tion of the images of objects as seen in such mirrors is capable of detecting variations from the plane, so minute as to escape all other tests.

18°. A surface which is not plane, and which is not composed of planes, is a curved surface. Such is the surface of a sphere, or cylinder.

19°. The point, the line, the curve, the plane and the curved surface are the elements which go to make up geometric figures.

Where a single plane is the only surface concerned, the point and line lie in it and form a plane figure. But where more than one plane is concerned, or where a curved surface is concerned, the figure occupies space, as a cube or a sphere, and is called a spatial figure or a solid.

The study of spatial figures constitutes Solid Geometry, or the Geometry of Space, as distinguished from Plane Geometry.

20°. Given Point and Line. A point or line is said to be given when we are made to know enough about it to enable us to distinguish it from every other point or line; and the data which give a point or line are commonly said to determine it.

A similar nomenclature applies to other geometric ele

ments.

The statement that a point or line lies in a plane does not give it, but a point or line placed in the plane for future reference is considered as being given. Such a point is usually called an origin, and such a line a datum line, an initial line, a prime vector, etc.

21°. Def. 1.- A line considered merely as a geometric element, and without any limitations, is an indefinite line.

2.-A limited portion of a line, especially when any reference is had to its length, is a finite line, or a line-segment, or simply a segment.

That absolute sameness (14°) which characterizes every part of a line leads directly to the following conclusions :(1) No distinction can be made between any two segments of the same line equal in length, except that of position in the line.

(2) A line cannot return into, or cross itself.

(3) A line is not necessarily limited in length, and hence, in imagination, we may follow a line as far as we please without coming to any necessary termination.

This property is conveniently expressed by saying that a line extends to infinity.

3.—The hypothetical end-points of any indefinite line are said to be points at infinity. All other points are finite

points.

22°. Notation. A point is denoted by a single letter whereever practicable, as "the point A."

An indefinite line is also denoted by a single letter as "the line L," but in this case the letter

has no reference to any point.

L

A segment is denoted by naming its end points, as the 'segment AB," where A and B are the end points. This is a biliteral, or two-letter notation.

A segment is also denoted by a single letter, when the limits of its length are supposed to be known, as the “segment a." This is a uniliteral, or one-letter notation.

The term 66 segment" involves the notion of some finite length. When length is not under consideration, the term "line" is preferred.

Thus the "line AB" is the indefinite line having A and B as two points upon it. But the "segment AB" is that portion of the line which lies between A and B.

23°. In dealing with a line-segment, we frequently have to consider other portions of the indefinite line of which the segment is a part.

As an example, let it be required to divide the segment AB

A

C B

into two parts whereof one shall be twice as long as the other.

To do this we put C in such a position that it may be twice as far from one of the end-points of the segment, A say, as it is from the other, B. But on the indefinite line through A and B we may place C' so as to be twice as far from A as from B. So that we have two points, C and C', both satisfying the condition of being twice as far from A as from B.

Evidently, the point C' does not divide the segment AB in the sense commonly attached to the word divide. But on account of the similar relations held by C and C' to the endpoints of the segment, it is convenient and advantageous to consider both points as dividing the segment AB.

When thus considered, C is said to divide the segment internally and C' to divide it externally in the same manner.

24°. Axiom.-Through a given point only one line can pass in a given direction.

Let A be the given point, and let the segment AP mark the given direction. Then, of all the lines

P

Р

A that can pass through the point A, only one can have the direction AP, and this one must lie along and coincide with AP so as to form with it virtually but one line. Cor. 1. A finite point and a direction determine one line.

Cor. 2. Two given finite points determine one line. For, if A and P be the points, the direction AP is given, and hence the line through A and having the direction AP is given.

Cor. 3. Two lines by their intersection determine one finite point. For, if they determined two, they would each pass through the same two points, which, from Cor. 2, is impossible.

Cor. 4. Another statement of Cor. 2 is-Two lines which have two points in common coincide and form virtually but one line.

25°. Axiom.—A straight line is the shortest distance between two given points.

Although it is possible to give a reasonable proof of this axiom, no amount of proof could make its truth more apparent.

The following will illustrate the axiom. Assume any two points on a thread taken as a physical line. By separating these as far as possible, the thread takes the form which we call straight, or tends to take that form. Therefore a straight finite line has its end-points further apart than a curved line of equal length. Or, a less length of line will reach from one given point to another when the line is straight than when it is curved.

Def.—The distance between two points is the length of the segment which connects them or has them as end-points.

26°. Superposition.—Comparison of Figures.-We assume that space is homogeneous, or that all its parts are alike, so that the properties of a geometric figure are independent of its position in space. And hence we assume that a figure may be supposed to be moved from place to place, and to be turned around or over in any way without undergoing any change whatever in its form or properties, or in the relations existing between its several parts.

The imaginary placing of one figure upon another so as to compare the two is called superposition. By superposition we are enabled to compare figures as to their equality or. inequality. If one figure can be superimposed upon another so as to coincide with the latter in every part, the two figures are necessarily and identically equal, and become virtually one figure by the superposition.

27°. Two line-segments can be compared with respect to length only. Hence a line is called a magnitude of one dimension.

Two segments are equal when the end-points of one can be

made to coincide with the end-points of the other by superposition.

28°. Def.—The sum of two segments is that segment which is equal to the two when placed in line with one end-point in each coincident.

D

Let AB and DE be two segments, and on the line of which AB is a segment let BC be equal to DE. Then AC is the sum of AB C and DE.

A

B

This is expressed symbolically by writing

=

AC=AB+DE,

where denotes the placing of the segments AB and DE in line so as to have one common point as an end-point for each. The interpretation of the whole is, that AC is equal in length to AB and DE together.

denotes equality in length, and

29°. Def.— The difference between two segments is the segment which remains when, from the longer of the segments, a part is taken away equal in length to the shorter.

Thus, if AC and DE be two segments of which AC is the longer, and if BC is equal to DE, then AB is the difference between AC and DE.

This is expressed symbolically by writing

AB-AC-DE,

which is interpreted as meaning that the segment AB is shorter than AC by the segment DE.

Now this is equivalent to saying that AC is longer than AB by the segment DE, or that AC is equal to the sum of AB and DE.

Hence when we have AB=AC - DE

we can write

AC AB+DE.

We thus see that in using these algebraic symbols, =, +, and, a term, as DE, may be transferred from one side of