The Elements of Hydrostatics

15

ground E C rests. CONCLUSION. A solid body then, heavier than water, sinks to the bottom, which we had to prove.

IIII DEMONSTRATION. #### IIII PROPOSITION.

A SOLID BODY EQUAL IN WEIGHT TO WATER, HOLDS EVERY PLACE THAT ONE GIVES IT WITHIN WATER.

GIVEN. Let A be a solid body, equal in weight to the water B C. REQUIRED. We must prove that A, placed in the water B C, holds every place that one gives it there.

PREPARATION. Let D be a surface vessel full of water, equal in size and identical to A. PROOF. Since A is equal in weight by the given to the water D, and that D is equal in size to A, so D is also equal in weight to A; Let us now pour out the water D that is in the surface vessel D, 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 D; The surface vessel D will then not sink deeper nor rise higher from A than from the water D, by the 3rd postulate; But the water D held every place in B C that one gave it by the 1st proposition, the solid body A then, holds every place in the water B C that one gives it.

A geometric diagram shows a large rectangle with a 'B' at the top left and a 'C' at the bottom right. Inside the rectangle are two smaller squares, the left one labeled 'D' and the right one labeled 'A', representing the displacement of the solid body.

CONCLUSION. A solid body then equal in weight to water, holds every place that one gives it within water, which we had to prove.

V DEMONSTRATION. #### V PROPOSITION.

A SOLID BODY LIGHTER THAN WATER THAT LIES THEREIN, IS EQUAL IN WEIGHT TO THE WATER EQUAL IN SIZE TO ITS PART THAT IS WITHIN THE WATER.

GIVEN. Let A B be a solid body lighter than the water C D where it lies, and its surface vessel be A B, and its part within the water be E B. REQUIRED. We must prove that the solid body A B, is equal in weight to the water that is equal in size to the part E B that is within the water C D. PROOF. Let us draw the solid body A B out of the surface vessel A B, and fill the surface vessel again with water, until it has sunk as deep as it was first with the body. Which being so, the water E B that is in the surface vessel A B, will (because the surface of all water of the surface vessel with a part outside the water stands, is always in the surface of the surrounding water, since the surface vessel does not weigh) be equal in weight to the given body A B, Reason, that two weights that cause a vessel to sink equally deep are also equal in weight, by the 3rd postulate.