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BOOK I.

RECTILINEAR FIGURES.

INTRODUCTORY REMARKS.

A ROUGH block of marble, under the stone-cutter's hammer, may be made to assume regularity of form.

If a block be cut in the shape represented in this diagram,

It will have six flat faces.

Each face of the block is called a Surface.

If these surfaces be made smooth by polishing, so that, when a straight-edge is applied to any one of them, the straight-edge in every part will touch the surface, the surfaces are called Plane Surfaces.

The sharp edge in which any two of these surfaces meet is called a Line.

The place at which any three of these lines meet is called a Point.

If now the block be removed, we may think of the place occupied by the block as being of precisely the same shape and size as the block itself; also, as having surfaces or boundaries which separate it from surrounding space. We may likewise think of these surfaces as having lines for their boundaries or limits; and of these lines as having points for their extremities or limits.

A Solid, as the term is used in Geometry, is a limited portion of space.

After we acquire a clear notion of surfaces as boundaries of solids, we can easily conceive of surfaces apart from solids, and suppose them of unlimited extent. Likewise we can conceive of lines apart from surfaces, and suppose them of unlimited length; of points apart from lines as having position, but no extent.

DEFINITIONS.

1. DEF. Space or Extension has three Dimensions, called Length, Breadth, and Thickness.

2. DEF. A Point has position without extension.

3. DEF. A Line has only one of the dimensions of extension, namely, length.

The lines which we draw are only imperfect representations of the true lines of Geometry.

A line may be conceived as traced or generated by a point in motion.

4. DEF. A Surface has only two of the dimensions of extension, length and breadth. A surface may be conceived as generated by a line in motion.

5. DEF. A Solid has the three dimensions of extension, length, breadth, and thickness. Hence a solid extends in all directions.

A solid may be conceived as generated by a surface in motion. Thus, in the diagram, let the upright B

'F surface A B C D move to the right to A4-7-------the position E F H K. The points

| D _ A, B, C, and D will generate the lines

----

I H

K----------A E, B F, C K, and D H respectively. C And the lines A B, BD, DC, and AC will generate the surfaces A F, BH, DK, and A K respectively. And the surface A B C D will generate the solid A H.

The relative situation of the two points A and H involves three, and only three, independent elements. To pass from A to H it is necessary to move East (if we suppose the direction A E to be due East) a distance equal to A E, North a distance equal to EF, and down a distance equal to FH.

· These three dimensions we designate for convenience length, breadth, and thickness.

6. The limits (extremities) of lines are points.

The limits (boundaries) of surfaces are lines.

The limits (boundaries) of solids are surfaces. 7. DEF. Extension is also called Magnitude. When reference is had to extent, lines, surfaces, and solids are called magnitudes.

8. DEF. A Straight line is a line which has the same direction throughout its whole extent.

9. DEF. A Curved line is a line which changes its direction at every point.

10. DEF. A Broken line is a series of connected straight lines.

When the word line is used a straight line is meant; and when the word curve is used a curved line is meant.

11. DEF. A Plane Surface, or a Plane, is a surface in which, if any two points be taken, the straight line joining these points will lie wholly in the surface.

12. DEF. A Curved Surface is a surface no part of which is plane.

13. Figure or form depends upon the relative position of points. Thus, the figure or form of a line (straight or curved) depends upon the relative position of points in that line; the figure or form of a surface depends upon the relative position of points in that surface.

When reference is had to form or shape, lines, surfaces, and solids are called figures.

14. DEF. A Plane Figure is a figure, all points of which are in the same plane.

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

Points, lines, surfaces, and solids, with their relations, are the geometrical conceptions, and constitute the subject matter of Geometry.

16. Plane Geometry treats of plane figures. Plane figures are either rectilinear, curvilinear, or mixtilinear.

Plane figures formed by straight lines are called rectilinear figures; those formed by curved lines are called curvilinear figures ; and those formed by straight and curved lines are called mixtilinear figures.

17. Def. Figures which have the same form are called Similar Figures. Figures which have the same extent are called Equivalent Figures. Figures which have the same form and extent are called Equal Figures.

ON STRAIGHT LINES.

18. If the direction of a straight line and a point in the line be known, the position of the line is known; that is, a straight line is determined in position if its direction and one of its points be known.

Hence, all straight lines which pass through the same point in the same direction coincide.

Between two points one, and but one, straight line can be drawn; that is, a straight line is determined in position if two of its points be known.

Of all lines between two points, the shortest is the straight line; and the straight line is called the distance between the two points.

The point from which a line is drawn is called its origin.

19. If a line, as C B, 4__4_B, be produced through C, the portions C B and C A may be regarded as different lines having opposite directions from the point C.

Hence, every straight line, as A B, A_ _B, has two opposite directions, namely from A toward B, which is expressed by saying line A B, and from B toward A, which is expressed by saying line B A.

20. If a straight line change its magnitude, it must become · longer or shorter. Thus by prolonging A B to 0,4_R__, AC=AB + BC; and conversely, BC=AC-A B.

If a line increase so that it is prolonged by its own magnitude several times in succession, the line is multiplied, and the resulting line is called a multiple of the given line. Thus, if A B= BC=C D, etc., 4_ _4_D_E, then A C=2 A B, A D= 3 A B, etc.

It must also be possible to divide a given straight line into an assigned number of equal parts. For, assumed that the nth part of a given line were not attainable, then the double, triple, quadruple, of the nth part would not be attainable. Among these multiples, however, we should reach the nth multiple of this nth part, that is, the line itself. Hence, the line itself would not be attainable ; which contradicts the hypothesis that we have the given line before us.

Therefore, it is always possible to add, subtract, multiply, and divide lines of given length.

21. Since every straight line has the property of direction, it must be true that two straight lines have either the same direction or different directions.

Two straight lines which have the same direction, without coinciding, can never meet; for if they could meet, then we should have two straight lines passing through the same point in the same direction. Such lines, however, coincide.

§ 18

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