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. ARITH.
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            <div xml:id="echoid-div117" type="math:theorem" level="3" n="57">
              <p>
                <s xml:id="echoid-s505" xml:space="preserve">
                  <pb o="37" rhead="THEOREM. ARITH." n="49" file="0049" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0049"/>
                g. in
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s506" xml:space="preserve">Nunc ex ſpeculatione præcedentis theorematis, eadem erit proportio
                  <var>.n.
                    <lb/>
                  t.</var>
                ad
                  <var>.o.u.</var>
                quæ eſt
                  <var>.n.s.</var>
                ad
                  <var>.o.r.</var>
                </s>
                <s xml:id="echoid-s507" xml:space="preserve">quare pro-
                  <lb/>
                ductum
                  <var>.k.</var>
                ex definitione ſimile erit
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/>
                producto
                  <var>.m.</var>
                cum vtraque ſint rectan-
                  <lb/>
                gula, vnde proportio
                  <var>.k.</var>
                ad
                  <var>.m.</var>
                ad pro-
                  <lb/>
                portionem
                  <var>.n.t.</var>
                ad
                  <var>.o.u.</var>
                ex .18. ſexti du-
                  <lb/>
                pla erit. </s>
                <s xml:id="echoid-s508" xml:space="preserve">Igitur proportio
                  <var>.k.</var>
                ad
                  <var>.m.</var>
                æ-
                  <lb/>
                qualis erit proportioni
                  <var>.x.</var>
                ad
                  <var>.y.</var>
                et
                  <var>.p.</var>
                  <lb/>
                ad
                  <var>.q.</var>
                et
                  <var>.i.</var>
                ad
                  <var>.l.</var>
                & permutando ſic ſe ha-
                  <lb/>
                bebit
                  <var>.k.</var>
                ad
                  <var>.i.</var>
                ſicut
                  <var>.m.</var>
                ad
                  <var>.l.</var>
                ſed
                  <var>.x.p.</var>
                ad
                  <var>.i.</var>
                  <lb/>
                ſicſe habere probatum eſt vt
                  <var>.y.q.</var>
                ad
                  <var>.l</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s509" xml:space="preserve">Quare ex eadem .24. quinti ſic ſe habe
                  <lb/>
                bit
                  <var>.x.p.k.</var>
                ad
                  <var>.i.</var>
                ſicut
                  <var>.y.q.m.</var>
                ad
                  <var>.l.</var>
                ſed
                  <var>.y.q.
                    <lb/>
                  m.</var>
                æqualis eſt
                  <var>.l</var>
                . </s>
                <s xml:id="echoid-s510" xml:space="preserve">Itaque
                  <var>.x.p.k.</var>
                pariter
                  <var>.i.</var>
                  <lb/>
                æqualis erit.</s>
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                <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a">
                  <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0049-01"/>
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            <div xml:id="echoid-div119" type="math:theorem" level="3" n="58">
              <head xml:id="echoid-head74" xml:space="preserve">THEOREMA
                <num value="58">LVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s511" xml:space="preserve">ALIVD quoque problema, nec tamen definitum, veteres propoſuerunt,
                  <lb/>
                nempe an aliquis numerus in .4. eiuſmodi partes diuidi poſſit, vt ſumma qua-
                  <lb/>
                dratorum duarum partium dupla ſit ſummæ quadratorum reliquarum duarum.</s>
              </p>
              <p>
                <s xml:id="echoid-s512" xml:space="preserve">Verum huius effectio & ſpeculatio non erit difficilis,
                  <reg norm="cum" type="context">cũ</reg>
                ſit eadem quæ præmiſsis
                  <lb/>
                proximè duobus theorematibus allata fuit, ſumpta nempe ſumma radicum quarun
                  <lb/>
                cunque ſic ſe habentium, prout dictum fuit. </s>
                <s xml:id="echoid-s513" xml:space="preserve">Verbigratia .44. cuius partes erunt.
                  <lb/>
                16. 12. 14. 2.
                  <reg norm="tunc" type="context">tũc</reg>
                progrediemur regula de tribus dicentes. </s>
                <s xml:id="echoid-s514" xml:space="preserve">Si .44 numerum propoſi-
                  <lb/>
                tum valet, quid .16. pars maior? </s>
                <s xml:id="echoid-s515" xml:space="preserve">nempe valebit partem maiorem numeri propoſi-
                  <lb/>
                ti reſpondentem .16. idem de cæteris dico.</s>
              </p>
              <p>
                <s xml:id="echoid-s516" xml:space="preserve">Porrò ſpeculatio eadem eſt cum ſuperioribus.</s>
              </p>
            </div>
            <div xml:id="echoid-div120" type="math:theorem" level="3" n="59">
              <head xml:id="echoid-head75" xml:space="preserve">THEOREMA
                <num value="59">LIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s517" xml:space="preserve">CVR diuidens propoſitum numerum in duas eiuſmodi partes, vt productum
                  <lb/>
                radicum quadratarum ipſarum partium æquale ſit alteri numero propoſito,
                  <lb/>
                cuius
                  <reg norm="tamen" type="context">tamẽ</reg>
                quadratum maius
                  <reg norm="non" type="context">nõ</reg>
                ſit quadrato dimidij primi numeri propoſiti. </s>
                <s xml:id="echoid-s518" xml:space="preserve">Rectè
                  <lb/>
                ſecundum numerum propoſitum in ſeipſum multiplicat, &
                  <reg norm="eundem" type="context">eundẽ</reg>
                ex quadrato di-
                  <lb/>
                midij primi detrahit,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                quadratam radicem ſubtrahit ex dimidio ipſius pri-
                  <lb/>
                mi, ex quo datur minor pars quæſita, quaipſi dimidio coniuncta, maior pars ha-
                  <lb/>
                betur.</s>
              </p>
              <p>
                <s xml:id="echoid-s519" xml:space="preserve">Exempli gratia, ſi proponatur numerus, 20. propoſito modo, in duas partes
                  <lb/>
                eiuſmodi diuidendus, vt productum radicum æquale ſit (verbigratia) 8. </s>
                <s xml:id="echoid-s520" xml:space="preserve">Dimi-
                  <lb/>
                dium priminumeri in ſeipſum multiplicabimus, cuius quadratum erit .100. ex
                  <lb/>
                quo quadratum ſecundi numeri, nempe .64. detrahemus,
                  <reg norm="remanebitque" type="simple">remanebitq́;</reg>
                .36. cuius radi
                  <lb/>
                ce quadrata coniuncta .10. dimidio inquam primi numeri propoſiti, dabitur nume
                  <lb/>
                rus .16. pars maior, & ſubtracta à dimidio, dabitur minor pars, nempe .4.</s>
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