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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IO. BAPT. BENED.
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            <div xml:id="echoid-div142" type="math:theorem" level="3" n="72">
              <pb o="48" rhead="IO. BAPT. BENED." n="60" file="0060" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0060"/>
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            <div xml:id="echoid-div144" type="math:theorem" level="3" n="73">
              <head xml:id="echoid-head89" xml:space="preserve">THEOREMA
                <num value="73">LXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s630" xml:space="preserve">HOC etiam problema à me inuentum eſt, nempe ſi duæ radices quadratæ in
                  <lb/>
                ſummam collectæ fuerint, & ex dimidio eiuſmodi ſummæ detracta fuerit mi
                  <lb/>
                nor radix,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                quadratum duplicatum
                  <reg norm="eique" type="simple">eiq́;</reg>
                ſummæ coniungatur du-
                  <lb/>
                plum producti ipſius reſidui in dimidium ſummæ radicum, atque huic ſummæ du-
                  <lb/>
                plum producti eiuſdem reſidui in radicem minorem coniunctum fuerit; </s>
                <s xml:id="echoid-s631" xml:space="preserve">vltima hæc
                  <lb/>
                ſumma differentia erit duorum quadratorum propoſitorum.</s>
              </p>
              <p>
                <s xml:id="echoid-s632" xml:space="preserve">Exempli gratia duæ radices quadraræ ſint .5. et .11. harum ſumma erit .16. & dimi
                  <lb/>
                dium .8. differentia minoris ab ipſo dimidio erit .3: duplum quadrati huius differen
                  <lb/>
                tiæ erit .18: </s>
                <s xml:id="echoid-s633" xml:space="preserve">duplum producti huius differentię in dimidium ſummę radicum erit .48.
                  <lb/>
                item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium
                  <lb/>
                ſumma erit .96. tantaq́ue erit differentia ſuorum quadratorum, quorum vnum
                  <lb/>
                erit .25. alterum verò .121.</s>
              </p>
              <p>
                <s xml:id="echoid-s634" xml:space="preserve">Pro cuius rei ſcientia, duæ quadratæ radices ſint
                  <var>.h.o.</var>
                et
                  <var>.o.d.</var>
                directæ inter ſe con-
                  <lb/>
                iunctæ, quæ ſumma per medium in puncto
                  <var>.e.</var>
                diuidatur, tum cogitetur
                  <var>.e.b.</var>
                æqualis
                  <lb/>
                  <var>o.e.</var>
                perpendicularis
                  <var>.h.d.</var>
                  <reg norm="ducanturque" type="simple">ducanturq́;</reg>
                lineæ
                  <var>.b.h</var>
                :
                  <var>b.o.</var>
                et
                  <var>.b.d</var>
                . </s>
                <s xml:id="echoid-s635" xml:space="preserve">Iam ex .4. primi
                  <var>.b.h.</var>
                æqua
                  <lb/>
                lis erit
                  <var>.b.d.</var>
                & quadratum
                  <var>.b.h.</var>
                æquale quadrato
                  <var>.h.o.</var>
                & quadrato
                  <var>.o.b.</var>
                ſimul cum du
                  <lb/>
                plo producti
                  <var>.o.e.</var>
                in
                  <var>.o.h.</var>
                ex .12. ſecundi Eucli. </s>
                <s xml:id="echoid-s636" xml:space="preserve">Sed ex .13.
                  <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                quadratum
                  <var>.b.d.</var>
                  <lb/>
                minus eſt quadrato
                  <var>.o.d.</var>
                cum quadrato
                  <var>.o.b.</var>
                ex duplo producti
                  <var>.o.e.</var>
                in
                  <var>.o.d.</var>
                at duplum
                  <lb/>
                eiuſmodi producti æquale eſt duplo qua-
                  <lb/>
                drati
                  <var>.o.e.</var>
                & duplo producti
                  <var>.o.e.</var>
                in
                  <var>.e.d.</var>
                ex
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/>
                tertia eiuſdem, itaque duo quadrata ſcili-
                  <lb/>
                cet
                  <var>.o.b.</var>
                et
                  <var>.o.d.</var>
                maiora erunt duobus qua-
                  <lb/>
                dratis, nempe
                  <var>.o.b.</var>
                et
                  <var>.o.h.</var>
                collectis cum du
                  <lb/>
                plo producti
                  <var>.o.e.</var>
                in
                  <var>.o.h.</var>
                ex duplo quadrati
                  <lb/>
                  <var>o.e.</var>
                vna
                  <reg norm="cum" type="context">cũ</reg>
                duplo producti
                  <var>.o.e.</var>
                in
                  <var>.e.d</var>
                . </s>
                <s xml:id="echoid-s637" xml:space="preserve">Qua
                  <lb/>
                re
                  <reg norm="differentia" type="context">differẽtia</reg>
                ſummæ duorum quadratorum
                  <lb/>
                  <var>o.b.</var>
                et
                  <var>.o.d.</var>
                à ſumma duorum
                  <var>o.b.</var>
                et
                  <var>.o.h.</var>
                du
                  <lb/>
                plum erit quadrati
                  <var>.o.e.</var>
                cum duplo produ-
                  <lb/>
                cti
                  <var>.o.e.</var>
                in
                  <var>.e.d.</var>
                & duplo producti
                  <var>.o.e.</var>
                in
                  <var>.o.h.</var>
                  <lb/>
                Quòd ſi ex ſingulis duabus ſummis quadratorum demptum fuerit quadratum
                  <var>.o.b.</var>
                  <lb/>
                eadem producta & quadrata ipſius
                  <var>.o.e.</var>
                remanebunt, tanquam differentia duorum
                  <lb/>
                quadratorum
                  <var>.o.u.</var>
                et
                  <var>.h.c</var>
                .</s>
              </p>
              <div xml:id="echoid-div144" type="float" level="4" n="1">
                <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a">
                  <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0060-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div146" type="math:theorem" level="3" n="74">
              <head xml:id="echoid-head90" xml:space="preserve">THEOREMA
                <num value="74">LXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s638" xml:space="preserve">CVR ſumma duorum
                  <reg norm="extremorum" type="context">extremorũ</reg>
                quatuor terminorum
                  <reg norm="proportionalium" type="context">proportionaliũ</reg>
                arith-
                  <lb/>
                meticè, æqualis eſt ſummæ duorum mediorum, vbi nota hac in re neceſſa-
                  <lb/>
                rium non eſſe proportionalitatem continuam exiſtere.</s>
              </p>
              <p>
                <s xml:id="echoid-s639" xml:space="preserve">Exempli gratia, ſi darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea
                  <lb/>
                dem eſſet primi ad ſecundum quæ tertij ad quartum, ſumma primi cum quarto eſſet
                  <lb/>
                26.
                  <reg norm="tantaque" type="simple">tantaq́;</reg>
                ſecundi cum tertio.</s>
              </p>
              <p>
                <s xml:id="echoid-s640" xml:space="preserve">Cuius ſpeculationis cauſa, primus
                  <reg norm="maiorque" type="simple">maiorq́;</reg>
                numerus ſignificetur linea
                  <var>.e.o.</var>
                ſecun-
                  <lb/>
                dus
                  <var>.s.q.</var>
                tertius
                  <var>.u.c.</var>
                quartus
                  <var>.g.t.</var>
                differentia porrò inter
                  <var>.e.o.</var>
                et
                  <var>.s.q.</var>
                ſit
                  <var>.i.o.</var>
                quæ æqualis
                  <lb/>
                erit differentiæ
                  <var>.r.c.</var>
                qua quartus à tertio ſuperatur ex hypotheſi. </s>
                <s xml:id="echoid-s641" xml:space="preserve">Itaque aſſero ſum
                  <lb/>
                mam
                  <var>.e.o.</var>
                cum
                  <var>.g.t.</var>
                nempe
                  <var>.a.o.</var>
                æqualem eſſe ſummę
                  <var>.q.s.</var>
                et
                  <var>.u.c.</var>
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                  <var>.q.p</var>
                . </s>
                <s xml:id="echoid-s642" xml:space="preserve">Nam in
                  <var>.a.o.</var>
                </s>
              </p>
            </div>
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