94. Similar Solids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, alike placed.

95. A Prism, is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms.

96. A Prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c.

97. A Right or Upright Prism, is that which has the planes of the sides perpendicular to the planes of the ends or base.

98. A Parallelopiped, or Parallelopipedon, is

a prism bounded by six parallelograms, every opposite two of which are equal, alike, and parallel.

99. A Rectangular Parallelopipedon, is that whose bounding planes are all rectangles, which are perpendicular to each other.

100. A Cube, is a square prism, being bounded by six equal square sides or faces, and are perpendicular to each other.

101. A Cylinder, is a round prism, having circles for its ends; and is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis.

102. The Axis of a Cylinder, is the right line joining the centres of the two parallel circles, about which the figure is described.

103 A Pyramid, is a solid, whose base is any right-lined plane figure, and its sides triangles, having all their vertices meeting together in a point above the base, called the Vertex of the pyramid.

104. A pyramid, like the prism, takes particular names from the figure of the base.

105. A Cone, is a round pyramid, having a circular base, and is conceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle.

VOL. I.

Xx

106. The

106. The Axis of a cone, is the right line, joining the vertex, or fixed point, and the centre of the circle about which the figure is described.

107 Similar Cones and Cylinders, are such as have their altitudes and the diameters of their bases proportional.

108. A Sphere, is a solid bounded by one curve surface, which is every where equally distant from a certain point within, called the Centre. It is conceived to be generated by the rotation of a semicircle about its diameter, which remains fixed.

109. The Axis of a Sphere, is the right line about which the semicircle revolves; and the centre is the same as that of the revolving semicircle.

110. The Diameter of a Sphere, is any right line passing through the centre, and terminated both ways by the surface.

111. The Altitude of a Solid, is the perpendicular draw from the vertex to the opposite side or base.

THEOREM XCV.

A Perpendicular is the Shortest Line which can be drawn from any Point to a Plane.

LET AB be perpendicular to the plane DE; then any other line, as xc, drawn from the same point A to the plane, will be longer than the line AB.

In the plane draw the line Bc, joining the points B, C.

D

BCE

Then, because the line AB is perpendicular to the plane DE, the angle в is a right angle (def. 89), and consequently greater than the angle e; therefore the line AB, opposite to the less angle, is less than any other line AC, opposite the greater angle (th. 21). Q. E. D.

THEOREM XCVI.

A Perpendicular Measures the Distance of any Point from a Plane.

THE distance of one point from another is measured by a right line joining them, because this is the shortest line which can be drawn from one point to another. So, also, the distance from a point to a line, is measured by a perpendicular, because this line is the shortest which can be drawn

from

from the point to the line. In like manner, the distance from a point to a plane, must be measured by a perpendicular drawn from that point to the plane, because this is the shortest line which can be drawn from the point to the plane.

THEOREM XCVII

The common Section of Two Planes, is a Right Line.

LET ACBDA, AEBFA, be two planes cutting each other. and A, B, two points in which the two planes meet: drawing the line AB, this line will be the common intersection of the two planes.

For because the right line AB touches the two planes in the points A and B, it touches them in all other points (def.

C

D

20): this line is therefore common to the two planes.

B

That

is, the common intersection of the two planes is a right line.

THEOREM XCVIII.

Q. E. D.

If a Line be Perpendicular to two other Lines, at their Common Point of Meeting; it will be Perpendicular to the Plane of those Lines.

LET the line AB make right angles with the lines AC, AD; then will it be perpendicular to the plane CDE which passes through these lines.

B

E

A

If the line AB were not perpendicular to the plane CDE, another plane might pass through the point A, to which the line AB would be perpendicular. But this is imposible; for, since the angles BAC, BAD, are right angles, this other plane must pass through the points c, D. Hence, this plane passing through the two points A, C of the line AC, and through the two points A, D, of the line AD, it, will pass through both these two lines, and therefore be the same plane with the former.

Q. E D.

THEOREM

THEOREM XCIX.

If Two Lines be Perpendicular to the Same Plane, they will be Parallel to each other.

LET the two lines AB, CD, be both perpendicular to the same plane EBDF; then will AB be parallel to CD.

For, join B, D, by the line ed in the plane. Then, because the lines AB, CD, are perpendicular to the plain EF, they

EB

C

are both perpendicular to the line BD (def. 89) in that plane; and consequently they are parallel to each other (corol. th. 13). QE D*

Corol. If two lines be parallel, and if one of them be perpendicular to any piane, the other will also be perpendicular to the same plane.

THEOREM C.

If Two planes Cut each other at Right Angles, and a Line be drawn in one of the Planes Perpendicular to their Common Intersection, it will be Perpendicular to the other Plane.

LET the two planes ACBD, AEBF, Cut each other at right angles; and the line CG be perpendicular to their common section AB; then will CG be also perpendicular to the other plane AEBF.

D

F

B

For, draw EG perpendicular to AB. Then because the two lines GC, GE, are perpendicular to the common intersection AB, the angle CGE is the angle of inclination of the two planes (def. 91). But since the two planes cut each other perpendicularly, the angle of inclination CGE is a right angle. And since the line CG is perpendicular to the two lines GA, GE, in the plane AEBF, it is therefore perpendicular to that plane (th. 98).

Q. E. D.

* This demonstration of Theorem xcix. does not appear to me to be conclusive. EDITOR.

THEOREM

THEOREM CI.

If one Plane Meet another Plane, it will make Angles with that other Plane, which are together equal to two Right Angles.

LET the plane ACBC meet the plane AEBF; these planes make with each other two angles whose sum is equal to two right angles.

For, through any point G, in the common section AB, draw CD, EF, perpendicular to AB. Then, the line CG makes with EF two angles together equal to two right angles. But these two angles are (by def. 91) the angles of inclination of the two planes. Therefore the two planes make angles with each other, which are together equal to two right angles.

Corol. In like manner it may be demonstrated, that planes which intersect, have their vertical or opposite angles equal; also, that parallel planes have their alternate angles equal; and so on, as in parallel lines.

THEOREM CII.

If Two Planes be Parallel to each other; a Line which is Perpendicular to one of the Planes, will also be Perpendicular to the other.

LET the two planes CD, EF, be parallel, and let the line AB be perpendicular to the plane CD; then shall it also be perpendicular to the other plane EF.

For, from any point G, in the plane EF, draw GH perpendicular to the plane CD, and draw AM, BG.

EB G

A H

Then, because BA, GH, are both perpendicular to the plane CD, the angles A and н are both right angles. And because the planes CD, EF, are parallel, the perpendiculars BA, GH, are equal (def. 92). Hence it follows that the lines BG, AH, are parallel (def. 9). And the line AB being perpendicular to the line AH, is also perpendicular to the parallel line BG (cor. th. 12).

In like manner it is proved, that the line AB is perpendicular to all other lines which can be drawn from the point B

in