338

OF PLANES AND SOLIDS.

DEFINITIONS.

DEF. 88. The Common Section of two Planes, is the line in which they meet, or cut each other.

89. A Line is Perpendicular to a Plane, when it is perpendicular to every line in that plane which meets it.

90. One Plane is Perpendicular to Another, when every line of the one, which is perpendicular to the line of their common section, is perpendicular to the other.

91. The Inclination of one Plane to another, or the angle they form between them, is the angle contained by two lines, drawn from any point in the common section, and at right angles to the same, one of these lines in each plane.

92. Parallel Planes, are such as being produced ever so far both ways, will never meet, or which are every where at an equal perpendicular distance.

93. A Solid Angle, is that which is made by three or more plane angles, meeting each other in the same point.

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.

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

100. A Cube, is a square prism, being bound. ed 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 cir cumferences 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.

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 drawn 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 ac, drawn from the same point A to the plane, will be longer than the line AB.

In the plane draw the line BC, joining D the points w

BCE

Then, because the line AB is perpendicular to the plane DE, the angle B is a right angle (def. 90), and consequently greater than the angle c; 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 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

C

B

G

טן

D

touches them in all other points (def. 20); this line is therefore common to the two planes. That is, the common intersection of the two planes is a right line.

Q. E. D.

Corol. From the same point in a plane, there cannot be drawn two perpendiculars to the plane on the same side of it. For, if it were possible, each of these lines would be perpendicular to the straight line which is the common intersection of the plane and another plane passing through the two perpendiculars, which is impossible.

THEOREM XCVIII.

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.

E

B

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 impossible; 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 XCIX.

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 co be also perpendicular to the other plane AEBF.

For, draw EG perpendicular to AB. Then, because the two lines, GC, GE, are perpendicular to the common intersection

D

B

AB, the angle CGE is the angle of inclination of the two planes (def. 92). But since the two planes cut each other perpendicularly, the angle of inclination CGE is a right angle. And since the line co is perpendicular to the two lines GA, GE, in the plane AEBF, it is therefore perpendicular to that plane (th. 98). Q. E. D.

Corol. 1. Every plane, ACB, passing through a perpendicular CG to another plane AEBF, will be perpendicular to that other plane. For, if ACB be not perpendicular to the plane AEBF, some other plane on the same side of AEBF, and passing through AB, will be perpendicular to it. Then, if from the point G a straight line be drawn in this other plane perpendicular to the common intersection, it will be perpendicular to the plane AEBF. But (hyp.) CG is perpendicular to that plane. Therefore, there will be, from the same point G, two perpendicu lars to the same plane on the same side of it, which is impossible (cor. 97).

Corol. 2. If from any point & in the common intersection of the two planes ACB and AEBF perpendicular to each other, a line be drawn perpendicular to either plane, that line will be in the other plane.

THEOREM C.

Ir 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.

A

EB

F

For, join B, D, by the line BD in the plane. The plane ABD is perpendicular to the plane EF (cor. 1, th. 99); and therefore the line CD, drawn from a point in the common intersection of the two planes, perpendicular to EF, will be in the plane ABD (cor. 2, th. 99). But, because the lines AB, CD, are perpendicular to the plane EF, they are both perpendicular to the line BD in that plane, and they have been proved to be in the same plane ABD; consequently, they are parallel to each other (cor. th. 13). Q. E. D.

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