[Sidenote: To finde the proportion Figures ought to have to the
waters Gravity, that by help of the contiguous Air, they may
swim.]
_To finde what proportion severall Figures of different Matters
ought to have, unto the Gravity of the Water, that so they may be
able by vertue of the Contiguous Air to stay afloat._
[Illustration]
Let, therefore, for better illustration, D F N E be a Vessell,
wherein the water is contained, and suppose a Plate or Board, whose
thickness is comprehended between the Lines I C and O S, and let it be
of Matter exceeding the water in Gravity, so that being put upon the
water, it dimergeth and abaseth below the Levell of the said water,
leaving the little Banks A I and B C, which are at the greatest height
they can be, so that if the Plate I S should but descend any little
space farther, the little Banks or Ramparts would no longer consist,
but expulsing the Air A I C B, they would diffuse themselves over the
Superficies I C, and would submerge the Plate. The height A I B C is
therefore the greatest profundity that the little Banks of water admit
of. Now I say, that from this, and from the proportion in Gravity,
that the Matter of the Plate hath to the water, we may easily finde of
what thickness, at most, we may make the said Plates, to the end, they
may be able to bear up above water: for if the Matter of the Plate or
Board I S were, for Example, as heavy again as the water, a Board of
that Matter shall be, at the most of a thickness equall to the
greatest height of the Banks, that is, as thick as A I is high: which
we will thus demonstrate. Let the Solid I S be double in Gravity to
the water, and let it be a regular Prisme, or Cylinder, to wit, that
hath its two flat Superficies, superiour and inferiour, alike and
equall, and at Right Angles with the other laterall Superficies, and
let its thickness I O be equall to the greatest Altitude of the Banks
of water: I say, that if it be put upon the water, it will not
submerge: for the Altitude A I being equall to the Altitude I O, the
Mass of the Air A B C I shall be equall to the Mass of the Solid C I O
S: and the whole Mass A O S B double to the Mass I S; And since the
Mass of the Air A C, neither encreaseth nor diminisheth the Gravity of
the Mass I S, and the Solid I S was supposed double in Gravity to the
water; Therefore as much water as the Mass submerged A O S B,
compounded of the Air A I C B, and of the Solid I O S C, weighs just
as much as the same submerged Mass A O S B: but when such a Mass of
water, as is the submerged part of the Solid, weighs as much as the
said Solid, it descends not farther, but resteth, as by (_a_)
_Archimedes_[75], and above by us, hath been demonstrated: Therefore, I S
shall descend no farther, but shall rest. And if the Solid I S shall
be Sesquialter in Gravity to the water, it shall float, as long as its
thickness be not above twice as much as the greatest Altitude of the
Ramparts of water, that is, of A I. For I S being Sesquialter in
Gravity to the water, and the Altitude O I, being double to I A, the
Solid submerged A O S B, shall be also Sesquialter in Mass to the
Solid I S. And because the Air A C, neither increaseth nor diminisheth
the ponderosity of the Solid I S: Therefore, as much water in quantity
as the submerged Mass A O S B, weighs as much as the said Mass
submerged: And, therefore, that Mass shall rest. And briefly in
generall.
[75] Of Natation Lib. 1. Prop. 3.
[Illustration]