11. A Solution is the process of performing a problem or an example. It should usually be accompanied by a demonstration of the process.

12. A Scholium is a remark made at the close of a discussion, and designed to call attention to some particular feature or features of it.

ILL.—Thus, after having discussed the subject of multiplication and division in Arithmetic, the remark that "Division is the converse of multiplication," is a scholium.

13. A Point is a place without size. Points are designated by letters.

'F

E

ILL.-If we wish to designate any particular point (place) on the paper, we put a letter by it, and sometimes a dot on it. Thus, in Fig. 3, the ends of the line, which are points, are designated as "point A," "point D;" or, simply, as A and D. The points marked on the line are designated as "point B," "point C," or as B and C. F and E are two points above the line.

A

B

FIG. 3.

* A concept is a thing thought about:-a thought-object. Thus, in Arithmetic, number is the concept; in Botany, plants; in Geometry, as will appear in this section, points, lines, and solids. These may also be said to constitute the subject-matter of the science.

LINES.

14. A Line is the path of a point in motion. Lines are represented upon paper by marks made with a pen or pencil, the point of the pen or pencil representing the moving point. A line is designated by naming the letters written at its extremities, or somewhere upon it.

ILL.-In each case in Fig. 4, conceive a point to start from A and move along

the path indicated by the mark to B. The path thus traced is a line. Since a true point has no size, a line has no breadth, though the marks by which we represent lines have some breadth. The first and third lines in the figure are each designated as the line AB." The second line is considered as traced by a point starting from A and coming around to A again, so that B and A coincide. This line may be designated as the line AmnA, or AmnB. In the fourth case, there are three lines represented, which are designated, respectively, as AmB, AnB, and AcB; or, the last, as AB.

15. Lines are of Two Kinds, Straight and Curved. A straight line is also called a Right Line. A curved line is often called simply a Curve.

16. A Straight Line* is a line traced by a point which moves constantly in the same direction.

17. A Curved Line is a line traced by a point which constantly changes its direction of motion.

ILL'S. Thus in 1, Fig. 4, if the line AB is conceived as traced by a point moving from A to B, it is evident that this point moves in the same direction throughout its course; hence AB is a straight line. If a body, as a stone, be let fall, it moves constantly toward the centre of the earth; hence its path represents a straight line. If a weight be suspended by a string, the string represents a straight line. Considering the line represented by AiB, Fig. 4, as the path of a point moving from A to B, we see that the direction of motion is constantly changing. For example, if this were a line traced on a map, we

The word "line" used alone signifies "straight line."

would say, that, starting from A, the point begins to move nearly north, but keeps changing its direction more and more toward the east, until at 3 it moves directly cast; and from 3 it continues to change its course and moves more and more toward the south, till at i it is moving directly south. The same general truth is illustrated in 2 and 4, Fig. 4. The path of a ball thrown into the air, in any direction except directly up, represents a curved line. Most of the lines seen in nature are curved, as the edges of leaves, the shore of a river or lake, etc. Sometimes a path like that represented in Fig. 5 is called, though improperly, a Broken Line. It is not a line

at all; that is, not one line: it is a series of straight lines.

FIG. 5.

SURFACES.

18. A Surface is the path of a line in motion.*

19. Surfaces are of Two Kinds, Plane and Curved.

20. A Plane Surface, or simply a Plane, is a surface with which a straight line may be made to coincide in any direction. Such a surface may always be conceived as the path of a straight line in motion.

21. A Curved Surface is a surface in which, if lines are conceived to be drawn in all directions, some or all of them will be curved lines.

B

ILL'S-Let AB, Fig. 6, be supposed to move to the right, so that its extremities A and B move at the same rate and in the same direction, A tracing the line AD, and B, the line BC. The path of the line, the figure ABCD, is a surface. This page is a surface, and may be conceived as the path of a line sliding like a ruler from top to bottom of it, or from one side to the other. Such a path will have length and breadth, being in the latter respect unlike a line, which has only length.

A

FIG. 6.

As a second illustration, suppose a fine wire bent into the form of the curve AmB, Fig. 7, and its ends A and B stuck into a rod, XY. Now, taking the rod XY in the fingers and rolling it, it is evident that the path of the line represented by the wire AmB, will be the surface of a ball (sphere).

* Should it be said that irregular surfaces are not included in this definition, the sufficient reply is, that such surfaces are not subjects of Geometrical investigation, except approxi mately, by means of regular surfaces.

Again, suppose the rod XY be placed on the surface of this paper so

m

X A

FIG. 7.

BY

that the wire AmB shall stand straight up from the paper, just as it would be if we could take hold of the curve at m and raise it right up, letting XY lie as it does in the figure. Now slide the rod straight up or down the page, making both ends move at the same rate. The path of AmB will be like the surface of a half-round rod (a semi-cylinder).

Thus we see how surfaces plane and curved may be conceived as the paths of lines in motion.

Ex. 3. If you fasten one end of a cord at a point in the ceiling and hang a ball on the other end, and then make the ball swing around in a circle, what kind of a surface will the string describe?

[NOTE. The student is not necessarily expected to give the geometrical name of the surface, but rather to tell in his own way what it is like, so as to make it clear that he conceives the thing itself.]

Ex. 4. If you were to draw lines in all directions on the surface of the stove-pipe, might any of them be straight? Could all of them be straight? What kind of a surface is this, therefore?

Ex. 5. Can you draw a straight line on the surface of a ball? On the surface of an egg? What kind of surfaces are these?

Ex. 6. When the carpenter wishes to make the surface of a board perfectly flat, he takes a ruler whose edge is a straight line, and lays this straight edge on the surface in all directions, watching closely

to see if it always touches. Which of our definitions is he illustrating by his practice?

Ex. 7. When the miller wishes to make flat the surface of one of the large stones with which wheat is ground into flour, he sometimes. takes a ruler with a straight edge, and smearing the edge with paint, applies it in all directions to the surface, and then chips off the stone where the paint is left on it. What principles is he illustrating?

Ex. 8. How can you conceive a straight line to move so that it shall not generate a surface?

ANGLES.

22. A Plane Angle, or simply an Angle, is the opening between two lines which meet each other. The point in which the lines meet is called the vertex, and the lines are called the sides. An angle is designated by placing a letter at its vertex, and one at each of its sides. In reading, we name the letter at the vertex when there is but one vertex at the point, and the three letters when there are two or more vertices at the same point. In the latter case, the letter at the vertex is put between the other two.

opening between the two lines AB and AC, in which the figure 1 stands, is called the angle A; or, if we choose, we may call it the angle BAC. At L there are two vertices, so that were we to say the angle L, one would not know whether we meant the angle (corner) in which 4 stands, or that in which 5 stands. To avoid this ambiguity, we say the angle HLR for the former, and RLT for the latter. The angle ZAY is the corner in which 11 stands; that is, the opening between the two lines AY and AZ. In designating an angle by three letters, it is immaterial which letter stands first so that the one at the vertex is put between the other two. Thus, PQS and SQP are both designations of the angle in which 6

B

4/5

14/15

FIG. 10/