INTRODUCTION.

What dimensions has a solid?

What are the boundaries of a solid? Give examples. What are the dimensions of a surface? Give examples. > Give the boundaries of a surface. Illustrate.

What dimension has a line? What are its boundaries? Can we apply the word "dimension" to a point?

1. Think of a square lying horizontally. Raise it vertically. What solid is described? What surfaces do the sides of the square describe?

2. Think of a circle lying horizontally. Raise it vertically. What solid is described? What surface is described by the circumference? Revolve a rectangle about one of its sides as an axis. What solid is generated? What surfaces?

3. Imagine a semicircle revolved about its diameter. What solid is formed by this revolution? What surface is generated by the semi-circumference? What does the revolution of the diameter generate?

4. As you fill a vessel with water, what is the solid traced by the surface of the water?

>If a point move through space, what will it describe?

If a line move keeping parallel to its original position, what will it generate?

If a plane move at right angles to its original position, what will it generate?

Revolve a line about one end as a center. What surface is described by the line? What is described by the other end? Revolve a right triangle about each side in order. Describe each solid and each surface formed.

Revolve an obtuse triangle about each side in order. Describe the solids and the surfaces formed.

Can you imagine a hollow glass cube? Can you picture other hollow glass figures? Give examples.

Can you imagine the cube, were the glass cube shattered? Can you see a cube with your eyes closed? Do you see the upper surface? the lower surface? the upper front edge? the lower front edge? the other edges? the upper right front corner? the other corners?

Think of a cube bisected. What kind of surfaces bound the parts?

In how many ways can you think of bisecting a cylinder? How many ways of bisecting a sphere?

Think of other solids; conceive them bisected. What new solids and plane figures are thus formed?

7

Write short, clear definitions of solid, surface, line, and point.

If you can think of a cube apart from the material, of its sides, of its edges, and its corners, you have à geometrical concept of a cube. If you can think of a surface apart from the solid which it bounds, you have a geometrical concept of a surface. If you can think of a line apart from the surface which it bounds, you have a geometrical concept of a line.

If you can think of a point apart from the extremities of a line or the intersection of two or more lines, you have a notion of the geometrical point.

Can you conceive of a cylinder, its boundaries, its surfaces?
Can you form geometrical concepts of other solids?

POSTULATES.

Can any two points in the same plane be joined by a straight line? Can you think of any two points not in the same plane? Can you think of any three points not in the same plane?

Is it self-evident that any straight line may be produced to any length in either direction?

May a circle be drawn with any point as a center and with any finite straight line as a radius?

Can a figure be moved unaltered to a new position?

Is it possible to think of two equal geometric cubes being so placed that they will coincide? Can you state five postulates?

AXIOMS.

What is an axiom?

Give conclusion in the following examples and state the axiom applicable.

1. Tom and John are each the same age as I; therefore— 2. I have the same amount of money as Brown or Smith; therefore

3. AB and C B; therefore

=

4. Brown has as much money as Smith, and Jones as Robinson; .. Brown and Jones together have—

AB and C=D; .. A+C=?

5. Two armies are equal in number, each loses 500 men in battle; consequently—

6. Brown and Smith are each double the height of the dwarf Jones; .'.

7. M is 9 times N, R is also 9 times N; ..

> 10. My whole hand is larger than my thumb. State axiom.

11. One-third of an apple is less than the whole of it. State axiom.

12. I am older than you. In 5 years I shall still be older. State axiom.

13. Smith has less money than Jones, each spends $5. Draw conclusion and state axiom.

14. If, when a sheet of paper is placed on another, their edges exactly coincide- State conclusion and axiom.

15. If, when one line is placed on another, their extremities coincide and every point in the first line coincides with a corresponding point in the second line, the lines are equal. State axiom.

16. Is it possible for the extremities of two straight lines to coincide when the other parts of the lines do not coincide? 17. Can you state the axiom which your answer suggests? 18. How does the carpenter get a straight line between two points without using a straight edge? What is the shortest distance between any two points? Do you think your answer self-evident?

19. How many points are necessary to determine the direction of a straight line? How many straight lines can be drawn between the same two points?

20. In how many points can two straight lines intersect? Why? Can you give an axiom for your answer?

DEFINITIONS.

It is of vital importance that the pupil shall be able to give clear, exact definitions to all terms used in Geometry. Of course, he must fully understand, and be prepared to illustrate any definition given. The questions previously given were designed to lead the pupil so far as possible to formulate his own definitions. But many of the terms in Geometry are diffi

cult to define, and the pupil can compare his definitions with those here given.

Upon these definitions and upon the axioms and postulates rest the demonstrations of the truths of Geometry. But do not mistake the mere learning of these truths to be the object of the study. It is the ability to reason which we acquire-their demonstration.

1.

A solid is a limited portion of space. Its dimensions are length, breadth, and thickness,

The pupil can conceive space to be divided into a multitude of forms or shapes. Each form pictured is a solid. The geometrical solid contains no matter; it is the limited portion of space conceived by the mind.

2.

A surface is the boundary of a solid. It divides space into parts and can be conceived without the solid, so the definition is often given: A surface is that which has length and breadth without thickness.

(1) A plane surface or plane is a surface in which if any two points are joined by a straight line, every point in the line will lie in the surface.

(2) A surface, no part of which is plane, is called a curved surface.

3.

A line is the boundary of a surface. We can conceive the line without the surface and define it to be that which has length, but neither breadth nor thickness.

(1) A straight line has the same direction throughout its entire length.