Page images
PDF
EPUB

GEOMETRY OF SPACE.

BOOK VI.

THE PLANE. POLYEDRAL ANGLES.

1. DEFINITION. A plane has already been defined as a surface such that the straight line joining any two points in it lies wholly in the surface.

Thus, the surface MN is a plane, if, A and B being any two points in it, the straight line AB lies wholly in the surface.

The plane is understood to be indefinite in

M

Α

B

extent, so that, however far the straight line is produced, all its points lie in the plane. But to represent a plane in a diagram, we are obliged to take a limited portion of it, and we usually represent it by a parallelogram supposed to lie in the plane.

DETERMINATION OF A PLANE.

PROPOSITION I.-THEOREM.

2. Through any given straight line an infinite number of planes may be passed.

Let AB be a given straight line.

straight line may be drawn in any plane, and the position of that plane may be

changed until the line drawn in it is

B

brought into coincidence with AB. We shall then have one plane

کا

passed through AB; and this plane may be turned upon AB as an axis and made to occupy an infinite number of positions.

3. Scholium. Hence, a plane subjected to the single condition that it shall pass through a given straight line, is not fixed, or determinate, in position. But it will become determinate if it is required to pass through an additional point, or line, as shown in the next proposition.

A plane is said to be determined by given lines, or points, when it is the only plane which contains such lines or points.

PROPOSITION II.-THEOREM.

4. A plane is determined, 1st, by a straight line and a point without that line; 2d, by two intersecting straight lines; 3d, by three points not in the same straight line; 4th, by two parallel straight lines.

M

1st. A plane MN being passed through a given straight line AB, and then turned upon this line as an axis until it contains a given point C not in the line AB, is evidently determined; for, if it is then turned in either direction about AB, it will cease to contain the point C. The plane is therefore determined by the given straight line and the point without it.

B

2d. If two intersecting straight lines AB, AC, are given, a plane passed through AB and any point C (other than the point A) of AC, contains the two straight lines, and is determined by these lines.

3d. If three points are given, A, B, C, not in the same straight line, any two of them may be joined by a straight line, and then the plane passed through this line and the third point, contains the three points, and is thus determined by them.

C

B

D

E

4th. Two parallel lines, AB, CD, are by A definition (I. 42) necessarily in the same plane, and there is but one plane containing them, since a plane passed through one of them, AB, and any point E of the other, is determined in position. 5. Corollary. The intersection of two planes is a straight line. For, the intersection cannot contain three points not in the same straight line, since only one plane can contain three such points.

PERPENDICULARS AND OBLIQUE LINES TO PLANES.

6. Definition. A straight line is perpendicular to a plane when it is perpendicular to every straight line drawn in the plane through its foot, that is, through the point in which it meets the plane.

In the same case, the plane is said to be perpendicular to the line,

PROPOSITION III.-THEOREM.

7. From a given point without a plane, one perpendicular to the plane can be drawn, and but one.

A

M

E

B

P

F

B'

Let A be the given point, and MN the plane. If any straight line, as AB, is drawn from A to a point B of the plane, and the point B is then supposed to move in the plane, the length of AB will vary. Thus, if B move along a straight line BB' in the plane, the distance AB will vary according to the distance of B from the foot C of the perpendicular AC let fall from A upon BB'. Now, of all the lines drawn from A to points in the plane, there must be one minimum, or shortest line. There cannot be two equal shortest lines; for if AB and AB' are two equal straight lines from A to the plane, each is greater than the perpendicular AC let fall from A upon BB'; hence they are not minimum lines. There is therefore one, and but one, minimum line from A to the plane. Let AP be that minimum line; then, AP is perpendicular to any straight line EF drawn in the plane through its foot P. For, in the plane of the lines AP and EF, AP is the shortest line that can be drawn from A to any point in EF, since it is the shortest line that can be drawn from A to any point in the plane MN; therefore, AP is perpendicular to EF (I. 28). Thus AP is perpendicular to any, that is, to every, straight line drawn in the plane through its foot, and is therefore perpendicular to the plane. Moreover, by the nature of the proof just given, AP is the only perpendicular that can be drawn from A to the plane MN.

8. Corollary. At a given point P in a plane MN, a perpendicular can be erected to the plane, and but one.

For, let M'N' be any other plane, A' any point without it, and

A'P' the perpendicular from A' to

M

A B

P C

M'

[ocr errors]
[ocr errors]

N'

this plane. Suppose the plane M'N' to be applied to the plane MN with the point P' upon P, and let AP be the position then occupied by the perpendicular A'P'. We then have one perpendicular, AP, to the plane MN, erected at P. There can be no other: for let PB be any other straight line drawn through P; let the plane determined by the two lines PA, PB, intersect the plane MN in the line PC; then, since APC is a right angle, BPC is not a right angle, and therefore BP is not perpendicular to the plane.

9. Scholium. By the distance of a point from a plane is meant the shortest distance; hence it is the perpendicular distance from the point to the plane.

PROPOSITION IV.-THEOREM.

10. Oblique lines drawn from a point to a plane, at equal distances from the perpendicular, are equal; and of two oblique lines unequally distant from the perpendicular the more remote is the greater.

1st. Let AB, AC be oblique lines from the point A to the plane MN, meeting the plane at the equal distances PB, PC, from the foot of the perpendicular AP; then, AB AC. For, the right triangles APB, APC, are equal (I. 76).

2d. Let AD meet the plane at a distance, PD, from P, greater than PC; then, AD> AC. For, upon PD take PB PC, and join AB: then AD> AB (I. 35); but AB AC; therefore, AD > AC.

=

=

M

D

B

A

E

N

11. Corollary I. Conversely, equal oblique lines from a point to a plane meet the plane at equal distances from the perpendicular; and of two unequal oblique lines, the greater meets the plane at the greater distance from the perpendicular.

12. Corollary II. Equal straight lines from a point to a plane meet the plane in the circumference of a circle whose centre is the foot of the perpendicular from the point to the plane. Hence we derive a method of drawing a perpendicular from a given point A to a given plane MN: find any three points, B, C, E, in the plane, equidistant from A, and find the centre P of the circle passing through these points; the straight line AP will be the required perpendicular.

PROPOSITION V.-THEOREM.

13. If a straight line is perpendicular to each of two straight lines at their point of intersection, it is perpendicular to the plane of those lines.. Let AP be perpendicular to PB and PC, at their intersection P; then, AP is perpendicular to the plane MN which contains those lines.

For, let PD be any other straight line drawn through P in the plane MN. Draw any straight line BDC intersecting PB, PC PD, in B, C, D; produce AP to A' making PA' PA, and join A and A' to each of the points B, C, D.

=

A

2

M

[ocr errors]

Σαμ

N

[ocr errors]

A'

Since BP is perpendicular to AA', at its middle point, we have BA - BA', and for a like reason CA — CA'; therefore, the triangles ABC, A'BC, are equal (I. 80). If, then, the triangle ABC is turned about its base BC until its plane coincides with that of the triangle A'BC, the vertex A will fall upon A'; and as the point D remains fixed, the line AD will coincide with A'D; therefore, D and P are each equally distant from the extremities of AA', and DP is perpendicular to AA' or AP (I. 41). Hence AP is perpendicular to any line PD, that is, to every line, passing through its foot in the plane MN, and is consequently perpendicular to the plane.

14. Corollary I. At a given point P of a straight line AP, a plane can be passed perpendicular to that line, and but one. For, two perpendiculars, PB, PC, being drawn to AP in any two different planes APB, APC, passed through AP, the plane of the lines PB, PC, will

« PreviousContinue »