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EUCLI D's

ELEMENTS.

BOOK VII.

DEFINITION S.

1. A Solid is that which has 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 perpenpendicular 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

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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, that 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 felid Figure comprehended under divers Planes fet upon one Plane, and put together at one Point.

XIII. A Prism 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 Reft, the Semicircle is turned about till it returns to the ene Place from whence it began 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 a 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

whence it first began to move. And if the fixed Right Lines be equal to each other, that contain the Right Angle, then the Cone is a rectangular Cone; but, if it be lefs, it is an obtufe-angled Cone; if greater, an acute-angled Cone. XIX. The Axis of a Cone is that fixed Right Line, about which the Triangle is moved. XX. The Base of a Cone is the Circle described by the Right Line moved about.

XXI. A Cylinder is a Figure defcribed by the Motion of a Right-angled Parallelogram, one of the Sides containing the Right Angle, remaining fixed while the Parallelogram is turned about to the fame Place from whence it began to be moved. XXII. The Axis of a Cylinder is that fixed. Right Line, about which the Parallelogram is turned. XXIII. And the Bafes of a Cylinder are the Circles that are defcribed by the Motion of the two oppofite Sides of the Parallelogram.

XXIV. Similar Cones and Cylinders are fuch, whofe Axes and Diameters of their Bafes are proportional.

XXV. A Cube is a folid Figure contained under fix equal Squares.

XXVI. A Tetrahedron is a folid Figure contained under four equilateral Triangles.

XXVII. An Octahedron is a folid Figure con tained under eight equal equilateral Triangles. XXVIII. A Dodecahedron is a folid Figure con'tained under twelve equal equilateral and equiangular Pentagons.

XXIX. An Icojahedron is a folid Figure contained under twenty equal equilateral Triangles.

XXX. A Parallelopipedon is a Figure contained under fix quadrilateral Figures, whereof thofe which are oppofite are parallel...

PRO

PROPOSITION I

THEOREM.

One Part of a Right Line cannot be in a plane
Superficies, and another Part above it.

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OR, if poffible, let the Part A B of the Right
Line ABC be in a plane Superficies, and the
Part B C above the fame.

There will be fome Right Line in the aforefaid
Plane, which, with A B, will be but one ftrait Line.
Let this Line be D B.

Then the two given Right Lines A BC, AB D, have one common Segment A B, which is impoffible; for one Right Line will not meet another in more Points than one. Wherefore, one Part of a Right Line cannot be in a plane Superficies, and another Part above it; which was to be demonstrated.

PROPOSITION II.
THEOREM.

If two Right Lines cut each other, they are both
in one Plane; and every Triangle is in one
Plane.

L ET two Right Lines A B, CD, cut each other in the Point E. I say, they are both in one Plane; and every Triangle is one Plane.

For, take any Points F and G, in the Right Lines A B, CD; and join C B, F G; and let there be drawn FH, GK. In the firft Place, I fay, the Triangle E B C is in one Plane.

For, if one Part F HC, or G B K, of the Triangle EBC, be in one Plane, and the other Part in another Plane; then one Part of each of the Lines EC, EB, fhall be in one Plane, and the other Part in another Plane; which we have proved to be abfurd. Therefore the Triangle EBC is one Plane; but both the Right Lines EC, EB, are in the fame Plane as the Triangle

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Triangle B C E is; and A B, C D, are both in the fame Plane as EC, E B, are. Wherefore, the Right Lines A B, CD, are both in one Plane; and every Triangle is in one Plane; which was to be demonftrated.

PROPOSITION II.
THEOREM.

If two Planes cut each other, their common Sec-
tion will be a Right Line.

ET two Planes A B, B C, cut each other, whofe common Section is the Line DB; I fay, DB is a Right Line.

For, if it be not, draw the Right Line D E B in the Plane A B, from the Point D to the Point B, and the Right Line D F B in the Plane B C..

Then two Right Lines DE B, DFB, have the * Ax. 1o. fame Terms, and include a Space, which is * abfurd. Therefore DE B, DF B, are not Right Lines. In the fame manner we demonftrate, that no other Line drawn from the Point D to the Point B, is a Right Line, befides D B, the common Section of the Planes A B, BC. If, therefore, two Planes cut each other, their common Section will be a Right Line; which was to be demonftrated.

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PROPOSITION IV.
THEOREM.

If to two Right Lines, cutting one another, a third
ftands at Right Angles in the common Section,
it fhall be alfo at Right Angles to the Plane
drawn through the faid Lines.

L ET the Right Line E F ftand at Right Angles to the two Right Lines AB, CD, in the common 'Section E. I fay, E F is alfo at Right Angles to the Plane drawn through A B, CD.

For,

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