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

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1
Let the two unequal Sections, by which paſs equal quantities
of
Water in equal times be A, the greater; and B, the leſſer:
I
ſay, that the Section A, ſhall have the ſame Proportion
to
the Section B, that reciprocally the velocity through B, hath to
the
velocity through A; for ſuppoſing that as the Water that
paſſeth
through A, is to that which paſſeth through B, ſo is the
8[Figure 8]
line
E to the line F: therefore the quantity of water which paſ­
ſeth
through A, being equal to that which paſſeth through B,
the
line E ſhall alſo be equal to the line F: Suppoſing moreover,
That
as the Section A, is to the Section B, ſo is the line F, to the
line
G; and becauſe the quantity of water which paſſeth
through
the Section A, is to that which paſſeth through the
Section
B, in a proportion compoſed of the proportions of the
Section
A, to the Section B, and of the velocity through A, to the
velocity
through B; therefore the line E, ſhall be the line to F, in
a
proportion compounded of the ſame proportions; namely, of
the
proportion of the Section A, to the Section B, and of the ve­
locity
through A, to the velocity through B; but the line E, hath
to
the line G, the proportion of the Section A, to the Section B,
therefore
the proportion remaining of the line G, to the line F,
ſhall
be the proportion of the velocity through A, to the velocity
through
B; therefore alſo the line G, ſhall be to the line E, as
the
velocity by A, to the velocity by B: And converſly, the ve­
locity
through B, ſhall be to the velocity through A, as the line
E
, to the line G; that is to ſay, as the Section A, to the Section B,
and
therefore in two Sections, &c.
which was to be demonſtrated.

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