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]

Table of figures

< >
[91. Figure]
[92. Figure]
[93. Figure]
[94. Figure]
[95. Figure]
[96. Figure]
[97. Figure]
[98. Figure]
[99. Figure]
[100. Figure]
[101. Figure]
[102. Figure]
[103. Figure]
[104. Figure]
[105. Figure]
[106. Figure]
[107. Figure]
[108. Figure]
[109. Figure]
[110. Figure]
[111. Figure]
[112. Figure]
[113. Figure]
[114. Figure]
[115. Figure]
[116. Figure]
[117. Figure]
[118. Figure]
[119. Figure]
[120. Figure]
< >
page |< < (67) of 445 > >|
THEOREM. ARIT.
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div195" type="math:theorem" level="3" n="104">
              <p>
                <s xml:id="echoid-s902" xml:space="preserve">
                  <var>
                    <pb o="67" rhead="THEOREM. ARIT." n="79" file="0079" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0079"/>
                  r.f.</var>
                hoc eſt
                  <var>.o.r.</var>
                ad
                  <var>.m.s.</var>
                ex .11. quinti. </s>
                <s xml:id="echoid-s903" xml:space="preserve">Itaque ex communi ſcientia ſic ſe habe-
                  <lb/>
                bit
                  <var>.d.i.</var>
                ad
                  <var>.d.b.</var>
                vt
                  <var>.e.d.</var>
                ad
                  <var>.e.b</var>
                : cum
                  <var>.e.d.</var>
                æqualis ſit
                  <var>.t.a</var>
                . </s>
                <s xml:id="echoid-s904" xml:space="preserve">Ita etiam vt
                  <var>.e.n.</var>
                ad
                  <var>.n.b</var>
                : cum
                  <var>.n.
                    <lb/>
                  e.</var>
                æqualis ſit
                  <var>.o.r</var>
                . </s>
                <s xml:id="echoid-s905" xml:space="preserve">Iam ſi ſic ſe habeat
                  <var>.d.i.</var>
                ad
                  <var>.d.b.</var>
                vt
                  <var>.d.e.</var>
                ad
                  <var>.e.b.</var>
                permutando
                  <reg norm="quoque" type="simple">quoq;</reg>
                ſic
                  <lb/>
                ſe habebit
                  <var>.d.i.</var>
                ad
                  <var>.d.e.</var>
                vt
                  <var>.d.b.</var>
                ad
                  <var>.b.e.</var>
                & compon endo ita
                  <var>.i.d.e.</var>
                ad
                  <var>.e.d.</var>
                vt
                  <var>.d.b.e.</var>
                ad
                  <var>.e.
                    <lb/>
                  b.</var>
                & permutando ſic
                  <var>.i.d.e.</var>
                ad
                  <var>.d.b.e.</var>
                vt. de
                  <var>.a.d.e.b.</var>
                nempe vt
                  <var>.e.n.</var>
                ad
                  <var>.n.b.</var>
                & permutan
                  <lb/>
                do ita
                  <var>.i.d.e.</var>
                ad
                  <var>.e.n.</var>
                vt
                  <var>.d.b.e.</var>
                ad
                  <var>.b.n.</var>
                & componendo ita
                  <var>.i.d.e.n.</var>
                ad
                  <var>.n.e.</var>
                vt
                  <var>.d.b.e.</var>
                et
                  <var>.b.
                    <lb/>
                  n.</var>
                ad
                  <var>.b.n.</var>
                & permutando ſic
                  <var>.i.d.e.n.</var>
                ad
                  <var>.d.b.e.</var>
                et
                  <var>.b.n.</var>
                nempe ad
                  <var>.a.c</var>
                :
                  <var>f.r</var>
                :
                  <var>m.s</var>
                : vt
                  <var>.e.n.</var>
                ad
                  <var>.
                    <lb/>
                  n.b.</var>
                hoc eſt. ut
                  <var>.o.r.</var>
                ad
                  <var>.m.s.</var>
                quod erat propoſitum.</s>
              </p>
              <div xml:id="echoid-div195" type="float" level="4" n="1">
                <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a">
                  <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0078-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div197" type="math:theorem" level="3" n="105">
              <head xml:id="echoid-head122" xml:space="preserve">THEOREMA
                <num value="105">CV</num>
              .</head>
              <p>
                <s xml:id="echoid-s906" xml:space="preserve">CVR deſideranti ſummam quorumcunque terminorum progreſſionis conti-
                  <lb/>
                nuæ geometricæ cognoſcere. </s>
                <s xml:id="echoid-s907" xml:space="preserve">Rectè minimus terminus ex maximo detrahen
                  <lb/>
                dus eſt,
                  <reg norm="reſiduumque" type="simple">reſiduumq́;</reg>
                per denominantem progreſſionis dempta vnitate diuidendum,
                  <lb/>
                  <reg norm="prouenientique" type="simple">prouenientiq́;</reg>
                maximum terminum addendum, ex quo oritur ſumma quæſita.</s>
              </p>
              <p>
                <s xml:id="echoid-s908" xml:space="preserve">Exempli gratia, ſi darentur quatuor termini continui proportionales .8. 12. 18.
                  <lb/>
                27. primum hoc eſt minimum .8. ex vltimo .27. detraheremus: </s>
                <s xml:id="echoid-s909" xml:space="preserve">
                  <reg norm="remaneretque" type="simple">remaneretq́;</reg>
                .19. qui
                  <lb/>
                per denominantem progreſſionis, dempta vnitate, diuideretur. </s>
                <s xml:id="echoid-s910" xml:space="preserve">Quo loco animad
                  <lb/>
                uertendum eſt, quamlibet
                  <reg norm="denominationem" type="context">denominationẽ</reg>
                cuiuſcunque proportionis numerorum
                  <lb/>
                ſupra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eſt, &
                  <lb/>
                idipſum de ſuperparticularibus, & ſuperpartientibus eſt intelligendum, vt in præ-
                  <lb/>
                ſenti proportio ſeſquialtera inter duos terminos cogitanda eſt, nempe inter vnum
                  <lb/>
                & dimidium, atque vnum. </s>
                <s xml:id="echoid-s911" xml:space="preserve">Seſquitertia autem inter vnum & tertiam partem,
                  <lb/>
                & vnum. </s>
                <s xml:id="echoid-s912" xml:space="preserve">Seſquiquinta inter vnum cum quinta parte, & vnum. </s>
                <s xml:id="echoid-s913" xml:space="preserve">De ſuperpartien
                  <lb/>
                tibus idem aſſero quod de proportione
                  <reg norm="ſuperbipartiente" type="context">ſuperbipartiẽte</reg>
                tertias appellata, vt .5.
                  <lb/>
                ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, ſuperbipartiens quar-
                  <lb/>
                tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans ſcilicet, ſem
                  <lb/>
                per ſit vnitas, alter verò denominans. </s>
                <s xml:id="echoid-s914" xml:space="preserve">Idem de cæteris. </s>
                <s xml:id="echoid-s915" xml:space="preserve">Quare in præſenti exem
                  <lb/>
                plo, detracta vnitate ex denominante progreſſionis, ſupererit tantummodo dimi-
                  <lb/>
                dium, quo diuiſo .19. proueniet .38. qui numerus æqualis erit ſummæ
                  <reg norm="reliquorum" type="context">reliquorũ</reg>
                  <lb/>
                omnium terminorum, cui coniuncto vltimo termino .27. dabitur ſumma quæſita .65.</s>
              </p>
              <p>
                <s xml:id="echoid-s916" xml:space="preserve">Pro cuius ſpeculatione, quatuor termini ſignificentur, quatuor lineis
                  <var>.m.s</var>
                :
                  <var>f.r</var>
                :
                  <var>c.a.
                    <lb/>
                  b.i.</var>
                primus autem terminus
                  <var>.m.s.</var>
                ex vltimo
                  <var>.b.i.</var>
                detrahatur,
                  <reg norm="reſiduumque" type="simple">reſiduumq́;</reg>
                ſit
                  <var>.n.i.</var>
                & ex
                  <lb/>
                ſecundo
                  <var>.f.r.</var>
                cuius reſiduum ſit
                  <var>.o.r.</var>
                proportio verò progreſſionis ea ſit, quæ
                  <var>.g.h.</var>
                ad
                  <var>.
                    <lb/>
                  y.</var>
                quo vnitas repræſentatur (ex quo ſic ſe habebit
                  <var>.g.h.</var>
                ad
                  <var>.y.</var>
                vt
                  <var>.f.r.</var>
                ad
                  <var>.m.s.</var>
                ) qua
                  <var>.y.</var>
                de
                  <lb/>
                tracta ex
                  <var>.g.h.</var>
                ſuperſit
                  <var>.h</var>
                . </s>
                <s xml:id="echoid-s917" xml:space="preserve">Tum erecta
                  <lb/>
                cogitetur linea
                  <var>.n.u.x.</var>
                indefinita per
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/>
                pendicularis
                  <var>.b.i.</var>
                à puncto
                  <var>.n.</var>
                quę diui
                  <lb/>
                datur in puncto
                  <var>.x.</var>
                ita vt
                  <var>.n.x.</var>
                æqualis
                  <lb/>
                ſit vnitati
                  <var>.y.</var>
                & in puncto
                  <var>.u.</var>
                ita. vt
                  <var>.n.
                    <lb/>
                  u.</var>
                æqualis ſit
                  <var>.h.</var>
                ex quo eadem erit
                  <lb/>
                proportio
                  <var>.n.u.</var>
                ad
                  <var>.n.x.</var>
                vt
                  <var>.h.</var>
                ad
                  <var>.y.</var>
                  <reg norm="nem- pe" type="context">nẽ-
                    <lb/>
                  pe</reg>
                  <var>.o.r.</var>
                ad
                  <var>.m.s</var>
                . </s>
                <s xml:id="echoid-s918" xml:space="preserve">Nam cú ſic ſe habeat
                  <var>.
                    <lb/>
                  f.r.</var>
                ad
                  <var>.m.s.</var>
                hoc eſt ad
                  <var>.f.o.</var>
                vt
                  <var>.g.h.</var>
                ad
                  <var>.y</var>
                  <lb/>
                hoc eſt ad
                  <var>.g.</var>
                permutando
                  <reg norm="quoque" type="simple">quoq;</reg>
                ſic
                  <lb/>
                ſe habebit
                  <var>.f.r.</var>
                ad
                  <var>.g.h.</var>
                vt
                  <var>.f.o.</var>
                ad
                  <var>.g</var>
                . </s>
                <s xml:id="echoid-s919" xml:space="preserve">Ita
                  <lb/>
                que ex .19. quinti
                  <var>.o.r.</var>
                ad
                  <var>.h.</var>
                vt
                  <var>.f.r.</var>
                ad
                  <var>.g.h.</var>
                ex quo ex .11. eiuſdem
                  <var>.o.r.</var>
                ad
                  <var>.h.</var>
                vt
                  <var>.f.o.</var>
                ad </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>