G F, and that G F is equal in size to A, so G F is heavier than A. Let us now pour out the water G F that is in the surface vessel E F, and lay in it the body A, which will fill that place exactly, since A by the preparation is identical and equal in size to G F; But as said before, the body A is lighter than the poured-out water; The surface vessel E F will then not sink as deep from A as it did from the water G F, by the 3rd postulate; But by as much as the surface vessel E F sinks shallower, by so much must the body A necessarily stick out above the water. CONCLUSION. A solid body then, lighter than water, does not sink entirely underneath, but a part remains sticking out; which we had to prove.

III. DEMONSTRATION.          III. PROPOSITION.

A SOLID BODY HEAVIER THAN WATER SINKS TO THE BOTTOM.

GIVEN. Let A be a solid body heavier than the water B C, whose surface is B D, and bottom is E C.

REQUIRED. We must prove that A, placed in the water B C, will sink to the bottom E C. PREPARATION. Let F G be a surface vessel filled with water, equal in size and identical to A, whose surface F H is in the surface B D.

A geometric diagram shows a container of water with the bottom labeled E and C. The water surface is marked with B and D. Inside the water, a rectangular body is labeled I. Above the surface, or indicating specific points, are the letters F, G, and H. To the right of the container, a separate rectangular body is labeled A.

PROOF. Since A is heavier than the water F G as stated, and F G is equal in size to A, so A is heavier than F G. Let us now pour out the water F G that is in the surface vessel F G, and lay in it the body A, which will fill that place exactly, since A by the stated is identical and equal in size to F G; But as we have said before, A is heavier than the poured-out water; The surface vessel F G will then sink deeper from A than it did from the water F G by the 3rd postulate. We have then shown that the body A will sink. There remains to be proven that it will also sink to the bottom E C, as follows: If (if it were possible) it did not sink to E C, but remained on the path between the two as where I is, and let us remove the solid body that is in the vessel I, and fill it with water, the same will remain in that place by the 1st proposition: But this water is lighter than that body, a heavier one and a lighter one would remain in the same place, which is impossible and contrary to the 3rd postulate. The body A, then, cannot remain between the surface B D and the bottom E C, it must then necessarily sink until it is at the bottom