This paper presents new developments in the fundamental theory for the hydrostatics of floating bodies, such as a ship. In it, we show that a proof that the center of buoyancy is equal to the center of hydrostatic pressure, a new derivation of the metacenter radius, and theoretical treatments of the hydrostatic stability of floating bodies based on the above two new theories.

In Chapter 1, we prove that " the center of buoyancy of a ship is equal to the center of hydrostatic pressure ". This subject is an unsolved problem in physics and naval architecture, even though the buoyancy taught by Archimedes' principle can be obtained clearly by the surface integral of hydrostatic pressure. As a breakthrough, we dared to create the left-right asymmetric pressure field by inclining the ship with heel angle. In that state, the force and moment due to hydrostatic pressure were calculated correctly with respect to the tilted coordinate system fixed to the floating body. By doing so, we succeeded in determining the center of pressure for the shape of rectangular and arbitrary cross-sections. Then, by setting the heel angle to zero, it was proved that the center of hydrostatic pressure is equal to the well-known center of buoyancy, i.e., the centroid of the cross-sectional area under the water surface in the upright state. Furthermore, we showed an extension to the center of buoyancy for a 3-D floating body.

In Chapter 2, we develop a new theory on the derivation of the transverse metacentric radius which governs the stability of ships. As a new development in its derivation process, it was shown that the direction of movement of the center of buoyancy due to lateral inclination of ship is the direction of the half angle of the heel angle. By finding it, we were able to derive a metacentric radius worthy of its name by showing that the metacentric radius correctly represents the radius centered on the metacenter, which is the center of inclination.

In Chapters 3 and 4, theoretical treatments on the hydrostatic stability of ships are presented. As the simplest hull form, a columnar ship with rectangular cross-section, which is made of homogeneous squared timber with arbitrary breadth and arbitrary material, is chosen. In Chapter 3, the conditions under which the ship is stable in the upright state with horizontal deck are analyzed by means of ship's hydrostatics. And in Chapter 4, the stable attitude in an inclined state of the ship, which is not stable in the upright state with horizontal deck, is analyzed. By doing so, the dependence of the stable conditions and of the inclined attitude on the breadth and material of the ship will be clarified.

In Appendices, we prove that the center of hydrostatic pressure is equal to the well-known center of buoyancy for four shapes separately which are a triangular prism and a semi-submerged circular cylinder as floating bodies, and a submerged circular cylinder and an arbitrary shaped submerged body as submerged bodies.