Archimedes Texts

40
supposed equal to the Section A, in magnitude; but of velocity
equal to the Section B, and let it be G, and as the Section A is

to the Section B, so let the line F be to the line D; and as the
velocity A, is to the velocity by B, so let the line D be to the
line R: Therefore the Water which passeth thorow A, shall be
to that which passeth through G (in regard the Sections A and
G are of equal bigness, but of unequal velocity) as the velocity
through A, to the velocity through G; But as the velocity
through A, is to the velocity through G, so is the velocity through
A, to the velocity through B; namely, as the line D, to the
line R: therefore the quantity of the Water which passe the
through A, shall be to the quantity which passeth through G, as
the line D is to the line R; but the quantity which passeth
through G, is to that which passeth through B, (in regard the
Sections G, and B, are equally swift) as the Section G to the Se
ction B; that is, as the Section A, to the Section B; that is, as
the line F, to the line D: Therefore by the equal and perturbed
proportionality, the quantity of the Water which passeth through
A, hath the same proportion to that which passeth through B,
that the line F hath to the line R; but F to R, hath a proportion
compounded of the proportions of F to D, and of D to R; that
is, of the Section A to the Section B; and of the velocity through
A, to the velocity through B. Therefore also the quantity of
Water which passeth through the Section A, shall have a propor
tion to that which passeth through the Section B, compounded of
the proportions of the Section A, to the Section B; and of
the velocity through A, to the velocity through B: And
therefore in two Sections of Rivers, the quantity of Water which
passeth by the first, &c. which was to be demonstrated.

The same followeth, though the quantity of the Water which
passeth through the Section A, be equal to the quantity of
Water which passeth through the Section B, as is manifest by the
same demonstration.