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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            <div xml:id="echoid-div87" type="math:theorem" level="3" n="40">
              <p>
                <s xml:id="echoid-s355" xml:space="preserve">
                  <pb o="26" rhead="IO. BAPT. BENED." n="38" file="0038" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0038"/>
                numerum inquam, cui differentia duorum quæſitorum æquanda eſt, in ſeipſum
                  <lb/>
                multiplicare, atque huic quadrato, ſecundum numerum propoſitum iungere, cui,
                  <lb/>
                productum numerorum quæſitorum æquale eſſe debet, & ex hac ſumma eruere qua
                  <lb/>
                dratam radicem, quæ coniuncta dimidio primi numeri propoſiti, dabit maiorem
                  <lb/>
                duorum numerorum & ex eadem radice detracto dimidio primi numeri, minorem
                  <lb/>
                numerum duorum quæſitorum.</s>
              </p>
              <p>
                <s xml:id="echoid-s356" xml:space="preserve">Exempli gratia, ſi proponeretur .12. cui differentia vnius numeri ab altero æqua-
                  <lb/>
                ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-
                  <lb/>
                torum ſimul
                  <reg norm="æquandum" type="context">æquãdum</reg>
                eſſet. </s>
                <s xml:id="echoid-s357" xml:space="preserve">Dimidium primi numeri in ſeipſum multiplicaremus,
                  <lb/>
                  <reg norm="proueniretque" type="simple">proueniretq́;</reg>
                  <reg norm="quadratum" type="context">quadratũ</reg>
                .36. cui coniuncto ſecundo, nempe .64. totum eſſet .100.
                  <lb/>
                ex quo detracta quadrata radice .10. etipſi coniuncto ſenario, dimidio primi nume
                  <lb/>
                ri, & ex eadem detracto eodem dimidio .6. pro maiore numero proueniret .16. &
                  <lb/>
                pro minore .4.</s>
              </p>
              <p>
                <s xml:id="echoid-s358" xml:space="preserve">Cuius rei ſpeculatio hæc eſt. </s>
                <s xml:id="echoid-s359" xml:space="preserve">Sit
                  <var>.e.o.</var>
                differentia cognita duorum incognitorum
                  <lb/>
                numerorum
                  <var>.a.o.</var>
                et
                  <var>.a.e.</var>
                quorum productum datum ſiue cognitum ſit
                  <var>.a.s</var>
                : conſide-
                  <lb/>
                remus nunc
                  <var>.e.i.</var>
                dimidium
                  <var>.e.o.</var>
                datæ differentiæ, & ex compoſito
                  <var>.a.i.</var>
                imaginetur
                  <lb/>
                quadratum
                  <var>.a.x.</var>
                in quo protracta ſit
                  <var>.t.u.</var>
                æquidiſtans lateri
                  <var>.a.i.</var>
                & tam ab ipſa
                  <var>.a.i.</var>
                re
                  <lb/>
                mota, quam
                  <var>.x.i.</var>
                ab
                  <var>.s.e.</var>
                vnde
                  <var>.t.e.</var>
                quadratum erit
                  <var>.e.i.</var>
                  <lb/>
                dimidiæ ſcilicet differentiæ datæ
                  <var>.e.o.</var>
                et
                  <var>.t.n.</var>
                rectan-
                  <lb/>
                  <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a" number="53">
                    <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0038-01"/>
                  </figure>
                gulum æquale erit rectangulo
                  <var>.n.c.</var>
                vt cuilibet licet
                  <lb/>
                per ſe conſiderare, vnde ſequitur gnomonem
                  <var>.e.r.t.</var>
                  <lb/>
                æqualem eſſe producto
                  <var>.a.s.</var>
                ideo cognitus, qui
                  <reg norm="quidem" type="context">quidẽ</reg>
                  <lb/>
                gnomon, ſi coniunctus fuerit quadrato
                  <var>.e.t.</var>
                cognito
                  <lb/>
                ex radice
                  <var>.e.i.</var>
                cognita (vt dimidia toralis differentię
                  <var>.
                    <lb/>
                  e.o.</var>
                datæ) habebimus quadratum totale
                  <var>.a.x.</var>
                cogni-
                  <lb/>
                tum, & ita eius radicem
                  <var>.a.i.</var>
                cognitam & reliqua om
                  <lb/>
                nia conſequenter quæ quidem ſpeculatio eadem eſt
                  <lb/>
                quæ .6. ſecundi ſeu .8. noni Euclidis.</s>
              </p>
              <p>
                <s xml:id="echoid-s360" xml:space="preserve">Poteris tamen ex modo & rationibus præceden-
                  <lb/>
                ti theoremate allatis, hocipſum concludere.</s>
              </p>
            </div>
            <div xml:id="echoid-div89" type="math:theorem" level="3" n="41">
              <head xml:id="echoid-head57" xml:space="preserve">THEOREMA
                <num value="41">XLI</num>
              .</head>
              <p>
                <s xml:id="echoid-s361" xml:space="preserve">CVR ij, qui aliquo propoſito numero, inuenturi ſunt duos numeros inter ſe
                  <lb/>
                differentes, quorum quadratorum ſumma altero numero propoſito æqualis
                  <lb/>
                ſit, rectè primum numerum propoſitum in ſeipſum multiplicant, quod quadratum
                  <lb/>
                exſecundo numero
                  <reg norm="detrahunt" type="context">detrahũt</reg>
                , & dimidium reſidui ſumunt, quod productum erit
                  <lb/>
                multiplicationis duorum numerorum interſe, in reliquis præcedentis theorematis
                  <lb/>
                ordinem ſequuntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s362" xml:space="preserve">Exempli gratia, ſi proponeretur .12. tanquam numerus, cui differentia duorum
                  <lb/>
                numerorum quæſitorum æquanda eſt, proponerentur præterea .272. quibus ſum-
                  <lb/>
                ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret ſanè
                  <lb/>
                primum numerum, nempe .12. in ſeipſum multiplicare, cuius
                  <reg norm="quadratum" type="context">quadratũ</reg>
                hoc loco
                  <lb/>
                eſſet .144. atque hoc detrahere ex ſecundo numero, ſupereſſet .128. ſumpto
                  <lb/>
                deinde dimidio huiuſce numeri, népe .64. producto in quam duorum numerorum
                  <lb/>
                  <reg norm="quæſitorum" type="context">quæſitorũ</reg>
                . </s>
                <s xml:id="echoid-s363" xml:space="preserve">Cum hoc .64. proſtea et duodenario primo propoſito numero, præceden
                  <lb/>
                tis theorematis ordinem ſequeremur.</s>
              </p>
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