Galilei, Galileo, The systems of the world, 1661

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="065/01/184.jpg" pagenum="176"/>
              and I will give you an anſwer. </s>
              <s>Tell me therefore, how much do
                <lb/>
              you think ſufficeth to make that motion ſwifter than this?</s>
            </p>
            <p type="main">
              <s>SIMP. </s>
              <s>I will ſay for example, that if that motion by the
                <lb/>
              gent were a million of times ſwifter than this by the ſecant, the
                <lb/>
              pen, yea, and the ſtone alſo would come to be extruded.</s>
            </p>
            <p type="main">
              <s>SALV. </s>
              <s>You ſay ſo, and ſay that which is falſe, onely for
                <lb/>
              want, not of Logick, Phyſicks, or Metaphyſicks, but of
                <lb/>
              try; for if you did but underſtand its firſt elements, you would
                <lb/>
              know, that from the centre of a circle a right line may be drawn
                <lb/>
              to meet the tangent, which interſecteth it in ſuch a manner, that
                <lb/>
              the part of the tangent between the contact and the ſecant, may
                <lb/>
              be one, two, or three millions of times greater than that part of
                <lb/>
              the ſecant which lieth between the tangent and the circumference,
                <lb/>
              and that the neerer and neerer the ſecant ſhall be to the contact,
                <lb/>
              this proportion ſhall grow greater and greater
                <emph type="italics"/>
              in infinitum
                <emph.end type="italics"/>
              ; ſo
                <lb/>
              that it need not be feared, though the
                <emph type="italics"/>
              vertigo
                <emph.end type="italics"/>
              be ſwift, and the
                <lb/>
              motion downwards ſlow, that the pen or other lighter matter can
                <lb/>
              begin to riſe upwards, for that the inclination downwards always
                <lb/>
              exceedeth the velocity of the projection.</s>
            </p>
            <p type="main">
              <s>SAGR. </s>
              <s>I do not perfectly apprehend this buſineſſe.</s>
            </p>
            <p type="main">
              <s>SALV. </s>
              <s>I will give you a moſt univerſal yet very eaſie demon­</s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg367"/>
                <lb/>
              ſtration thereof. </s>
              <s>Let a proportion be given between B A [
                <emph type="italics"/>
              in Fig.
                <emph.end type="italics"/>
                <lb/>
              3.] and C: And let B A be greater than C at pleaſure. </s>
              <s>And let
                <lb/>
              there be deſcribed a circle, whoſe centre is D. </s>
              <s>From which it is
                <lb/>
              required to draw a ſecant, in ſuch manner, that the tangent may
                <lb/>
              be in proportion to the ſaid ſecant, as B A to C. </s>
              <s>Let A I be
                <lb/>
              ſuppoſed a third proportional to B A and C. </s>
              <s>And as B I is to
                <lb/>
              I A, ſo let the diameter F E be to E G; and from the point G,
                <lb/>
              let there be drawn the tangent G H. </s>
              <s>I ſay that all this is done as
                <lb/>
              was required; and as B A is to C, ſo is H G to G E. </s>
              <s>And in
                <lb/>
              gard that as B I is to I A, ſo is F E to E G; therefore by
                <lb/>
              ſition, as B A is to A I; ſo ſhall F G be to G E. </s>
              <s>And becauſe C
                <lb/>
              is the middle proportion between
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              A and A I; and G H is a
                <lb/>
              middle term between F G and G E; therefore, as B A is to C,
                <lb/>
              ſo ſhall F G be to G H; that is H G to G E, which was to be
                <lb/>
              demonſtrated.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg367"/>
                <emph type="italics"/>
              A geometrical
                <lb/>
              demonſtration to
                <lb/>
              prove the
                <lb/>
              bility of extruſion
                <lb/>
              by means of the
                <lb/>
              terreſtrial
                <emph.end type="italics"/>
              vertigo.</s>
            </p>
            <p type="main">
              <s>SAGR. </s>
              <s>I apprehend this demonſtration; yet nevertheleſſe, I
                <lb/>
              am not left wholly without hæſitation; for I find certain
                <lb/>
              ſed ſcruples role to and again in my mind, which like thick and
                <lb/>
              dark clouds, permit me not to diſcern the cleerneſſe and neceſſity
                <lb/>
              of the concluſion with that perſpicuity, which is uſual in
                <lb/>
              matical Demonſtrations. </s>
              <s>And that which I ſtick at is this. </s>
              <s>It is
                <lb/>
              true that the ſpaces between the tangent and the circumference do
                <lb/>
              gradually diminiſh
                <emph type="italics"/>
              in infinitum
                <emph.end type="italics"/>
              towards the contact; but it is alſo
                <lb/>
              true on the contrary, that the propenſion of the moveable to </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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