Archimedes Texts

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.

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

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.