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-div333" type="section" level="3" n="14">
              <pb o="139" rhead="DE PERSPECT." n="151" file="0151" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0151"/>
              <p>
                <s xml:id="echoid-s1675" xml:space="preserve">Ad habendam deinde quantitatem diſtantiæ, aut interualli ſimul cum ſitu, in fa-
                  <lb/>
                cie
                  <var>.q.d.k.</var>
                quem latus
                  <var>.p.l.</var>
                perpendiculariter reſpicit. </s>
                <s xml:id="echoid-s1676" xml:space="preserve">Imaginemur à puncto
                  <var>.u.</var>
                ſuper
                  <lb/>
                  <var>q.a.</var>
                cad ere lineam perpendicularem
                  <var>.u.o.</var>
                quæ illico reperitur cum triangulum
                  <var>.a.
                    <lb/>
                  u.q.</var>
                ex lateribus datis & cognitis conſtet,
                  <reg norm="quodquidem" type="context">quodquidẽ</reg>
                triangulum, medietas eſt qua-
                  <lb/>
                drilateri, ſeu. rumbi
                  <var>.q.a.b.u.</var>
                cui vnaquæque dictarum quatuor facierum perpendi-
                  <lb/>
                cularis exiſtit ex .4. ct .18. lib. 11. & ob id linea
                  <var>.u.o.</var>
                extenſa in ſuperficie dicti quadri-
                  <lb/>
                lateri, & perpendicularis lineæ
                  <var>.q.a.</var>
                perpendicularis erit faciei
                  <var>.q.d.k.</var>
                & ex .29.
                  <lb/>
                primi, angulus
                  <var>.b.u.o.</var>
                rectus erit, ut
                  <reg norm="etiam" type="context">etiã</reg>
                angulus
                  <var>.o.u.l.</var>
                ex .2. definitione lib. 11. vnde
                  <lb/>
                ex .4. eiuſdem lib
                  <var>.o.u.</var>
                perpendicularis erit faciei
                  <var>.b.p.l</var>
                . </s>
                <s xml:id="echoid-s1677" xml:space="preserve">Ha bebimus ergo ſitum in fa-
                  <lb/>
                cie
                  <var>.q.d.k.</var>
                qui reſpicietur ad angulos rectos à linea
                  <var>.p.l.</var>
                quiquidem erit in perpendi-
                  <lb/>
                culari à puncto
                  <var>.o.</var>
                ad
                  <var>.q.a.</var>
                ducta.</s>
              </p>
              <p>
                <s xml:id="echoid-s1678" xml:space="preserve">Quòd autem
                  <var>.a.o.</var>
                ſit latus exagoni æquilateris circumſcrip tibilis ab eodem circu
                  <lb/>
                lo, qui vnam ex faciebus triangularibus æquilateribus propoſiti corporis circunſcri-
                  <lb/>
                bere pot eſt, ita oſtenditur. ſit
                  <reg norm="comprehenſum" type="context">cõprehenſum</reg>
                imaginatione, triangulum
                  <var>.a.q.u.</var>
                ſepara
                  <lb/>
                tim, cuius latus
                  <var>.a.u.</var>
                æquale eſt vni ex lateribus
                  <reg norm="triangulorum" type="context">triangulorũ</reg>
                eiuſdem corporis ex .33.
                  <lb/>
                primi, quo dlibet verò aliorum duorum æquale perpendicularibus dictorum trian-
                  <lb/>
                gulorum, in quo triangulo
                  <var>.a.u.q.</var>
                ducta ſit perpendicularis
                  <var>.u.o.</var>
                ab vna
                  <reg norm="extremitatum" type="context">extremitatũ</reg>
                  <lb/>
                lateris maioris, ad vnum ex minoribus lateribus, quę perpendicularis intra triangu-
                  <lb/>
                lum cadet, quia dictum triangulum oxigonium eſt. </s>
                <s xml:id="echoid-s1679" xml:space="preserve">quod autem attinet ad duos angu
                  <lb/>
                los
                  <var>.a.</var>
                et
                  <var>.u.</var>
                cum æquales ſint ex quinta lib. primi; </s>
                <s xml:id="echoid-s1680" xml:space="preserve">17. nos certiores facit; </s>
                <s xml:id="echoid-s1681" xml:space="preserve">quod verò an­
                  <lb/>
                gulus
                  <var>.q.</var>
                ſit
                  <reg norm="etiam" type="context">etiã</reg>
                acutus: </s>
                <s xml:id="echoid-s1682" xml:space="preserve">30. lib. tertii nos cer-
                  <lb/>
                tos reddit,
                  <reg norm="quia" type="simple">ꝗa</reg>
                  <var>.a.u.</var>
                minor eſt diametro
                  <reg norm="ſphae­ ræ" type="simple">ſphę­
                    <lb/>
                    <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a" number="206">
                      <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0151-01"/>
                    </figure>
                  ræ</reg>
                datum corpus circumſcribentis, cum
                  <var>.q.</var>
                  <lb/>
                dictæ ſphęrę ſuperficiem tangat.</s>
              </p>
              <p>
                <s xml:id="echoid-s1683" xml:space="preserve">Ad probandum
                  <var>.a.o.</var>
                ęqualem eſſe lateri
                  <lb/>
                exagoni dicti, ſatis erit probare
                  <var>.a.q.</var>
                ſeſqui
                  <lb/>
                alteram eſſe ad
                  <var>.a.o.</var>
                quia ſi in ſubſcripto
                  <lb/>
                hîc circulo ducemus duas ſemidiametros
                  <var>.
                    <lb/>
                  n.p.</var>
                et
                  <var>.n.l.</var>
                ad. angulos
                  <reg norm="trianguli" type="context">triãguli</reg>
                ęquilateri
                  <var>.p.</var>
                  <lb/>
                et
                  <var>.l.</var>
                & cum quodlibet laterum ipſius exago
                  <lb/>
                ni, ęquale ſit ſemidiametro circuli ex .15.
                  <lb/>
                lib. 4. habebimus ex .8. primi, angulum
                  <var>.n.
                    <lb/>
                  p.l.</var>
                æqualem angulo
                  <var>.q.p.l</var>
                . </s>
                <s xml:id="echoid-s1684" xml:space="preserve">Vnde ex .4. eiuſ
                  <lb/>
                dem
                  <var>.o.n.</var>
                ęqualiserit ipſi
                  <var>.o.q.</var>
                ideſt
                  <var>.q.a.</var>
                ſeſ
                  <lb/>
                quialtera erit ad
                  <var>.a.o</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1685" xml:space="preserve">Ad probandum nunc in triangulo
                  <var>.a.q.
                    <lb/>
                  u</var>
                :
                  <var>a.q.</var>
                ſeſquialteram eſſe ad
                  <var>.a.o.</var>
                eſt
                  <reg norm="quoque" type="simple">quoq;</reg>
                  <lb/>
                ſciendum primò omne latus trianguli ęquilateri in potentia ſeſquitertium eſſe ad
                  <lb/>
                perpendicularem eiuſdem trianguli, quod vndecima lib. 14. Eucli. breuiter demon
                  <lb/>
                ſtratum eſt.</s>
              </p>
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