[Sidenote: Natation and Submersion, collected from the thickness,
excluding the length and breadth of Plates.]
_The Expansion of Figure not only is not the Cause of the Natation
of those grave Bodies, which otherwise do submerge, but also the
determining what be those Boards of Ebony, or Plates of Iron or
Gold that will swimme, depends not on it, rather that same
determination is to be collected from the only thickness of those
Figures of Ebony or Gold, wholly excluding the consideration of
length and breadth, as having no wayes any share in this Effect._
It hath already been manifested, that the only cause of the Natation
of the said Plates, is the reduction of them to be less grave than the
water, by means of the connexion of that Air, which descendeth
together with them, and possesseth place in the water; which place so
occupyed, if before the circumfused water diffuseth it self to fill
it, it be capable of as much water, as shall weigh equall with the
Plate, the Plate shall remain suspended, and sinke no farther.
Now let us see on which of these three dimensions of the Solid
depends the terminating, what and how much the Mass of that ought to
be, that so the assistance of the Air contiguous unto it, may suffice
to render it specifically less grave than the water, whereupon it may
rest without Submersion. It shall undoubtedly be found, that the
length and breadth have not any thing to do in the said determination,
but only the height, or if you will the thickness: for, if we take a
Plate or Board, as for Example, of Ebony, whose Altitude hath unto the
greatest possible Altitude of the Rampart, the proportion above
declared, for which cause it swims indeed, but yet not if we never so
little increase its thickness; I say, that retaining its thickness,
and encreasing its Superficies to twice, four times, or ten times its
bigness, or dminishing it by dividing it into four, or six, or twenty,
or a hundred parts, it shall still in the same manner continue to
float: but encreasing its thickness only a Hairs breadth, it will
alwaies submerge, although we should multiply the Superficies a
hundred and a hundred times. Now forasmuch as that this is a Cause,
which being added, we adde also the Effect, and being removed, it is
removed; and by augmenting or lessening the length or breadth in any
manner, the effect of going, or not going to the bottom, is not added
or removed: I conclude, that the greatness and smalness of the
Superficies hath no influence upon the Natation or Submersion. And
that the proportion of the Altitude of the Ramparts of Water, to the
Altitude of the Solid, being constituted in the manner aforesaid, the
greatness or smalness of the Superficies, makes not any variation, is
manifest from that which hath been above demonstrated, and from this,
that, _The Prisms and Cylinders which have the same Base, are in
proportion to one another as their heights._ Whence Cylinders or
Prismes[76], namely, the Board, be they great or little, so that they be
all of equall thickness, have the same proportion to their
Conterminall Air, which hath for Base the said Superficies of the
Board, and for height the Ramparts of water; so that alwayes of that
Air, and of the Board, Solids, are compounded, that in Gravity equall
a Mass of water equall to the Mass of the Solids, compounded of Air,
and of the Board: whereupon all the said Solids do in the same manner
continue afloat. We will conclude in the third place, that,
[76] Prismes and Cylinders having the same Base, are to one
another as their heights.