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-div129" type="math:theorem" level="3" n="63">
              <p>
                <s xml:id="echoid-s548" xml:space="preserve">
                  <pb o="41" rhead="THEOREM. ARITH." n="53" file="0053" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0053"/>
                mentum eſt quadrati
                  <var>.q.d.</var>
                totalis. </s>
                <s xml:id="echoid-s549" xml:space="preserve">Quare duplicato
                  <var>.a.i.</var>
                & coniuncto
                  <var>.b.</var>
                cognoſci-
                  <lb/>
                mustotum
                  <var>.q.d.</var>
                & conſequenter
                  <var>.a.d.</var>
                ſuam radicem, hoc eſt ſummam duarum radi
                  <lb/>
                cum
                  <var>.a.g.</var>
                et
                  <var>.g.d.</var>
                quæ medio
                  <var>.a.i.</var>
                cognito, & quadrageſimoquinto theoremate ſingu-
                  <lb/>
                læ cognoſcuntur.</s>
              </p>
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            <div xml:id="echoid-div130" type="math:theorem" level="3" n="64">
              <head xml:id="echoid-head80" xml:space="preserve">THEOREMA
                <num value="64">LXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s550" xml:space="preserve">CVR propoſitum aliquem num erum in duas eiuſmodi partes diuiſuri, vt ſum-
                  <lb/>
                ma radicum dictarum partium æqualis ſit alteri numero propoſito. </s>
                <s xml:id="echoid-s551" xml:space="preserve">Rectè ſe-
                  <lb/>
                cundum numerum in ſeipſum multiplicant, ex quo quadrato, primum datum nu-
                  <lb/>
                merum detrahunt,
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                reſiduum in ſeipſum multiplicant, & ex eo quadrato
                  <lb/>
                quartam partem deſumunt,
                  <reg norm="quam" type="context">quã</reg>
                ex quadrato dimidij primi numeri detrahunt, radi-
                  <lb/>
                cemq́ue qua dratam reſidui cum iunxerint, & ex dimidio primi numeri detraxerint,
                  <lb/>
                partes quæſitæ proferuntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s552" xml:space="preserve">Exempli gratia, ſi proponeretur primus numerus .20. diuidendus et .6. ſecundus
                  <lb/>
                pro ſumma radicum, hunc ſecundum .6. in ſeipſum multiplicabimus,
                  <reg norm="dabiturque" type="simple">dabiturq́;</reg>
                nu-
                  <lb/>
                merus .36. ex quo quadrato primus numerus detrahetur,
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                numerus .16.
                  <lb/>
                qui quadratus dabit .256. cuius numeri quarta pars ſumetur, nempe .64. quæ ex qua
                  <lb/>
                drato dimidij primi numeri detrahetur, nempe .100.
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                .36. cuius radix qua
                  <lb/>
                drata .6. coniuncta & detracta ex .10. dabit .16. partem maiorem et .4. minorem.</s>
              </p>
              <p>
                <s xml:id="echoid-s553" xml:space="preserve">Cuius rei hæc ſpeculatio, primus numerus diuiſibilis ſignificetur linea
                  <var>.a.b.</var>
                diui-
                  <lb/>
                ſa in puncto
                  <var>.e.</var>
                in partes adhuc incognitas, et
                  <var>.a.c.</var>
                ſit productum
                  <var>.a.e.</var>
                in
                  <var>.e.b.</var>
                item
                  <var>.q.
                    <lb/>
                  p.</var>
                ſecundum numerum ſignificet, æqualem ſummæ radicum, quæ puncto
                  <var>.n.</var>
                diſtin-
                  <lb/>
                guantur. </s>
                <s xml:id="echoid-s554" xml:space="preserve">Poſtmodum totum quadratum
                  <var>.p.d.</var>
                erigatur (quod nobis eſt cognitum),
                  <lb/>
                in duo quadrata diuiſum
                  <var>.o.p.</var>
                et
                  <var>.o.d.</var>
                quorum ſumma
                  <var>.a.b.</var>
                cum detur, cognita rema-
                  <lb/>
                net ſumma
                  <reg norm="duorum" type="context">duorũ</reg>
                  <reg norm="ſupplementorum" type="context">ſupplementorũ</reg>
                  <var>.o.u.</var>
                et
                  <var>.o.q.</var>
                qua quadrata
                  <reg norm="cum" type="context">cũ</reg>
                fuerit dabit quadru
                  <lb/>
                  <reg norm="plum" type="context">plũ</reg>
                quadrati
                  <reg norm="ſupplementi" type="context">ſupplemẽti</reg>
                  <var>.o.q.</var>
                  <reg norm="nempe" type="context">nẽpe</reg>
                  <reg norm="quadruplum" type="context">quadruplũ</reg>
                producti
                  <var>.a.c.</var>
                etenim
                  <var>.a.c.</var>
                ex .19. theo
                  <lb/>
                remate huius libri quadratum eft ipſius
                  <var>.q.o.</var>
                  <reg norm="ſicque" type="simple">ſicq́;</reg>
                poterant etiam veteres quadrare
                  <lb/>
                dimidium differentiæ
                  <var>.a.b.</var>
                ab
                  <var>.p.d.</var>
                nempe quadrato tantummodo ſupplemento
                  <var>.q.
                    <lb/>
                  o</var>
                . </s>
                <s xml:id="echoid-s555" xml:space="preserve">Tunc habito
                  <var>.a.c.</var>
                eius ope tanquam producti
                  <var>.a.e.</var>
                in
                  <var>.e.b.</var>
                ex .45. theoremate ſingu
                  <lb/>
                læ partes cognoſcentur.</s>
              </p>
              <p>
                <s xml:id="echoid-s556" xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito ſupplemento
                  <var>.
                    <lb/>
                  q.o.</var>
                diſtinguendæ radices
                  <var>q.n.</var>
                et
                  <var>.n.p.</var>
                ex .45. theoremate, quibus cognitis, eorum
                  <lb/>
                etiam quadrata cognoſcuntur.</s>
              </p>
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