PART III.

MODERN GEOMETRY.

SECTION I.

SYMMETRY.

I. With respect to an axis.

Two points are symmetrical with respect to a line, when the line joining them is bisected at right angles by the given line; thus, A and B are symmetrical with respect to the line OP when AB is bisected at right angles by OP.

The line OP is called the axis of symmetry.

Two lines, surfaces or solids are symmetrical with respect to a line, when any point in one has a symmetrical point in the other.

The following propositions are easily deduced. The student should prove them:

1. If one of two symmetrical lines intersect the axis of

symmetry, the other intersects it in the same point.

2. They make equal angles with it.

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3. If two points of a straight line be symmetrical with two points of another, the lines are symmetrical throughout.

4. If one line be concave. or convex to the axis of symmetry, the other is also concave or convex.

5. Any diameter of a circle is an axis of symmetry.

6. A line through the opposite angles of a regular polygon of an even number of sides is an axis of symmetry.

7. A line through an angle, and the middle point of the opposite side of a regular polygon of an odd number of sides is an axis of symmetry.

8. Two symmetrical polygons are equal. Because every point of one has a symmetrical point in the other, either may be revolved about the axis until it coincides with the other; the corresponding symmetrical lines are homol

ogous.

9. Any diameter of a sphere is an axis of symmetry. 10. The axis of a cylinder or cone is an axis of symmetry. 11. A diagonal of a rhomboid is not an axis of symmetry.

NOTE. Let the student take various surfaces and solids (such as books and houses), and determine the various axes, centres and planes of symmetry.

II. With respect to a centre.

Two points are symmetrical with respect to another point when the line joining them is bisected at this point.

If AO-BO, then A and B are symmetrical with respect to 0. centre of symmetry.

O is called a

Two lines, surfaces or solids are symmetrical with respect to a centre, when any point of one has a sym

metrical point in the other.

The following propositions may be deduced:

1. If one of two symmetrical lines pass through the centre of symmetry, the other will also.

2. If two points of a straight line be symmetrical with two points of another, the

lines are symmetrical throughout, and the portions between two pairs of symmetrical points are equal.

3. A centre of a circle or of a regular polygon of an even number of sides is a centre of symmetry.

4. Two symmetrical polygons are equal; one of them may be revolved about the centre until it coincides with the other.

5. If a figure be symmetrical with respect to two axes cutting each other at right angles, it is symmetrical with respect to their intersection as a centre.

Let ABDEFG be a polygon, symmetrical with respect to OP, O'P'; it is symmetrical with respect to C.

Take any point H; join HC; produce it to K; draw KL, HL parallel to the axes. The student should fill up the proof of the following steps: KL and HL

will meet in a side of the poly

G

0'

B

M

H

E

P'

P

gon. The two triangles HMC, CNK will have HM, MC and HMC equal to CN, NK and CNK; therefore HC= CK, and

the same being true of any points, the polygon is symmetrical with respect to the centre C.

6. Two symmetrical diedral angles are equivalent. 7. Two symmetrical solid angles are equivalent.

8. Two symmetrical polyedrons are equivalent, for their faces are symmetrical, and therefore equivalent.

9. The intersection of the diagonals of a parallelopiped is a centre of symmetry.

10. The centre of a sphere is a centre of symmetry.

11. A cone or pyramid has no centre of symmetry.

III. With respect to a plane.

Two points are symmetrical with respect to a plane when the plane bisects at right angles the line joining them.

The plane is called the plane of symmetry.

Two lines, surfaces or solids are symmetrical with respect to a plane when any point in one has a symmetrical point in the other.

The following propositions may be deduced:

1. Two symmetrical finite lines are equal.

2. If one line intersect the plane of symmetry in a point, the other will in the same point, and make an equal angle with it.

3. Two symmetrical planes make equal angles with the plane of symmetry, and intersect it in the same line.

4. Symmetrical triangular pyramids are equivalent, Take the plane of the base as the plane of symmetry; hence the two pyramids will have the same base and equal altitudes.

5. Symmetrical polyedrons are equivalent.

Take two symmetrical points within

the polyedrons. Decompose the poly

edrons into triangular pyramids, of which these symmetrical points are the common vertices. The triangular pyramids will be symmetrical, two and two, and therefore equivalent. Hence the whole polyedrons will be equivalent.

6. Any plane of a great circle is a plane of symmetry of the sphere.

7. There are nine planes of symmetry of the cube.

8. If a figure have two planes of symmetry at right angles to each other, their intersection is an axis of symmetry.

9. If a figure have three planes of symmetry at right angles to each other, their intersection is a centre of symmetry.

10. How many planes of symmetry in the Capitol at Washington?

SECTION II.
LOCI.

A locus is the line or surface containing all points having a common property.

The idea of loci may best be obtained by examples:

1. What is the locus of a point in a plane at a given distance from another point?