Castelli, Benedetto - Of the mensuration of running waters

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1as the velocity through D F, to the velocity of L N, that is, to
the velocity through L N, that is, to the velocity through A F.
therefore the quantity of Water which runneth through D F,
to that which paſſeth through L N, ſhall have the proportion
that R hath to S; but the quantity of the Water that runneth
through L N, to that which runneth through A F; (the Sections
being equally ſwift) ſhall have the proportion that the Section
L N hath to the Section A F, that is, that the height B D hath to
the height B A, that is, that S hath to T. Therefore, by equal
proportion, the quantity of the Water which runneth by D F,
to that which runneth by A F, ſhall have the proportion of R to
T, that is, ſhall be compounded of the proportions of the height
D B, to the height A B; and of the velocity through D F, to
the velocity through A F. And therefore if a River increaſe in
quick height, the quantity of the Water that runneth after the
increaſe, to that which runneth before the increaſe, hath the
proportion compounded, &c. Which was to be demonſtrated.

COROLLARIE I.

Hence it followeth, that we having ſhewn, that the quantity of
the Water which runneth, whilſt the River is high, to that
which ran, whilſt it was low, hath the proportion compounded
of the velocity to the velocity, and of the height to the height.
And it having been demonſtrated, that the velocity to the velo
city is as the height to the height; it followeth, I ſay, that the
quantity of the Water that runneth, whilſt the River is high, to
that which runneth, whilſt it is low, hath duplicate proportion of
the height to the height, that is, the proportion that the ſquares
of the heights have.

COROLLARIE II.

Vpon which things dependeth the reaſon of that which I have
ſaid, in my ſecond Conſideration, that if by the diverſion of
5/9 of the Water that entereth by the Rivers into the Moor or
Fen, the Water be abated ſuch a meaſure, that ſame ſhall be
only one third of its whole height; but moreover diverting the 4/9, it
ſhall abate two other thirds, a moſt principal point; and ſuch,
that its not having been well underſtood, hath cauſed very great
diſorders, and there would now, more than ever, follow extream
dammage, if one ſhould put in execution the diverſion of the Sile
and other Rivers; and it is manifeſt, that in the ſame manner,
wherewith it hath been demonſtrated, that the quantity of the
Water increaſing quadruple, the height would increaſe onely

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