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

Table of figures

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              as the velocity through D F, to the velocity of L N, that is, to
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              the velocity through L N, that is, to the velocity through
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              A F.
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              therefore the quantity of Water which runneth through D
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              F,
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              to that which paſſeth through L N, ſhall have the proportion
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              that R hath to S; but the quantity of the Water that runneth
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              through L N, to that which runneth through
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              A F
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              ; (the Sections
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              being equally ſwift) ſhall have the proportion that the Section
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              L N hath to the Section A F, that is, that the height
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              B
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              D hath to
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              the height
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              B
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              A, that is, that S hath to T. Therefore, by equal
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              proportion, the quantity of the Water which runneth by D F,
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              to that which runneth by A F, ſhall have the proportion of R to
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              T, that is, ſhall be compounded of the proportions of the height
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              D
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              B,
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              to the height A
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              B
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              ; and of the velocity through D F, to
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              the velocity through A F. </s>
              <s>And therefore if a River increaſe in
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              quick height, the quantity of the Water that runneth after the
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              increaſe, to that which runneth before the increaſe, hath the
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              proportion compounded, &c. </s>
              <s>Which was to be demonſtrated.</s>
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              <s>
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              COROLLARIE I.
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              <s>Hence it followeth, that we having ſhewn, that the quantity of
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              the Water which runneth, whilſt the River is high, to that
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              which ran, whilſt it was low, hath the proportion compounded
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              of the velocity to the velocity, and of the height to the height.
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              <s>And it having been demonſtrated, that the velocity to the velo­
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              city is as the height to the height; it followeth, I ſay, that the
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              quantity of the Water that runneth, whilſt the River is high, to
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              that which runneth, whilſt it is low, hath duplicate proportion of
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              the height to the height, that is, the proportion that the ſquares
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              of the heights have.</s>
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              COROLLARIE II.
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              <s>Vpon which things dependeth the reaſon of that which I have
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              ſaid, in my ſecond Conſideration, that if by the diverſion of
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              5/9 of the Water that entereth by the Rivers into the Moor or
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              Fen, the Water be abated ſuch a meaſure, that ſame ſhall be
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              only one third of its whole height; but moreover diverting the 4/9, it
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              ſhall abate two other thirds, a moſt principal point; and ſuch,
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              that its not having been well underſtood, hath cauſed very great
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              diſorders, and there would now, more than ever, follow extream
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              dammage, if one ſhould put in execution the diverſion of the
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              Sile
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              and other Rivers; and it is manifeſt, that in the ſame manner,
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              wherewith it hath been demonſtrated, that the quantity of the
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              Water increaſing quadruple, the height would increaſe onely </s>
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