Coroll. 1. An Angle at the Centre of a Circle is to four Right Angles, as an Arc on which it ftands is to the whole Circumference; for as the Angle BAC is to a Right Angle, fo is the Arc BC to a Quadrant of the Circle: Wherefore, if the Confequents be quadrupled, the Angle BAC fhall be to four Right Angles as the Arc BC is to the whole Circumference.

2. The Arcs IL, BC, of unequal Circles, which fubtend equal Angles, whether at their Centres, or Circumferences, are fimilar; for I L, is to the whole Circumference IL E, as the Angle I A L is to four Right Angles; but as I AL, or BAC, is to four Right Angles, fo is the Arc BC to the whole Circumference BCF. Therefore, as IL is to the whole Circumference ILE, fo is BC to the whole Circumference BCF; and fo the Arcs IL, BC, are fimilar.

3. Two Semidiameters AB, AC, cut off fimilar Arcs IL, BC, from concentric Circumferences.

The END of the SIXTH BOOK.

EUCLID's

ELEMEN'T S.

BOOK XI.

DEFINITION S.

1. A Solid is that which bas Length, Breadth, and Thickness.

II. The Term of a Solid is a Superficies.
III. A Right Line is perpendicular to a Plane,
when it makes Right Angles with all the
Lines that touch it, and are drawn in the faid
Plane.

IV. A Plane is perpendicular to a Plane, when all the Right Lines in one Plane, drawn at Right Angles to the common Section of the two Planes, are at Right Angles to the other Plane. V. The Inclination of a Right Line to a Plane, is the acute Angle contained under that Line, and another Right one drawn in the Plane from that End of the inclining Line which is in the Plane, to the Point where a Right Line falls from the other End of the inclining Line perpendicular to the Plane.

VI. The Inclination of a Plane to a Plane, is the

acute Angle contained under the Right Lines drawn in both the Planes to the fame Point of

their common Interfection, and making Right Angles with it.

VII. Planes are faid to be inclined fimilarly, when the faid Angles of Inclination are equal. VIII. Parallel Planes are fuch, which, being produced, never meet.

IX. Similar folid Figures are fuch, as are contained under equal Numbers of fimilar Planes. X. Equal and fimilar folid Figures are those, thai are contained under equal Numbers of fimilar and equal Planes.

XI. A folid Angle is the Inclination of more than two Right Lines, that touch one another, and are not in the fame Superficies: Or, a folid Angle is that which is contained under more than two plane Angles, which are not in the fame Superficies, but being all at one Point. XII. A Pyramid is a folid Figure comprehended under divers Planes fet upon one Plane, and put together at one Point.

XIII. A Prifm is a folid Figure contained under Planes, whereof the two oppofite are equal, fimilar, and parallel, and the other Parallelograms. XIV. A Sphere is a folid Figure, made when the Diameter of a Semicircle remaining at Rest, the Semicircle is turned about till it returns to the fame Place from whence it begun to move. XV. The Axis of a Sphere is that fixed Line, about which the Semicircle is turned.

XVI. The Centre of a Sphere is the fame with that of the Semicircle. XVII. The Diameter of a Sphere is a Right Line drawn through the Centre, and terminated on either Side by the Superficies of the Sphere. XVIII. A Cone is a Figure defcribed when one of the Sides of a Right-angled Triangle, containing the Right Angle, remaining fixed, the Triangle is turned about until it returns to the Place from whence