... the third of the figures prescribed, because the angles at y and g are equal; and [the lines] y b, g b are therefore equal. Wherefore also s r, c r, and p z, o u, and z t, u k. Since [it] is less than double of o to a n, it is plain that p z is less than double of z t. Let then p ω be double of ω t, and let k ω, being joined, be produced to e. The centre of gravity of the whole, then, will be k; that of the part between the fluid, ω; and that of the part outside, on the line k e, let it be e. But k z will be perpendicular to the surface of the fluid, wherefore the line through the points e, ω is parallel to k z. Therefore the segment does not remain, but will incline so that its base does not touch the surface of the fluid at a single [point], since, now touched at one [point], it inclines. It is manifest, therefore, that the segment will stand so that its axis makes with the surface of the fluid an angle greater than the angle x.
Let the segment have to the fluid, in gravity, the proportion that the square on x o has to that on b d, and let it be released into the fluid thus inclined. Let it be cut through the axis by a plane at right angles to the surface of the fluid; let the section of the solid be a p o l, a section of a right-angled cone [a parabola]; the surface of the fluid, o i; the axis of the segment and diameter of the section, b d. Let b d be cut as before, and let p n be drawn parallel to i o, touching the section at p; and p t parallel to b d; and p s perpendicular upon b d. It must be demonstrated that the segment does not remain thus inclined, but inclines until its base touches the surface of the fluid at a single point. Let the things arranged before in the upper figure also be presupposed, and let c o be drawn perpendicular upon b d, and let a x, being joined, be produced to q. And a x will be equal to x q; and let o y be drawn parallel to a q. And since the segment is supposed to have to the fluid, in gravity, the proportion that the square on x a has to that on b d, and the immersed segment has this same proportion to the whole, that is, [the square] on a t p to that on b d, the line p t will be equal to x o. And since the diameters of the segments i b o, a b q are equal, and the segments ...

... k c, and let it be c. It will be manifest, by the preceding theorem, that the segment does not remain but will incline so that its base does not touch the surface of the fluid even at a single [point]. But that it will stand so that the axis makes with the surface of the fluid an angle not less than the angle f, will be demonstrated. For let it stand, if possible, so that it makes an angle not less than f, and let the rest be arranged as in the third figure. Similarly it will be shown that t m is equal to x, and therefore to i h. And since h l is less than f, it is therefore not greater; nor is s r, nor n a, greater than o g; and since i h is hemiolic [one and a half times] of p y, but p y less than g o, and the [line] that it has is equal to p c, while h a is not less than o g, therefore a h is greater than p y. Therefore a h is greater than double of t a. Let a h be double of y t, and let y k, being joined, be produced. It is plain, as in the former [cases], that the segment does not remain, but will turn so that its axis makes with the surface of the fluid an angle less than the angle f.
Archimedes, On Floating Bodies, the second book, in fluid.