He elsewhere cites this as a Proposition, therefore I make it of that number.
And here I will not omit to note, as a second Corrollary dependent upon the things demonstrated, that,
THEOREME VIII.
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, namely, the Board, be they great or little, so that they be Prismes and Cylinders having the same Base, are to one another as their heights. 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,