The Elements of Hydrostatics (1586) - Page 23

VI DEMONSTRATION. VII PROPOSITION.

BEING two waters of unequal weight, and a solid body lighter than any of them: As the weight of the heaviest water, to the weight of the lightest, so the size of this solid body within the water placed in the lightest water, to its size within the water placed in the heaviest.

GIVEN. Let A B be a water, heavier than the water C D, and E F be a solid body lighter than any of those two waters, which first placed in the water A B, so that there is under the water the part G F, but the same body E F placed in the water C D, which there is H I, so sinks under the water the part K I. REQUIRED. We must prove that as the weight of the water A B, to the weight of the water C D, so the size K I, to G F.

PROOF. The water of the water A B equal in size to G F, is equal in weight to the body E F, and the water of the water C D equal in size to K I, is equal in weight to the body H I by the 5th proposition, but the body E F or H I is all one same body by the given, therefore the water of the water A B equal in size to G F, is equal in weight to the water of the water C D equal in size to K I; But being two equal-weighted waters, as their size to size, so changes their weight to weight, as necessarily follows from the admitted 5th definition, therefore as the weight of the water A B, to the weight of the water C D, so the size K I, to the size G F.

A geometric diagram illustrates a principle of hydrostatics. Two rectangular containers, labeled A B on the left and C D on the right, are filled with water. Inside each container is a rectangular body. In the left vessel, the body is labeled E (top), G (at the water level), and F (bottom submerged part). In the right vessel, the body is labeled H (top), K (at the water level), and I (bottom submerged part).

CONCLUSION. Being then two waters of unequal weight and a solid body, &c.

VII DEMONSTRATION. VIII PROPOSITION.

EVERY solid body's weight is so much lighter in the water than in the air, as the weight of the water equal in size to it.

GIVEN. Let A be a solid body, and B C a water.