PREFACE

Indeed, with a foot of water assumed at 72 pounds, it made a foot of sea water 86 17/77 pounds. Wine 70 1/3. Wax 68 11/14. Oil 66. Honey 104 2/3. Brick 127. Earth 95 1/2. Salt 110 1/3. Marble 252. Tin 532 1/2. Iron 576. Bronze 648. Silver 744. Lead 828. Mercury 977 1/2. Gold 1368. Sand 120. Common stone 184. Wheat 55. These figures represent the density of various substances expressed in pounds per cubic foot. Mersenne uses water as his base reference to calculate the specific gravity of other materials.

XV. From Hydraulic phenomena, it can be concluded by what means divers Urinatores: Specialized salvage divers who practiced breath-holding techniques to recover lost cargo. ought to descend more or less one hundred fathoms into the sea. They do this to fish out ships, cannons, and whatever has suffered shipwreck. These matters were briefly explained in corollary 2 of proposition 49, and will be narrated more fully in the treatise on Navigation.

XVI. Regarding page 103 of the Hydraulics and thereafter, where the properties of the Ellipse are discussed: add that any circle is to an Ellipse as the square of the circle's diameter is to the rectangle original: "rectangulum", referring to the product of the major and minor axes. of the axes, according to the 5th and 6th propositions of Archimedes on conoids and spheroids. Therefore, any circle whose diameter equals the major diameter of the Ellipse is to that Ellipse as the square of the major diameter of the Ellipse is to the rectangle formed by both diameters.

But as the square of the major is to the rectangle under both, so is the minor to the major; and as the square of the minor is to the rectangle under both, so is the minor to the major. Therefore, as the major is to the minor, so is the circle from the major diameter to the Ellipse. And as the minor is to the major, so is the circle from the minor diameter to the Ellipse. Therefore, if the area of the circle is given, the area of the Ellipse will be given, and vice versa.

As for the measurement of spheroids, any Cone is one third of a cylinder having the same base and the same height. Yet a cylinder is formed from the plane of the circular base multiplied by the height. Therefore, if the minor diameter of the spheroid is given, the area of the circle described by that diameter is given. When this is multiplied by the height of half the major diameter, a cylinder is generated. One third of this is a cone having the same height as the half-spheroid and the same base.

A geometric diagram shows an ellipse with a horizontal major axis marked A-B and a vertical minor axis marked C-D. These axes intersect at a central point E. Two points, F and G, represent the foci on the major axis. A triangle is constructed within the ellipse with its base on the vertical axis C-D and its vertex at point A. A curved line also connects points C, A, and D to illustrate the relationship between the circular and elliptical sections.

But a half-spheroid is double a cone of this kind. Therefore, the entire mass of the spheroid equals four times that cone, according to proposition 29 of Archimedes on conoids. To understand these things more easily, let the diagram from page 103 of the Hydraulics be repeated. In this diagram, let A B and C D be the diameters of the proposed Ellipse with Center E. With the interval A E, the circumference will cut A B at the foci F and G. Thus, the circle with diameter C D is to the proposed Ellipse as C D is to A B. And the circle with diameter A B is to the same as A B is to C D. Furthermore, the cone whose base is the circle with diameter C D and whose height is A E, is [one fourth] of the entire spheroid ABCD.