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"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter, this perpendicular will be a mean proportional between the two parts into which it divides the diameter."

Circle With a Perpendicular Drawn to the Diameter

"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter,…

Illustration showing a conchoid, "a curve, shell-like in flexure (whence the name), invented by Nicomedes in the 2nd century B.C., and used by him for finding two mean proportionals."

Conchoid

Illustration showing a conchoid, "a curve, shell-like in flexure (whence the name), invented by Nicomedes…

"The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum."

Frustum of Cone to Find Volume

"The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones whose…

Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum of three pyramids (cones) whose common altitude is the altitude of the frustum and whose bases are respectively the upper base, the lower base, and a mean proportional between them."

Volume of Frustum of a Pyramid

Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum…

The frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the two bases of the frustum.

Triangular Pyramid Frustum

The frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude…

Right triangle inscribed in semicircle. Illustration shows that the perpendicular from any point in the circumference to the diameter of a circle is the mean proportional between the segments of the diameter.

Right Triangle Inscribed in Semicircle Shows Mean Proportional

Right triangle inscribed in semicircle. Illustration shows that the perpendicular from any point in…