Galilei, Galileo, The systems of the world, 1661

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="065/01/022.jpg" pagenum="16"/>
              perpendicular ſhould be taken near to the end C, and in the
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              clination, far from it.</s>
            </p>
            <p type="main">
              <s>SALV. </s>
              <s>You ſee then, that the Propoſition which ſaith, that
                <lb/>
              the motion by the perpendicular is more ſwift than by the
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              nation, holds not true univerſally, but onely of the motions,
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              which begin from the extremity, namely from the point of reſt:
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              without which reſtriction, the Propoſition would be ſo deficient,
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              that its very direct contrary might be true; namely, that the
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              tion in the inclining plane is ſwifter than in the perpendicular:
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              for it is certain, that in the ſaid inclination, we may take a ſpace
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              paſt by the moveable in leſs time, than the like ſpace paſt in the
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              perpendicular. </s>
              <s>Now becauſe the motion in the inclination is in
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              ſome places more, in ſome leſs, than in the perpendicular;
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              fore in ſome places of the inclination, the time of motion of the
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              moveable, ſhall have a greater proportion to the time of the motion
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              of the moveable, by ſome places of the perpendicular, than the
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              ſpace paſſed, to the ſpace paſſed: and in other places, the
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              portion of the time to the time, ſhall be leſs than that of the
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              ſpace to the ſpace. </s>
              <s>As for example: two moveables departing
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              from their quieſcence, namely, from the point C, one by the
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              pendicular C B, [in
                <emph type="italics"/>
              Fig.
                <emph.end type="italics"/>
              4.] and the other by the inclination C A,
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              in the time that, in the perpendicular, the moveable ſhall have
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              paſt all C B, the other ſhall have paſt C T leſſer. </s>
              <s>And therefore
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              the time by C T, to the time by C B (which is equal) ſhall have
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              a greater proportion than the line C T to C B, being that the
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                <emph type="italics"/>
              ſame
                <emph.end type="italics"/>
              to the
                <emph type="italics"/>
              leſs,
                <emph.end type="italics"/>
              hath a greater proportion than to the
                <emph type="italics"/>
              greater.
                <emph.end type="italics"/>
                <lb/>
              And on the contrary, if in C A, prolonged as much as is
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              ſite, one ſhould take a part equal to C B, but paſt in a ſhorter
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              time; the time in the inclination ſhall have a leſs proportion to
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              the time in the perpendicular, than the ſpace to the ſpace. </s>
              <s>If
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              therefore in the inclination and perpendicular, we may ſuppoſe
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              ſuch ſpaces and velocities, that the proportion between the ſaid
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              ſpaces be greater and leſs than the proportion of the times; we
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              may eaſily grant, that there are alſo ſpaces, by which the times
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              of the motions retain the ſame proportion as the ſpaces.</s>
            </p>
            <p type="main">
              <s>SAGR. </s>
              <s>I am already freed from my greateſt doubt, and
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              ceive that to be not onely poſſible, but neceſſary, which I but
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              now thought a contradiction: but nevertheleſs I underſtand not
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              as yet, that this whereof we now are ſpeaking, is one of theſe
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              poſſible or neceſſary caſes; ſo as that it ſhould be true, that the
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              time of deſcent by C A, to the time of the fall by C B, hath the
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              ſame proportion that the line C A hath to C B; whence it may
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              without contradiction be affirmed, that the velocity by the
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              nation C A, and by the perpendicular C B, are equal.</s>
            </p>
            <p type="main">
              <s>SALV. </s>
              <s>Content your ſelf for this time, that I have removed </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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