conceive of any thing of more than three dimensions. Therefore, every thing which has extent and form belongs to one of these three classes.

The extent of a line is called its LENGTH; of a surface, its AREA; and of a solid, its VOLUME.

34. Whatever has only extent and form is called a MAGNITUDE.

GEOMETRY is the science of magnitude.

Geometry is used whenever the size, shape, or position of any thing is investigated. It establishes the principles upon which all measurements are made. It is the basis of Surveying, Navigation, and Astronomy.

In addition to these uses of Geometry, the study is cultivated for the purpose of training the student's powers of language, in the use of precise terms; his reason, in the various analyses and demonstrations; and his inventive faculty, in the making of new solutions and demonstrations.

THE POSTULATES.

We may

35. Magnitudes may have any extent. conceive lines, surfaces, or solids, which do not extend beyond the limits of the smallest spot which represents a point; or, we may conceive them of such extent as to reach across the universe. The astronomer knows that his lines reach to the stars, and his planes extend beyond the sun. These ideas are expressed in the following

Postulate of Extent.-A magnitude may be made to have any extent whatever.

36. Magnitudes may, in our minds, have any form, from the most simple, such as a straight line, to that of the most complicated piece of machinery. We may

conceive of surfaces without solids, and of lines without surfaces.

It is a useful exercise to imagine lines of various forms, extending not only along the paper or blackboard, but across the room. In the same way, surfaces and solids may be conceived of all possible forms.

The form of a magnitude consists in the relative position of the parts, that is, in the relative directions of the points. Every change of form consists in changing the relative directions of the points of the figure.

Every geometrical conception, however simple or complex, is composed of only two kinds of elementary thoughts-directions and distances. The directions determine its form, and the distances its extent.

Postulate of Form.-The points of a magnitude may be made to have from each other any directions whatever, thus giving the magnitude any conceivable form.

These two are all the postulates of geometry. They rest in the very ideas of space, form, and magnitude.

37. Magnitudes which have the same form while they differ in extent, are called SIMILAR.

Any point, line, or surface in a figure, and the similarly situated point, line, or surface in a similar figure, are called HOMOLOGOUS.

Magnitudes which have the same extent, while they differ in form, are called EQUIVALENT.

MOTION AND SUPERPOSITION.

38. The postulates are of constant use in geometrical reasoning.

Since the parts of a magnitude may have any position, they may change position. By this idea of mo

tion, the mutual derivation of points, lines, surfaces, and solids may be explained.

The path of a point is a line, the path of a line may be a surface, and the path of a surface may be a solid. The time or rate of motion is not a subject of geometry, but the path of any thing is itself a magnitude.

39. By the idea of motion, one magnitude may be mentally applied to another, and their form and extent compared.

This is called the method of superposition, and is the most simple and useful of all the methods of demonstration used in geometry. The student will meet with many examples.

EQUALITY.

40. When two equal magnitudes are compared, it is found that they may coincide; that is, each contains the other. Since they coincide, every part of one will have its corresponding equal and coinciding part in the other, and the parts are arranged the same in both.

Conversely, if two magnitudes are composed of parts respectively equal and similarly arranged, one may be applied to the other, part by part, till the wholes coincide, showing the two magnitudes to be equal.

Each of the above convertible propositions has been stated as an axiom, but they appear rather to constitute the definition of equality.

FIGURES.

41. Any magnitude or combination of magnitudes which can be accurately described, is called a geometrical FIGURE.

Figures are represented by diagrams or drawings, and such representations are, in common language, called figures. A small spot is commonly called a point, and a long mark a line. But these have not only

extent and form, but also color, weight, and other properties; and, therefore, they are not geometrical points and lines.

It is the more important to remember this distinction, since the point and line made with chalk or ink are constantly used to represent to the eye true mathematical points and lines.

42. The figure which is the subject of a proposition, together with all its parts, is said to be GIVEN. The additions to the figure made for the purpose of demonstration or solution, constitute the CONSTRUCTION.

43. In the diagrams in this work, points are designated by capital letters. Thus,

the points A and B are at the extremities of the line.

Figures are usually designated by naming some of their points, as the line AB, and the figure CDEF, or simply the figure DF.

When it is more convenient to designate a figure by a single letter, the small letters are used. Thus, the line a, or the figure b.

A

a

b

LINES.

44. A STRAIGHT LINE is one which has the same di

rection throughout its whole extent.

A straight line may be regarded as the path of a point moving in one direction, turning neither up nor down, to the right or left.

45. A CURVED LINE is one which constantly changes its direction. The word curve is used for a curved line.

46. A line composed of straight lines, is called BROKEN. A line may be composed of curves, or of both curved and straight parts.

THE STRAIGHT LINE.

47. Problem.-A straight line may be made to pass through any two points.

48. Problem. There may be a straight line from any point, in any direction, and of any extent.

These two propositions are corollaries of the postulates.

49. From a point, straight lines may extend in all directions. But we can not conceive that two separate straight lines can have the same direction from a common point. This impossibility is expressed by the following

Axiom of Direction. In one direction from a point, there can be only one straight line.

50. Corollary. From one point to another, there can be only one straight line

51. Theorem.—If a straight line have two of its points common with another straight line, the two lines must coincide throughout their mutual extent.

For, if they could separate, there would be from the point of separation two straight lines having the same direction, which is impossible (49).