Castelli, Benedetto, Of the mensuration of running waters, 1661

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    <archimedes>
      <text>
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          <chap>
            <pb xlink:href="068/01/055.jpg" pagenum="41"/>
            <p type="head">
              <s>PROPOSITION III.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              In two Sections unequal, through which paſs equal
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              quantities of Water in equal times, the Sections
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              have to one another, reciprocal proportion to their
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              velocitie.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let the two unequal Sections, by which paſs equal quantities
                <lb/>
              of Water in equal times be A, the greater; and B, the leſſer:
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              I ſay, that the Section A, ſhall have the ſame Proportion
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              to the Section B, that reciprocally the velocity through B, hath to
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              the velocity through A; for ſuppoſing that as the Water that
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              paſſeth through A, is to that which paſſeth through B, ſo is the
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                <figure id="id.068.01.055.1.jpg" xlink:href="068/01/055/1.jpg" number="8"/>
                <lb/>
              line E to the line F: therefore the quantity of water which paſ­
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              ſeth through A, being equal to that which paſſeth through B,
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              the line E ſhall alſo be equal to the line F: Suppoſing moreover,
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              That as the Section A, is to the Section B, ſo is the line F, to the
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              line G; and becauſe the quantity of water which paſſeth
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              through the Section A, is to that which paſſeth through the
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              Section B, in a proportion compoſed of the proportions of the
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              Section A, to the Section B, and of the velocity through A, to the
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              velocity through B; therefore the line E, ſhall be the line to F, in
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              a proportion compounded of the ſame proportions; namely, of
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              the proportion of the Section A, to the Section B, and of the ve­
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              locity through A, to the velocity through B; but the line E, hath
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              to the line G, the proportion of the Section A, to the Section B,
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              therefore the proportion remaining of the line G, to the line F,
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              ſhall be the proportion of the velocity through A, to the velocity
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              through B; therefore alſo the line G, ſhall be to the line E, as
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              the velocity by A, to the velocity by B: And converſly, the ve­
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              locity through B, ſhall be to the velocity through A, as the line
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              E, to the line G; that is to ſay, as the Section A, to the Section B,
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              and therefore in two Sections, &c. </s>
              <s>which was to be demonſtrated.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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