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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.
A segment is denoted by paming 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 “seg. ment a.
This is a uniliteral, or one-letter notation. The term 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-seginent, 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
into two parts whereof one shall C
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
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 geonietric 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. 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
Ċ and DE.
AC=AB+DE, where = denotes equality in length, and + 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.
299. 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.
=, t, and –, a term, as DE, may be transferred from one side of
the equation to another by changing its sign from + to vice versa.
Owing to the readiness with which these symbolic expressions can be manipulated, they seem to represent simple algebraic relations, hence beginners are apt to think that the working rules of algebra must apply to them as a matter of necessity. It must be remembered, however, that the formal rules of algebra are founded upon the properties of numbers, and that we should not assume, without examination, that these rules apply without modification to that which is not number.
This subject will be discussed in Part II.
30°. Def.--That point, in a line-segment, which is equidistant from the end-points is the middle point of the segment.
It is also called the internal point of bisection of the segment, or, when spoken of alone, simply the point of bisection.
1, If two segments be in line and have one common end
point, by what name will you call the distance between
their other end-points ? 2. Obtain any relation between “ the sum and the differ
ence” of two segments and "the relative directions” of
the two segments, they being in line. 3. A given line-segment has but one middle point. 4. In Art. 23°, if C becomes the middle point of AB, what
becomes of C'? 5. In Art. 30° the internal point of bisection is spoken of.
What meaning can you give to the “external point of bisection”?