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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IO. BAPT. BENED.
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            <div xml:id="echoid-div495" type="section" level="3" n="5">
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                <head xml:id="echoid-head380" style="it" xml:space="preserve">Demonstrationes quarundam propoſitionum de quibus agit
                  <lb/>
                Cardanus capite primo libro .16. de
                  <lb/>
                ſubtilitate.</head>
                <head xml:id="echoid-head381" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3171" xml:space="preserve">EA quæ Cardanus in primo cap. lib. 16. de ſubtilitate ita ſcribit, quod ſi diame-
                    <lb/>
                  tros producatur extra quantumlibet, alia verò diametro in centro ſecetur ad
                    <lb/>
                  rectos, ex huius fine
                    <reg norm="&c." type="unresolved">&c.</reg>
                  quæ quidem ſecundum illum eſt vndecima proprietas cir
                    <lb/>
                  culi, quoniam te id non intelligere ſcribis,
                    <reg norm="idemque" type="simple">idemq́;</reg>
                  dicis etiam de duodecima, & ſi-
                    <lb/>
                  militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, ſeu
                    <lb/>
                  ellipſi, & hyperboli, tibi breuiter reſpondebo.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3172" xml:space="preserve">Circa vndecimam proprietatem circuli verum dicit. </s>
                  <s xml:id="echoid-s3173" xml:space="preserve">Imaginemur circulum
                    <var>.p.
                      <lb/>
                    d.q.</var>
                  à duabus diametris, inuicem ad angulos rectos coniunctis, diuiſum
                    <var>.p.d.</var>
                  et
                    <var>.d.g.</var>
                  di
                    <lb/>
                  uidatur enim quarta
                    <var>.q.d.</var>
                  per quot partes æquales volueris, mediantibus punctis
                    <var>.b.a.
                      <lb/>
                    o.</var>
                    <reg norm="ducanturque" type="simple">ducanturq́;</reg>
                  ab ijſdem punctis tot perpendiculares diametro
                    <var>.d.g.</var>
                  quæ ſint
                    <var>.b.m.a.n.</var>
                    <lb/>
                  et
                    <var>.o.s.</var>
                  quæ quidem erunt parallelæ diametro
                    <var>.q.p.</var>
                  coniungatur deinde extremitas
                    <var>.d.</var>
                    <lb/>
                  diametri
                    <var>.d.g.</var>
                  cum primo puncto
                    <var>.b.</var>
                  & protrahatur
                    <var>.d.b.</var>
                  vſque ad concurſum cum diz
                    <lb/>
                  metro
                    <var>.p.q.</var>
                  protracto in puncto, h. </s>
                  <s xml:id="echoid-s3174" xml:space="preserve">Nunc dico
                    <var>.q.h.</var>
                  quæ adiacet diametro
                    <var>.q.p.</var>
                  æqua-
                    <lb/>
                  lem eſſe omnibus dictis perpendicularibus, quapropter coniungantur puncta
                    <var>.m.a</var>
                  :
                    <lb/>
                    <var>n.o.</var>
                  et
                    <var>.s.q.</var>
                  & producantur vſque ad adiacentem diametro
                    <var>.q.p.</var>
                  in punctis
                    <var>.c.</var>
                  et
                    <var>.e.</var>
                  vn
                    <lb/>
                  de habebimus angulos
                    <var>.b.a.o.q.</var>
                  inuicem æquales ex .26. tertij, cum verò
                    <var>.o.s.a.n.</var>
                  et
                    <lb/>
                    <var>b.m.</var>
                  parallelæ ſint ipſi
                    <var>.p.h</var>
                  . </s>
                  <s xml:id="echoid-s3175" xml:space="preserve">tunc anguli
                    <var>.b.h.c</var>
                  :
                    <var>a.c.e</var>
                  : et
                    <var>.o.e.q.</var>
                  æquales erunt angulis
                    <var>.d.
                      <lb/>
                    b.m</var>
                  :
                    <var>m.a.n.</var>
                  et
                    <var>.n.o.s.</var>
                  ex .29. primi: </s>
                  <s xml:id="echoid-s3176" xml:space="preserve">quare anguli
                    <var>.h.c.e.q.</var>
                  erunt inuicem æquales, vnde
                    <lb/>
                  ex .28. eiuſdem
                    <var>.b.h</var>
                  :
                    <var>m.c</var>
                  :
                    <var>n.e.</var>
                  et
                    <var>.s.q.</var>
                  erunt
                    <reg norm="inuicem" type="context">inuicẽ</reg>
                  parallelę, & ex .34.
                    <var>e.q.</var>
                  æqualis erit
                    <var>.
                      <lb/>
                    o.s.</var>
                  et
                    <var>.e.c.</var>
                  æqualis
                    <var>.n.a.</var>
                  et
                    <var>.m.b.</var>
                  æqualis
                    <var>.c.h.</var>
                  verum eſt igitur propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3177" xml:space="preserve">Duodecima vero
                    <reg norm="proprietas" type="simple">ꝓprietas</reg>
                  eſt, ut ſi fuerit circulus
                    <var>.a.b.e.q.</var>
                  cuius duo diametriad
                    <lb/>
                  rectos coniuncti ſint
                    <var>.a.e.</var>
                  et
                    <var>.q.b.</var>
                  & diameter
                    <var>.a.e.</var>
                  protractus indeterminatè ad partem
                    <lb/>
                  e. </s>
                  <s xml:id="echoid-s3178" xml:space="preserve">tunc ſi ab extremo
                    <var>.b.</var>
                  diametri
                    <var>.q.b.</var>
                  ducta fuerit
                    <var>.b.n.u.</var>
                  extra circulum, ſeu
                    <var>.b.u.n.</var>
                  in
                    <lb/>
                  tra circulum, vt in ſubiecta figura patet, ita vt ſecta ſit à circunferentia circuli in
                    <reg norm="pum" type="context">pũ</reg>
                    <lb/>
                  cto
                    <var>.n.</var>
                  vel à diametro in puncto
                    <var>.u.</var>
                  ſemper id quod fit ex
                    <var>.u.b.</var>
                  in
                    <var>.b.n.</var>
                  æquale erit qua-
                    <lb/>
                  drato inſcriptibili in dicto circulo, hoc autem diuerſimodè cognoſci poteſt, tribus
                    <lb/>
                  enim modis ego inueni, quorum primus ita ſe habet. </s>
                  <s xml:id="echoid-s3179" xml:space="preserve">Nam ſi punctus
                    <var>.u.</var>
                  fuerit ex-
                    <lb/>
                  tra circulum, ducantur
                    <var>.b.e.</var>
                  et
                    <var>.e.n.</var>
                  & habebimus duos triangulos
                    <var>.b.n.e.</var>
                  et
                    <var>.b.e.u.</var>
                  ſimi
                    <lb/>
                  les inuicem, eo, quod angulus
                    <var>.b.</var>
                  communis ambobus exiſtit, & angulus
                    <var>.b.n.e.</var>
                  æqua
                    <lb/>
                  lis eſt angulo
                    <var>.b.e.u.</var>
                  quod ita probatur, nam angulus
                    <var>.b.n.e.</var>
                  cum angulo
                    <var>.b.a.e.</var>
                  (ducta
                    <lb/>
                  cum fuerit
                    <var>.b.a.</var>
                  ) æquatur duobus rectis ex .21. tertij, ſed ex quinta primi angulus
                    <var>.b.
                      <lb/>
                    e.a.</var>
                  ęqualis eſt angulo
                    <var>.b.a.e</var>
                  : </s>
                  <s xml:id="echoid-s3180" xml:space="preserve">quare angulus
                    <var>.b.n.e.</var>
                  cum angulo
                    <var>.b.e.a.</var>
                  ęquatur duobus
                    <lb/>
                  rectis, ſed ex .13. eiuſdem angulus
                    <var>.b.n.e.</var>
                  cum angulo etiam
                    <var>.e.n.u.</var>
                  æquatur duobus re
                    <lb/>
                  ctis, ergo angulus
                    <var>.e.n.u.</var>
                  æquatur angulo
                    <var>.b.e.a</var>
                  . </s>
                  <s xml:id="echoid-s3181" xml:space="preserve">quare angulus
                    <var>.b.n.e.</var>
                  æquatur
                    <reg norm="etiam" type="context">etiã</reg>
                  an-
                    <lb/>
                  gulo
                    <var>.b.e.u.</var>
                  vnde ex .32. eiuſdem reliquus angulus
                    <var>.b.u.e.</var>
                  æqualis erit reliquo angulo
                    <lb/>
                    <var>b.e.n.</var>
                  latera igitur erunt proportionalia ex .4. ſexti, vnde ita ſe habebit
                    <var>.u.b.</var>
                  ad
                    <var>.b.
                      <lb/>
                    e.</var>
                  vt
                    <var>.b.e.</var>
                  ad
                    <var>.b.n.</var>
                  ex .16. ſexti igitur
                    <reg norm="verum" type="context">verũ</reg>
                  erit propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3182" xml:space="preserve">Sed ſi punctus
                    <var>.u.</var>
                  intra circulum fuerit, triangulus
                    <var>.b.e.n.</var>
                  ſimilis erit triangulo
                    <var>.b.u.
                      <lb/>
                    e.</var>
                  nam angulus
                    <var>.b.</var>
                  ambobus communis erit. </s>
                  <s xml:id="echoid-s3183" xml:space="preserve">Angulus vero
                    <var>.b.n.e.</var>
                  ęqualis eſt angulo
                    <var>.
                      <lb/>
                    b.e.u.</var>
                  ex .26. tertij, </s>
                  <s xml:id="echoid-s3184" xml:space="preserve">quare ex .32. primi reliquus angulus
                    <var>.b.e.n.</var>
                  æqualis erit reliquo </s>
                </p>
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