Archimedes Texts

Hence it is manifest, that Sections of the same River (which
are no other than the vulgar measures of the River) have
betwixt themselves reciprocal proportions to their veloci
ties; for in the first Proposition we have demonstrated that the
Sections of the same River, discharge equal quantities of Water
in equal times; therefore, by what hath now been demonstrated
the Sections of the same River shall have reciprocal proportion
to their velocities; And therefore the same running water chan
geth measure, when it changeth velocity; namely, increaseth the
measure, when it decreaseth the velocity, and decreaseth the
measure, when it increaseth the velocity.

On which principally depends all that which hath been said
above in the Discourse, and observed in the Corollaries and Ap
pendixes; and therefore is worthy to be well understood and
heeded.

If a River fall into another River, the height of the
first in its own Chanel shall be to the height that it
shall make in the second Chanel, in a proportion
compounded of the proportions of the breadth of
the Chanel of the second, to the breadth of the
Chanel of the first, and of the velocitie acquired in
the Chanel of the second, to that which it had in
its proper and first Chanel.

Let the River A B, whose height is A C, and breadth C B,
that is, whose Section is A C B; let it enter, I say, into a
nother River as broad as the line E F, and let it therein make
the rise or height D E, that is to say, let it have its Section in
the River whereinto it falls D E F; I say, that the height A C
hath to the height D E the proportion compounded of the pro
portions of the breadth E F, to the breadth C B, and of the ve
locity through D F, to the velocity through A B. Let us sup
pose the Section G, equal in velocity to the Section A B, and in
breadth equal to E F, which carrieth a quantity of Water e
qual to that which the Section A B carrieth, in equal times,
and consequently, equal to that which D F carrieth. Moreover,
as the breadth E F is to the breadth C B, so let the line H be to