Of the Circle.

37. The area of a circle is equal to the product of half the radius, into the circumference. Thus, the area of the circle, whose centre is C, is equal to half the radius CA, multiplied by the circumference: that is,

B

area CA× circumference ABD.

A

38. The areas of circles are to each other as the squares described on their diameters: that is, the areas of two circles are to each

other as the squares

A

B C

described on the di

ametres AB and CD.

D

B

39. If from any point of the circumference of a circle, a line be drawn perpendicular to a diameter, the square of the perpendicular will be equal to the rectangle of the segments of the diameter. That is, if DB be perpendicular to AC, we have

DB2ABX BC.

A

QUEST.-37. What is the area of a circle equal to? If the diameter is 1, and circumference 3,1416, what is the area? 38. How are the areas of two circles to each other? If the diameters are 2 and 4, in what proportion are the areas? 39. What is the square of a perpendicular to a diameter equal to ?

Of Planes.

SECTION IX.

OF PLANES AND THEIR ANGLES.

1. A plane has been defined ( I. Art. 6), to be a surface which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide.

2. The common intersection of two planes is the line in which they cut each other. This line is always a straight line.

3. Two planes which intersect each other form an angle. This angle is measured by two lines, one in each plane, and both perpendicular to the common intersection at the same point.

Let AB be a plane coinciding with the plane of the paper, and ECF a plane intersecting it in the line FC. Now, if from any point of the common intersection A as C, we draw CD in the plane E

D

F

C

QUEST.-1. What is a plane? If a straight line be laid upon a plane, will it exactly coincide with the plane? 2. What is the line of intersection of two planes? Is this line straight or curved ? 3. May two planes form an angle? How is this angle measured?

Of Planes.

AB, and CE in the plane ECF, and both perpendicular to CF at C, then will the angle DCE measure the inclination between the two planes.

It should be remembered that the line EC is directly over the line CD.

4. If the angle ECD is a right angle, the planes will be perpendicular to each other.

5. A line is said to be perpendicular to a plane, when it inclines no more to one side than to the other. The point at which the perpendicular meets the plane is called the foot of the perpendicular.

B

P

6. When a line is perpendicular to a plane, it will be perpendicular to every line of the plane which it meets. Thus, if AP is perpendicular to the plane BC at the point A, it will be perpendicular to the lines AD, AB and AC, drawn in the plane. It should be remembered that the point P is directly over the point A.

D

QUEST.-4. If the angle formed by the measuring lines is a right angle, what angle will the planes make with each other? 5. When is a line said to be perpendicular to a plane? What is the point called at which the perpendicular meets the plane? 6. If a line is perpendicular to a plane, will it be perpendicular to other lines of the plane which it meets?

Of Planes.

7. If several lines are parallel to each other, and one of them is perpendicular to a plane, all the others will likewise be perpendicular to the plane.

D

P

8. A perpendicular is the shortest distance from a point to a plane. If oblique lines be drawn to the plane, those nearest the perpendicular will be less than those more remote. Thus, if PA be perpendicular to the plane DC, it will be the shortest line from P to the plane; and the line PF

F

near PA, will be less than PC, which is more remote.

9. Two planes are said to be parallel when they are

at the same perpendicular distance from each other. Par

allel planes will never meet each other, how far soever they may be produced. If the two planes AB and CD are paral

B

E G

A

D

F

H

QUEST.-7. If several lines are parallel to each other, and one of them is perpendicular to a plane, what position will the other lines also have with the plane? 8. What is the shortest line which can be drawn from a point to a plane? If a perpendicular and several oblique lines be drawn, which of the oblique lines will be the least? 9. When are two planes said to be parallel? Will parallel planes meet each other if produced? If two or more lines be drawn perpendicular to two parallel planes, will they be equal or unequal?

Of Planes.

lel, the lines EF and GH drawn perpendicular to them

will be equal, and parallel.

10. If two parallel planes are cut by a third plane, the lines of intersection will be parallel. Thus, if the two parallel planes AB and CD be cut by the plane FHGE, the lines of intersection FH and EG will be parallel.

11. A Solid Angle is the angular space included between several planes which meet at a point.

Thus, the solid angle S, is

formed by the union of the planes

ASB, BSC, CSD, and DSA.

A

B

Three planes, at least, are requi

site to form a solid angle.

QUEST.-10. If two parallel planes are cut by a third plane, what position will the lines of intersection have with each other? 11. What is a solid angle? What is the least number of planes which can form a sclid angle?