul collecta faciunt .53875. deinde multiplicetur vnumquodque </s>
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