SEGMENTS OF A

This article appears in Volume V11, Page 690 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: SCY-SHA

SEGMENTS OF A LINE
8. Any two points A and B in space determine on the line through them a finite

part

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

CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

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

SPECIAL

case

CASE
CASE, JOHN (d. 1600)

we have
AB+BA=o, (I)
AB-FBC+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

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to
barter
- BARTER (from Fr. barater, to
  truck
  - TRUCK
  , to exchange)
; the ultimate derivation is probably from the root which appears in the Gr. 冠 urmu', to bend)

subtraction of segments into addition by changing the sense, so that ABCB 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' . ADCD . AD CA . BD=CD . BD+DA . BD =CD . BDAD . BD AB . CD=AD . CD-I-DB . CD=AD . CDBD . CD.
It will be seen that the sum of the right-hand 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

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= BPI'
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 toI.
If, on the other hand, C is to the left of A, say at Q, we have
AC=AQ=AB+BQ=ABQB, hence CB=Q--I.
Here AB " As C moves along the line from an infinite distance to the left to an infinite distance at the right, the ratio always increases; it starts with the value1, reaches o at A, +1 at M, at B, now changes sign to co , and increases till at an infinite distance it reaches again the value1. It assumes therefore all possible values from m to+ xo , and each value only once, so that not only does every position of C determine a definite value of the ratio AC :CB, but also, conversely, to every positive or negative value of this ratio belongs onefrom algebraic identities is very simple. For example, if a, b, c, x be any four quantities, then
a (ab)(a c)(xa)
+(bc) (b ba)(xb)+
x
(ca)(cb)(x-c) (xa)b)(xc)' this may be proved, cumbrously, by multiplying up, or, simply, by decomposing the right-hand member of the identity into partial fractions. Now take a line ABCDX, and let AB = a, AC = b, AD =c, AX =x. Then obviously (ab) =ABAC =BC, paying regard to signs; (ac)=ABAD=DB, and so on. Substituting these values in the identity we obtain the following relation connecting the segments formed by five points on a line
:
AB AC AD AX BC.BD.BX+CD.CB.CX+DB.DC.DX=BX.CX.DX'
Conversely, if a metrical relation be given, its validity may be tested by reducing to an algebraic equation, which is an identity if the relation be true. For example, if ABCDX be five collinear points, prove
AD.AX BD.BX CD.CX
AB . AC'+BC . BA+CA . CB'=I'
Clearing of fractions by multiplying throughout by AB . BC . CA, we have to prove
AD . AX. BC BD. BX. CACD . CX. AB =AB . BC . CA.
Take A as origin and let AB = a, AC = b, AD =c, AX =x. Substituting for the segments in terms of a, b, c, x, we obtain on simplification
a2bab2 =ab2+a2b, an obvious identity.
An alternative method of testing a relation is illustrated in the following example:If A, B, C, D, E, F be six collinear points, then
AE.AF BE. BF CE.CF DE.DF
AB.AC.AD+BC. BD.BA+CD.CA.CB+DA . DB . DC=0' Clearing of fractions by multiplying throughout by AB . BC . CD: DA, and reducing to a common origin 0 (calling OA=a, OB=b, &c.), an equation containing the second and lower rowers of OA (=a), &c., is obtained. Calling OA=x, it is found that x=b, x=c,'x'=d are solutions. Hence the quadratic has three roots; consequently it is an identity.
The relations connecting five points which we have instanced above may be readily deduced from the six-point relation; the first by taking D at infity, and the second by taking F at infinity, and then making the obvious permutations of the points.]

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