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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THEOR. ARITH.
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            <div xml:id="echoid-div79" type="math:theorem" level="3" n="36">
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                  <pb o="23" rhead="THEOR. ARITH." n="35" file="0035" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0035"/>
                ipſius
                  <var>.a.x.</var>
                tam ſit multiplex ad vnitatem, quam cupimus numerum
                  <var>.a.e.</var>
                numero
                  <var>.
                    <lb/>
                  e.o.</var>
                multiplicem eſſe.</s>
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            <div xml:id="echoid-div81" type="math:theorem" level="3" n="37">
              <head xml:id="echoid-head53" xml:space="preserve">THEOREMA
                <num value="37">XXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s325" xml:space="preserve">CVR inuenire cupientes duos numeros, quorum quadrata in ſummam colle-
                  <lb/>
                cta, æqualia ſint numero propoſito, & ijſdem numeris multiplicatis ad-
                  <lb/>
                inuicem, productum alteri numero propoſito ſit æquale, rectè ſumant dimidium
                  <lb/>
                primi numeri propoſiti, cui ſumma quadratorum æquari debet,
                  <reg norm="hocque" type="simple">hocq́;</reg>
                dimidium
                  <lb/>
                in ſeipſum multiplicent, vnà etiam alterum numerum propoſitum in ſeipſum
                  <lb/>
                multiplicent, quod quadratum detrahunt de primo, & reſidui quadratam radicem,
                  <lb/>
                dimidio primi numeri propoſiti coniungunt, ex qua ſumma, quadratam radicem
                  <lb/>
                  <reg norm="eruunt" type="context">eruũt</reg>
                , quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero
                  <lb/>
                detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum
                  <reg norm="quae- ſitorum" type="simple">quę-
                    <lb/>
                  ſitorum</reg>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s326" xml:space="preserve">Exempli gratia, ſi proponerentur .34. pro primo numero cui æquari de-
                  <lb/>
                beret ſumma duorum quadratorum, quorum radicum productum æquale eſſe de-
                  <lb/>
                beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi
                  <lb/>
                numeri in ſeipſum multiplicari, cuius dimidij quadratum erit .289. è quo ſi detra-
                  <lb/>
                has quadratum ſecundi numeri, nempe .225. remanebit .64.
                  <reg norm="atque" type="simple">atq;</reg>
                huius ſi quadra-
                  <lb/>
                tam radicem ſumas nempe .8. quam dimidio primi numeri, nempe .17. coniun-
                  <lb/>
                gas, dabitur duorum quadratorum numerorum quęſitorum maior numerus .25. hac
                  <lb/>
                deinde radice è dimidio detracta, minus quadratum dabitur .9. ſcilicet, quorum
                  <lb/>
                radices .5. et .3. eſſent ij numeri, qui quæruntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s327" xml:space="preserve">Cuius ſpeculationis gratia, cogitemus primum numerum, cui quadratorum fum
                  <lb/>
                ma æquari debet, ſignificari linea
                  <var>.a.n.</var>
                tum concipiamus quæſita quadrata ſignifi-
                  <lb/>
                cari,
                  <reg norm="coniungique" type="simple">coniungiq́</reg>
                modo ſubſcripto
                  <var>.t.b.k.</var>
                ſecundum porrò numerum propoſitum,
                  <lb/>
                ſignificari producto
                  <var>.d.b</var>
                . </s>
                <s xml:id="echoid-s328" xml:space="preserve">Iam nil ſupereſt aliud quam vt quantitates
                  <var>.d.p.</var>
                et
                  <var>.b.p.</var>
                  <lb/>
                quæramus.</s>
              </p>
              <p>
                <s xml:id="echoid-s329" xml:space="preserve">Itaque cum in linea
                  <var>.a.n.</var>
                ſummæ quadratorum numerus detur, quadratum di-
                  <lb/>
                midij
                  <var>.o.a.</var>
                ſit
                  <var>.s.a.</var>
                quod nobis erit cognitum; </s>
                <s xml:id="echoid-s330" xml:space="preserve">ſit etiam
                  <var>.a.u.</var>
                numerus quadrati ma
                  <lb/>
                ioris, et
                  <var>.u.n.</var>
                minoris, et
                  <var>.a.z.</var>
                productum vnius in alterum; </s>
                <s xml:id="echoid-s331" xml:space="preserve">qui quidem numerus
                  <var>.a.
                    <lb/>
                  z.</var>
                æqualis erit
                  <lb/>
                quadrato nume
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/>
                ri
                  <var>.d.b.</var>
                ex .19.
                  <lb/>
                theoremate hu-
                  <lb/>
                ius libri. </s>
                <s xml:id="echoid-s332" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <lb/>
                  <var>a.z.</var>
                cognitum
                  <lb/>
                erit, cum eius
                  <lb/>
                radix
                  <var>.d.b.</var>
                ſit
                  <reg norm="ſe- cundus" type="context">ſe-
                    <lb/>
                  cũdus</reg>
                numerus
                  <lb/>
                propoſitus, quæ
                  <lb/>
                minor erit
                  <var>.a.s.</var>
                ex quinta ſecundi, aut ſeptima conſequentia poſt .16. noni Eucli-
                  <lb/>
                dis. </s>
                <s xml:id="echoid-s333" xml:space="preserve">Iam ſubtracta quantitate
                  <var>.z.a.</var>
                è quadrato
                  <var>.a.s.</var>
                cognoſcetur quadratum
                  <var>.t.x.</var>
                  <lb/>
                cuius radix æqualis erit
                  <var>.o.u.</var>
                ex poſtremo adductis, Itaque cognoſcemus
                  <var>.o.u.</var>
                qui
                  <lb/>
                numerus coniunctus dimidio
                  <var>.o.a.</var>
                cognito, dabit quadratum
                  <var>.a.u.</var>
                cognitum, at-
                  <lb/>
                queita
                  <var>.u.n.</var>
                pariter cognoſcetur, & eorum radices conſequenter.</s>
              </p>
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                <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a">
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