PROPOSITION III.
In two Sections unequal, through which paſs equal
quantities of Water in equal times, the Sections
have to one another, reciprocal proportion to their
velocitie.
quantities of Water in equal times, the Sections
have to one another, reciprocal proportion to their
velocitie.
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
41[Figure 41]
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.
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
41[Figure 41]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.