<s><emph type="italics"/>COROLLARIE.<emph.end type="italics"/></s>

 <s>Hence it is manife&longs;t, that Sections of the &longs;ame River (which <lb/>
are no other than the vulgar mea&longs;ures of the River) have <lb/>
betwixt them&longs;elves reciprocal proportions to their veloci<lb/>
ties; for in the fir&longs;t Propo&longs;ition we have demon&longs;trated that the <lb/>
Sections of the &longs;ame River, di&longs;charge equal quantities of Water <lb/>
in equal times; therefore, by what hath now been demon&longs;trated <lb/>
the Sections of the &longs;ame River &longs;hall have reciprocal proportion <lb/>
to their velocities; And therefore the &longs;ame running water chan<lb/>
geth mea&longs;ure, when it changeth velocity; namely, increa&longs;eth the <lb/>
mea&longs;ure, when it decrea&longs;eth the velocity, and decrea&longs;eth the <lb/>
mea&longs;ure, when it increa&longs;eth the velocity.</s>

 <s>On which principally depends all that which hath been &longs;aid <lb/>
above in the <emph type="italics"/>Di&longs;cour&longs;e,<emph.end type="italics"/> and ob&longs;erved in the <emph type="italics"/>Corollaries<emph.end type="italics"/> and <emph type="italics"/>Ap<lb/>
pendixes<emph.end type="italics"/>; and therefore is worthy to be well under&longs;tood and <lb/>
heeded.</s>

 <s>PROPOSITION IV.</s>

 <s><emph type="italics"/>If a River fall into another River, the height of the <lb/>
fir&longs;t in its own Chanel &longs;hall be to the height that it <lb/>
&longs;hall make in the &longs;econd Chanel, in a proportion <lb/>
compounded of the proportions of the breadth of <lb/>
the Chanel of the &longs;econd, to the breadth of the <lb/>
Chanel of the fir&longs;t, and of the velocitie acquired in <lb/>
the Chanel of the &longs;econd, to that which it had in <lb/>
its proper and first Chanel.<emph.end type="italics"/></s>

 <s>Let the River A B, who&longs;e height is A C, and breadth C B, <lb/>
that is, who&longs;e Section is A C B; let it enter, I &longs;ay, into a<lb/>
nother River as broad as the line E F, and let it therein make <lb/>
the ri&longs;e or height D E, that is to &longs;ay, let it have its Section in <lb/>
the River whereinto it falls D E F; I &longs;ay, that the height A C <lb/>
hath to the height D E the proportion compounded of the pro<lb/>
portions of the breadth E F, to the breadth C B, and of the ve<lb/>
locity through D F, to the velocity through A B. </s> <s>Let us &longs;up<lb/>
po&longs;e the Section G, equal in velocity to the Section A B, and in <lb/>
breadth equal to E F, which carrieth a quantity of Water e<lb/>
qual to that which the Section A B carrieth, in equal times, <lb/>
and con&longs;equently, equal to that which D F carrieth. </s> <s>Moreover, <lb/>
as the breadth E F is to the breadth C B, &longs;o let the line H be to