SEGMENTS OF A LINE
§ 8. Any two points A and B in space determine on the line through them a finite part, which may be considered as being described by a point moving from A to B. This we shall denote by AB, and distinguish it from BA, which is supposed as being described by a point moving from B to A, and hence in a direction or in a " sense " opposite to AB. Such a finite line, which has a definite sense, we shall call a " segment," so that AB and BA denote different segments, which are said to be equal in length but of opposite sense. The one sense is often called positive and the other negative.
P
In introducing the word " sense " for direction in a line, we have the word direction reserved for direction of the line itself, so that different lines have different directions, unless they be parallel, whilst in each line we have a positive and negative sense.
We may also say, with Clifford, that AB denotes the " step " of going from A to B.
§ q. If we have three points A, B, C in a line (fig. 2), the step AB will bring us from A to B, and the step
A B BC from B to C. Hence both steps are
—*— equivalent to the one step AC. This is expressed by saying that AC is the " sum " of AB and BC ; in symbols
A B AB+BC=AC,
where account is to be taken of the
A G B sense.
This equation is true whatever be the position of ' the three points on the line. As a special case we have
AB+BA=o, (I)
ABFBC+CA=o, (2) which again is true for any three points in a line. We further write
AB =—BA,
where — denotes negative sense.
We can then, just as in algebra, change subtraction of segments into addition by changing the sense, so that AB—CB is the same as AB+(—CB) or AB+BC. A figure will at once show the truth of this. The sense is, in fact, in every respect equivalent to the " sign " of a number in algebra.
§ Io. Of the many formulae which exist between points in a line we shall have to use only one more, which connects the segments between any four points A, B, C, D in a line. We have
BC=BD+DC, CA=CD+DA, AB=AD+DB; or multiplying these by AD, BD, CD respectively, we get
BC . AD =BD . AD+DC . AD =BD' . AD—CD . AD CA . BD=CD . BD+DA . BD =CD . BD—AD . BD AB . CD=AD . CDIDB . CD=AD . CD—BD . CD.
It will be seen that the sum of the righthand sides vanishes, hence that
BC AD+CA . BD+AB : CD =o (3)
for any four points on a line.
§ ii. If C is any point in the line AB, then we say that C divides the segment AB in the ratio AC/CB, account being taken of the sense of the two segments AC and CB. If C lies between A and B the ratio is positive, as AC and CB have the same sense. But if C lies without the segment AB, i.e. if C divides AB externally, then
the ratio is negative. Q A M B P To see how the value of
this ratio changes with
the whole line (fig. 3),
whilst A and B remain fixed. If C lies at the point A, then AC =o, hence the ratio AC:CB vanishes. As C moves towards B, AC increases and CB decreases, so that our ratio increases. At the middle point M of AB it assumes the value +I, and then increases till it reaches an infinitely large value, when C arrives at B. On passing beyond B the ratio becomes negative. If C is at P we have AC=AP=AB+BP, hence
AC AB BP 'AB
CB=PB+PB= BP—I'
In the last expression the ratio AB:BP is positive, has its greatest value co when C coincides with B, and vanishes when BC becomes infinite. Hence, as C moves from B to the right to the point at infinity, the ratio AC:CB varies from — to—I.
If, on the other hand, C is to the left of A, say at Q, we have
AC=AQ=AB+BQ=AB—QB, hence CB=QI.
Here AB 

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