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5. The surface of a solid is no part of the sol simply the boundary or limit of the solid. A surf fore, has only two dimensions, length and breadth. if any number of flat surfaces be put together, they cide and form one surface.

6. A line is no part of a surface. It is simply a or limit of the surface. A line, therefore, has only o sion, length. So that, if any number of straight lin together, they will coincide and form one line.

7. A point is no part of a line. It is simply th the line. A point, therefore, has no dimension, b position simply. So that, if any number of poin together, they will coincide and form a single point.

8. A solid, in common language, is a limited p space filled with matter; but in Geometry we have do with the matter of which a body is composed; simply its shape and size; that is, we regard a solid ited portion of space which may be occupied by a body, or marked out in some other way. Hence,

A geometrical solid is a limited portion of space.

9. It must be distinctly understood at the outse points, lines, surfaces, and solids of Geometry are pu though they can be represented to the eye in only a way. Lines, for example, drawn on paper or on t board, will have some width and some thickness, a far fail of being true lines; yet, when they are used the mind in reasoning, it is assumed that they repre fect lines, without breadth and without thickness,

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named by a letter, as A (Fig. 2); a line is named by two
letters, placed one at each end,
as BF; a surface is represented
and named by the lines which
bound it, as BCDF; a solid is
represented by the faces which
bound it.

A

B

F

FIG. 2.

11. By supposing a solid to diminish gradually until it vanishes we may consider the vanishing point, a point in space, independent of a line, having position but no extent.

12. If a point moves continuously in space, its path is a line. This line may be supposed to be of unlimited extent, and may be considered independent of the idea of a surface.

13. A surface may be conceived as generated by a line moving in space, and as of unlimited extent. A surface can then be considered independent of the idea of a solid.

14. A solid may be conceived as generated by a surface in motion.

The

D

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Thus, in the diagram, let the upright surface ABCD move to the right to the position EFGH. points A, B, C, and D will generate the lines AE, BF, CG, and DH. respectively. The lines AB, BC, CD, and AD will generate the surfaces AF, BG, CH, and AH, respectively. The surface ABCD will generate the solid AG.

B

FIG 3

15. Geometry is the science which treats of position, form, and magnitude.

16. Points, lines, surfaces, and solids, with their relations,

A straight line, or right line, is a line same direction throughout its A.

whole extent, as the line AB.

18. A curved line is a line

no part of which is straight, as the line CD.

19. A broken line is a series

of different successive straight

lines, as the line EF

[graphic]

E

G

FIG. 4.

20. A mixed line is a line composed of straight a lines, as the line GH.

A straight line is often called simply a line, and line, a curve.

21. A plane surface, or a plane, is a surface in any two points be taken, the straight line joining th will lie wholly in the surface.

22. A curved surface is a surface no part of which

23. Figure or form depends upon the relative p points. Thus, the figure or form of a line (straight o depends upon the relative position of the points in t the figure or form of a surface depends upon the rela tion of the points in that surface.

24. With reference to form or shape, lines, surf solids are called figures.

With reference to extent, lines, surfaces, and s called magnitudes.

25. A plane figure is a figure all points of which a same plane.

26. Plane figures formed by straight lines are ca tilinear figures; those formed by curved lines an curvilinear figures; and those formed by straight an lines are called mixtilinear figures.

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-B

D

H

27. Figures which have the same shape are called similar figures. Figures which have the same size are called equivalent figures. Figures which have the same shape and size are called equal figures.

28. Geometry is divided in two parts, Plane Geometry and Solid Geometry. Plane Geometry treats of figures all points of which are in the same plane. Solid Geometry treats of figures all points of which are not in the same plane.

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STRAIGHT LINES.

29. Through a point an indefinite number of straight lines may be drawn. These lines will have different directions.

30. If the direction of a straight line and a point in the line are known, the position of the line is known; in other words, a straight line is determined if its direction and one of its points are known. Hence,

All straight lines which pass through the same point in the same direction coincide, and form but one line.

31. Between two points one, and only one, straight line can be drawn; in other words, a straight line is determined if two of its points are known. Hence,

Two straight lines which have two points common coincide throughout their whole extent, and form but one line.

32. Two straight lines can intersect (cut each other) in only one point; for if they had two points common, they would coincide and not intersect.

33. Of all lines joining two points the shortest is the straight line, and the length of the straight line is called the distance

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as prolonged indefinitely both ways. Such a line is indefinite straight line.

35. Often only the part of the line between two fi is considered. This part is then called a segment of For brevity, we say "the line AB" to designate of a line limited by the points A and B.

36. Sometimes, also, a line is considered as procee a fixed point and extending in only one direction. point is then called the origin of the line.

37. If any point C be taken in a given straight lin two parts CA and CB are said to have opposite directions from the point C.

A

FIG. 5.

38. Every straight line, as AB, may be considere ing opposite directions, namely, from A towards B expressed by saying "line AB"; and from B towards is expressed by saying "line BA."

39. If the magnitude of a given line is changed, i longer or shorter.

Thus (Fig. 5), by prolonging AC to B we add and AB AC+ CB. By diminishing AB to C, w

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CB from AB, and AC AB – CB.

If a given line increases so that it is prolonged b

magnitude several times in

succession, the line is multi-4

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B

+

FIG. 6.

Thus (F

is called a multiple of the given line.

AB BC= CD DE, then AC-2AB, AD=3

=

AE 4 AB. Also, AB AC, AB= AD, and A Hence,

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