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-div83" type="math:theorem" level="3" n="38">
              <p>
                <s xml:id="echoid-s343" xml:space="preserve">
                  <pb o="25" rhead="THEOREM. ARIT." n="37" file="0037" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0037"/>
                noni,
                  <reg norm="hocque" type="simple">hocq́;</reg>
                rectangulum
                  <var>.g.r.</var>
                quadratum eſt primi numeri propoſiti ex .19. theo-
                  <lb/>
                remate huius libri,
                  <reg norm="itaque" type="simple">itaq;</reg>
                cognitum erit. </s>
                <s xml:id="echoid-s344" xml:space="preserve">vnà etiam gnomon
                  <var>.u.g.t.</var>
                cognoſcetur,
                  <lb/>
                quare totum quadratum
                  <var>.g.y.</var>
                  <reg norm="eiusque" type="simple">eiusq́;</reg>
                radix
                  <var>.b.g.</var>
                manifęſta erit, cui coniuncta
                  <var>.q.b.</var>
                  <lb/>
                data, maius quadratum
                  <var>.q.g.</var>
                cognoſcetur, ex qua
                  <var>.b.g.</var>
                detracta
                  <var>.b.i.</var>
                data, cogno-
                  <lb/>
                ſcetur
                  <var>.i.g.</var>
                quadratum minus conſequenter, etiam eorum radices notæ erunt.</s>
              </p>
              <div xml:id="echoid-div83" type="float" level="4" n="1">
                <figure xlink:label="fig-0036-02" xlink:href="fig-0036-02a">
                  <image file="0036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0036-02"/>
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            <div xml:id="echoid-div85" type="math:theorem" level="3" n="39">
              <head xml:id="echoid-head55" xml:space="preserve">THEOREMA
                <num value="39">XXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s345" xml:space="preserve">
                  <emph style="sc">ALia</emph>
                etiam ratione idipſum definiri poteſt, prætermiſſa antiquorum via,
                  <lb/>
                nempe multiplicatis in ſemetipſis primo & ſecundo, numeris propoſitis, qua-
                  <lb/>
                  <reg norm="druplicatoque" type="simple">druplicatoq́;</reg>
                quadrato primi, qua ſumma coniuncta cum quadrato ſecundi nume-
                  <lb/>
                ri, & ex hac altera ſumma eruta radice quadrata, ex qua detracto ſecundo nume-
                  <lb/>
                ro, & è reliquo ſumpto dimidio, quod erit
                  <reg norm="quadratum" type="context">quadratũ</reg>
                minus, quo detracto ex radi-
                  <lb/>
                ce poſtremo iuncta, ſupererit quadrarum maius.</s>
              </p>
              <p>
                <s xml:id="echoid-s346" xml:space="preserve">Exempli gratia, ſi proponeretur numerus .8. cui productum duorum numerorum
                  <lb/>
                quæſitorum æquandum eſt, proponeretur idem .12. cui differentia quadratorum
                  <lb/>
                duorum numerorum æqualis eſſe debet. </s>
                <s xml:id="echoid-s347" xml:space="preserve">Iubeo primum numerum, nempe .8. in ſe
                  <lb/>
                ipſum multiplicari, ex quo exurget .64. pro numero ſui quadrati, quod quadru-
                  <lb/>
                plicari volo,
                  <reg norm="eritque" type="simple">eritq́;</reg>
                productum .256. quod cenſeo
                  <reg norm="coniungendum" type="context">coniũgendum</reg>
                cum quadrato ſe-
                  <lb/>
                cundi numeri propoſiti, nempe .144.
                  <reg norm="eritque" type="simple">eritq́;</reg>
                ſumma .400. ex quaſumetur radix, ſci
                  <lb/>
                licet .20. & ex hac detrahetur ſecundus numerus .12.
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                dimidium, nempe
                  <num value="4">.
                    <lb/>
                  4.</num>
                pro quadrato minore, quo in ſummam collecto cum, 12. dabit quadratum
                  <lb/>
                maius .16.</s>
              </p>
              <p>
                <s xml:id="echoid-s348" xml:space="preserve">Cuius ſpeculationis cauſa, quadratum maius per lineam
                  <var>.q.g.</var>
                minus per
                  <var>.g.p.</var>
                ſi-
                  <lb/>
                gnificetur: </s>
                <s xml:id="echoid-s349" xml:space="preserve">ſuper integram autem
                  <var>.q.p.</var>
                erigatur quadratum integrum
                  <var>.d.p.</var>
                diuiſum,
                  <lb/>
                vt quadratum
                  <var>.f.g.</var>
                vigeſimiſeptimi theorematis huius libri, (idipſum accideret di-
                  <lb/>
                uiſo quadrato modo octauæ ſecundi Euclidis) quæ quidem diuiſio, eſt via quatuor
                  <lb/>
                productorum
                  <var>.q.g.</var>
                in
                  <var>.g.p.</var>
                è quibus vnum ſit
                  <var>.g.r.</var>
                quod erit cognitum ex .19. theore
                  <lb/>
                mate cum ſit
                  <reg norm="quadratum" type="context">quadratũ</reg>
                primi numeri ppoſiti, ex quo illa quatuor cognita
                  <reg norm="erunt" type="context">erũt</reg>
                . </s>
                <s xml:id="echoid-s350" xml:space="preserve">Iam
                  <lb/>
                verò ſi cogitemus
                  <var>.q.p.</var>
                ſectam in puncto
                  <var>.t.</var>
                ita vt
                  <var>.q.t.</var>
                æqualis ſit
                  <var>.p.g.</var>
                dabitur differen
                  <lb/>
                tia
                  <var>.t.g.</var>
                cognita, vt radix quadrati
                  <var>.e.o.</var>
                cum ex præſup-
                  <lb/>
                poſito
                  <var>.r.n.</var>
                æqualis ſit
                  <var>.q.g.</var>
                et
                  <var>.r.e</var>
                :
                  <var>g.p.</var>
                ex quo etiam
                  <var>.q.t.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/>
                ita pariter
                  <var>.e.n.t.g.</var>
                æqualis erit. </s>
                <s xml:id="echoid-s351" xml:space="preserve">Collecto
                  <reg norm="itaque" type="simple">itaq;</reg>
                quadra
                  <lb/>
                to
                  <var>.e.o.</var>
                ipſius
                  <var>.t.g.</var>
                cum quadruplo
                  <var>.g.r</var>
                : cognitum erit
                  <lb/>
                quadratum
                  <var>.d.p.</var>
                ipſius
                  <var>.q.p.</var>
                </s>
                <s xml:id="echoid-s352" xml:space="preserve">quare cognoſcetur
                  <var>.q.p.</var>
                de
                  <lb/>
                quo numero detracta differétia quadratorum cognita
                  <var>.
                    <lb/>
                  t.g.</var>
                ſupererit aggregatum
                  <var>.p.g.</var>
                et
                  <var>.q.t.</var>
                cognitum. </s>
                <s xml:id="echoid-s353" xml:space="preserve">Qua-
                  <lb/>
                re ex conſequenti, dimidium aggregati, nempe
                  <var>.g.p.</var>
                  <lb/>
                cognoſcetur, tanquam minus duorum quadratorum.
                  <lb/>
                </s>
                <s xml:id="echoid-s354" xml:space="preserve">cui iuncta
                  <var>.g.t.</var>
                aut detracta
                  <var>.p.g.</var>
                ex
                  <var>.p.q.</var>
                quadratum
                  <var>.q.
                    <lb/>
                  g.</var>
                maius cognitum remanebit.</s>
              </p>
              <div xml:id="echoid-div85" type="float" level="4" n="1">
                <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a">
                  <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0037-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div87" type="math:theorem" level="3" n="40">
              <head xml:id="echoid-head56" xml:space="preserve">THEOREMA
                <num value="40">XL</num>
              .</head>
              <p>
                <s xml:id="echoid-s355" xml:space="preserve">CVR ijs, qui volunt duos eiuſmodi numeros inuenire, vt eorum maior mi-
                  <lb/>
                norem, numero propoſito ſuperet, & productum vnius in alterum, alteri nu-
                  <lb/>
                mero propoſito adęquetur, conſultiſsimum ſit dimidium primi numeri propoſiti, </s>
              </p>
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          </div>
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