1ſuperficies A E, the Cylinder C F ſhall be leſſe than A E: For
if they were equal, its Superficies, by the laſt Propoſition would
be greater than the Superficies A E, and
62[Figure 62]
much the more, if the ſaid Cylinder C F
were greater than A E. Let the Cylinder
I D be ſuppoſed equal to A E: There
fore, by the precedent Propoſition, the
Superficies of the Cylinder I D ſhall be
to the Superficies A E, as the height I F
to the Mean-proportional betwixt I F &
A B. But the Superficies A E being by
Suppoſition equal to C F and I D, ha
ving the ſame proportion to C F that the
height I F hath to C D: Therefore
C D is the Mean-Proportional between
I F and A B. Moreover, the Cylinder
I D being equal to the Cylinder A E,
they ſhall both have the ſame proporti
on to the Cylinder C F: But I D is to
C F, as the height I F is to C D: Therefore the Cylinder A E
ſhall have the ſame proportion to the Cylinder C F, that the line
I F hath to C D; that is, that C D hath to A B: Which was to be
demonſtrated.
if they were equal, its Superficies, by the laſt Propoſition would
be greater than the Superficies A E, and
62[Figure 62]much the more, if the ſaid Cylinder C F
were greater than A E. Let the Cylinder
I D be ſuppoſed equal to A E: There
fore, by the precedent Propoſition, the
Superficies of the Cylinder I D ſhall be
to the Superficies A E, as the height I F
to the Mean-proportional betwixt I F &
A B. But the Superficies A E being by
Suppoſition equal to C F and I D, ha
ving the ſame proportion to C F that the
height I F hath to C D: Therefore
C D is the Mean-Proportional between
I F and A B. Moreover, the Cylinder
I D being equal to the Cylinder A E,
they ſhall both have the ſame proporti
on to the Cylinder C F: But I D is to
C F, as the height I F is to C D: Therefore the Cylinder A E
ſhall have the ſame proportion to the Cylinder C F, that the line
I F hath to C D; that is, that C D hath to A B: Which was to be
demonſtrated.
Of Corn-ſacks
with a Board at
the Bottom, made
of the ſame Stuffe,
but different in
height, which are
the more capa
cious.
with a Board at
the Bottom, made
of the ſame Stuffe,
but different in
height, which are
the more capa
cious.
From hence is collected the Cauſe of an Accident, which the
Vulgar do not hearken to without admiration; and it is, how it
is poſſible that the ſame piece of ^{*}Cloth, being longer one way than
another, if a Sack be made thereof to hold Corn, as the uſual
manner is, with a Board at the bottom, will hold more, making
uſe of the leſſer breadth of the Cloth, for the height of the Sack,
Vulgar do not hearken to without admiration; and it is, how it
is poſſible that the ſame piece of ^{*}Cloth, being longer one way than
another, if a Sack be made thereof to hold Corn, as the uſual
manner is, with a Board at the bottom, will hold more, making
uſe of the leſſer breadth of the Cloth, for the height of the Sack,
and with the other encompaſſing the Board at the bottom, than if
it be made up the other way: As if for Example, the Cloth were
one way ſix foot, and the other way twelve, it will hold more,
when with the length of twelve one encompaſſeth the Board at the
bottom, the Sack being ſix foot high, than if it encompaſſed a
bottom of ſix foot, having twelve for its height. Now, by what
hath been demonſtrated, there is added to the Knowledge in ge
neral that it holds more that way than this, the Specifick, and
particular Knowledge of how much it holdeth more: which is,
That it will hold more in proportion as it is lower, and leſſer, as
it is higher. And thus in the meaſures afore taken, the Cloth be
ing twice as long as broad, when it is ſewed the length-ways it will
hold but half ſo much, as it will do the other way. And likewiſe
having a Mat to make a ^{*} Frale or Basket twenty five foot long,
and ſuppoſe ſeven broad; made up the long-way it will hold but
onely ſeven of thoſe meaſures, whereof the other way it will hold
five and twenty.