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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              <p>
                <s xml:id="echoid-s250" xml:space="preserve">
                  <pb o="17" rhead="THEOREM. ARITH." n="29" file="0029" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0029"/>
                numerorum, proueniat numerus æqualis numero producti duorum primorum nu-
                  <lb/>
                m erorum ſimul.</s>
              </p>
              <p>
                <s xml:id="echoid-s251" xml:space="preserve">Sint exempli gratia propoſiti numeri .2. et .8. qui mutuo diuiſi in primis dent pro
                  <lb/>
                uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-
                  <lb/>
                cta dabunt ſummam quatuor integrorum et quartæ partis vnius, ſumma autem qua
                  <lb/>
                dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar
                  <lb/>
                tam partem vnius diuiſus dabit .16. pro proueniente, quæ .16. æqualia erunt pro
                  <lb/>
                ducto binarii in octonarium.</s>
              </p>
              <p>
                <s xml:id="echoid-s252" xml:space="preserve">Cuius rei hæc erit ſpeculatio, ſint duæ lineæ
                  <var>.o.e.</var>
                et
                  <var>.o.n.</var>
                quæ duos numeros pro-
                  <lb/>
                poſitos ſignificent, inuicem ad angulum rectum
                  <var>.o.</var>
                coniunctæ, quarum quadrata
                  <lb/>
                ſint
                  <var>.o.a.</var>
                et
                  <var>.o.p.</var>
                ipſorum productum ſit
                  <var>.n.e.</var>
                tum
                  <var>.o.t.</var>
                ſit proueniens ex diuiſione
                  <var>.o.e.</var>
                  <lb/>
                per
                  <var>.o.n</var>
                . </s>
                <s xml:id="echoid-s253" xml:space="preserve">Hęc ſingulatim conſideremus (
                  <reg norm="nam" type="context">nã</reg>
                ſi in partibus ſimplicibus quod dicimus ac
                  <lb/>
                ciderit, id ipſum in compoſitis conſequenter eueniet) quamobrem ex definitione di
                  <lb/>
                uiſionis dabitur eadem proportio
                  <var>.o.e.</var>
                ad
                  <var>.o.t.</var>
                quæ eft
                  <var>.o.n.</var>
                ad vnitatem, quæ ſit
                  <var>.o.
                    <lb/>
                  x</var>
                . </s>
                <s xml:id="echoid-s254" xml:space="preserve">Nunc cogitemus
                  <reg norm="ſuperficiem" type="context">ſuperficiẽ</reg>
                  <reg norm="rectangulam" type="context">rectangulã</reg>
                  <var>.o.c.</var>
                  <reg norm="æqualem" type="context">æqualẽ</reg>
                quadrato
                  <var>.o.a</var>
                . </s>
                <s xml:id="echoid-s255" xml:space="preserve">tunc numerus
                  <var>.
                    <lb/>
                  c.t.</var>
                proueniens erit, ut patet, ex diuiſione numeri quadrati
                  <var>.o.a.</var>
                per
                  <reg norm="numerum" type="context">numerũ</reg>
                  <var>.o.t.</var>
                  <reg norm="eritque" type="simple">eritq́</reg>
                  <lb/>
                  <reg norm="eadem" type="context">eadẽ</reg>
                proportio
                  <var>.c.t.</var>
                ad
                  <var>.o.e.</var>
                quæ eſt
                  <var>.o.e.</var>
                ad
                  <var>.o.t.</var>
                ex ſecunda parte quintæ decimæ ſexti,
                  <lb/>
                aut .20. ſeptimi. </s>
                <s xml:id="echoid-s256" xml:space="preserve">
                  <reg norm="Iam" type="context">Iã</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                dictum eſt
                  <var>.o.e.</var>
                ad
                  <var>.o.t.</var>
                ſic ſe habere ſicut
                  <var>.o.n.</var>
                ad
                  <var>.o.x</var>
                . </s>
                <s xml:id="echoid-s257" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex .
                  <lb/>
                11. quinti ſic ſe habebit
                  <var>.c.t.</var>
                ad
                  <var>.o.e.</var>
                ſicut
                  <var>.o.n.</var>
                ad
                  <var>.o.x</var>
                . </s>
                <s xml:id="echoid-s258" xml:space="preserve">Sed ex prima ſexti, aut .18. vel .
                  <lb/>
                19. ſeptimi, ſic ſe habet
                  <reg norm="productum" type="simple">ꝓductum</reg>
                  <var>.n.e.</var>
                ad
                  <var>.e.x.</var>
                ſicut
                  <var>.o.n.</var>
                ad
                  <var>.o.x</var>
                . </s>
                <s xml:id="echoid-s259" xml:space="preserve">quare denuo ſic ſe ha-
                  <lb/>
                bebit numerus
                  <var>.c.t.</var>
                ad numerum
                  <var>.o.e.</var>
                ſicut nume-
                  <lb/>
                rus
                  <var>.n.e.</var>
                ad numerum
                  <var>.x.e</var>
                . </s>
                <s xml:id="echoid-s260" xml:space="preserve">Sed numerus
                  <var>.o.e.</var>
                cum
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/>
                numero
                  <var>.x.e.</var>
                ſpecie idem eſt, igitur ex .9. quinti nu
                  <lb/>
                merus
                  <var>.c.t.</var>
                numero
                  <var>.n.e.</var>
                æqualis erit.</s>
              </p>
              <div xml:id="echoid-div59" type="float" level="4" n="1">
                <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a">
                  <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0029-01"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s261" xml:space="preserve">Id ipſum de quadrato ipſius
                  <var>.o.n.</var>
                videlicet
                  <var>.p.o.</var>
                  <lb/>
                dico. </s>
                <s xml:id="echoid-s262" xml:space="preserve">Nam ſi proueniens
                  <var>.o.n.</var>
                diuiſo per
                  <var>.o.e.</var>
                ideſt
                  <var>.
                    <lb/>
                  o.i.</var>
                proportionale reſpondens ad
                  <var>.o.t.</var>
                cum
                  <var>.o.t.</var>
                  <lb/>
                  <reg norm="coniunctum" type="context">coniunctũ</reg>
                fuerit, et per
                  <reg norm="hanc" type="context">hãc</reg>
                ſummam diuiſa ſumma
                  <lb/>
                quadratorum
                  <var>.o.a.</var>
                et
                  <var>.o.p.</var>
                patet per ſe proueniens
                  <lb/>
                futurum eiuſdem numeri
                  <var>.c.t.</var>
                  <reg norm="ipſumque" type="simple">ipſumq́</reg>
                  <var>.c.t.</var>
                proue-
                  <lb/>
                niens ſemper ſuturum.</s>
              </p>
              <p>
                <s xml:id="echoid-s263" xml:space="preserve">Quo autem lucidius res hæc innoteſcat. </s>
                <s xml:id="echoid-s264" xml:space="preserve">Cogi
                  <lb/>
                temus proueniens quadrati
                  <var>.o.p.</var>
                diuiſi ab
                  <var>.o.i.</var>
                re-
                  <lb/>
                  <reg norm="ſpondentisque" type="simple">ſpondentisq;</reg>
                  <var>.o.t.</var>
                eſſe
                  <var>.i.u.</var>
                quod via prædicta inue-
                  <lb/>
                nitur æqualis eſſe numero
                  <var>.n.e.</var>
                ex quo conſe-
                  <lb/>
                quenter æquale
                  <var>.c.t</var>
                : cogitato deinde rectangu-
                  <lb/>
                lo
                  <var>.o.u.</var>
                æquali
                  <var>.o.p.</var>
                coniuncto
                  <var>.o.c</var>
                :totum
                  <var>.t.u.</var>
                æqua-
                  <lb/>
                le erit compoſito duorum quadratorum
                  <var>.o.a.</var>
                et
                  <var>.o.
                    <lb/>
                  p.</var>
                cum in nullo numerus
                  <var>.c.t.</var>
                mutetur, tam ex com-
                  <lb/>
                poſito
                  <var>.t.u.</var>
                  <reg norm="quam" type="context">quã</reg>
                ex ſimplici
                  <var>.o.c.</var>
                ex quo propoſiti ſe
                  <lb/>
                ſe ueritas profert.</s>
              </p>
            </div>
            <div xml:id="echoid-div61" type="math:theorem" level="3" n="27">
              <head xml:id="echoid-head43" xml:space="preserve">THEOREMA
                <num value="27">XXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s265" xml:space="preserve">
                  <emph style="sc">PRoposvervnt</emph>
                veteres nobile quidem problema, ſed quod tamen citra al-
                  <lb/>
                gebraticam effectionem, aut neſcierunt, aut noluerunt diſſoluere, quod nihi-
                  <lb/>
                lominus facillimum eſt.</s>
              </p>
            </div>
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