Ceva, Giovanni, Geometria motus, 1692

Table of figures

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              ad verticem parabolarum, vel trilineorum; erit rectangu­
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              lum ad parabolam ſibi inſcriptam vt aggregatum
                <expan abbr="exponẽ-tium">exponen­
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                tium</expan>
              vtriuſque poteſtatis ad exponentem altioris ipſarum
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              poteſtatum parabolæ; & ad trilineum vt aggregatum ex­
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              ponentium poteſtatum trilinei ad exponentem inferioris
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              poteſtatis eiuſdemmet trilinei. </s>
              <s id="s.000281">Sic enim in expoſita figu­
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              ra prædicta, ſi eſſet quadratum ex FG ad quadratum ex
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              IH, ſicut cubus ex FK ad cubum ex IH, eſſet rectangulum
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              GF in FK ad figuram GFK (quæ tunc foret trilineum, vt
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              5 ad 2; nam vbi poteſtas abſciſſarum maior eſt illa applica.
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              <s id="s.000282">tarum eſt ſemper GF trilineum. </s>
              <s id="s.000283">Simili modo, ſi ſit vt qua­
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              dratum ex FK ad quadratum ex KI ita cubocubus ex FG
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              ad cubocubum ex IH; hoc eſt ſi ſit cubus ex FG ad
                <expan abbr="cubũ">cubum</expan>
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              ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex ſimi­
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              libus ſimiles rationes) erit ſigura GFK parabola, ad quam
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              ſibi circumſcriptum rectangulum eandem habebit
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              ,
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              quam 4 ad 3, & ſic dicendum erit de omnibus alijs para­
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              bolis atque trilineis. </s>
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              DEMONSTRATIO.
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              <s id="s.000285">VErùm vt propoſitum oſtendamus, eſto quælibet ex
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              parabolis GFK, nimirum quadratocubus ex FG ad
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              quadratocubum ex IH habeat eandem rationem, quam̨
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              cubus ex FK ad cubum ex IK. Demonſtro, rectangulum
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              GF in FK habere eandem rationem ad parabolam GFK,
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              quam aggregatum exponentium 8 ad maiorem exponen­
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              tem 5. Primùm, quam rationem habet rectangulum GF in
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              FK ad parabolam GFK, eandem habebit rectangulum HI
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              in IK ad parabolam HIK (hoc enim demonſtrabimus in­
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              frà) permutandoque, erit rectangulum GF in FK ad re­
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              ctangulum HI in IK, vt parabola GFK ad parabolam HIK;
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              componuntur verò illa rectangula ex rationibus GF ad
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              IH, & FK ad IK, ergo etiam parabola ad parabolam com-</s>
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