Page:EB1911 - Volume 11.djvu/710

Rh 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. § 9. 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 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—

AB + BC = AC,

where account is to be taken of the sense.

This equation is true whatever be the position of the three points on the line. As a special case we have

and similarly

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.

§ 10. 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 It will be seen that the sum of the right-hand sides vanishes, hence that

for any four points on a line. § 11. 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. To see how the value of this ratio changes with C, we will move C along the whole line (fig. 3), whilst A and B remain fixed. If C lies at the point A, then AC = 0, 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 +1, 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

In the last expression the ratio AB : BP is positive, has its greatest value ∞ 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 −1.

If, on the other hand, C is to the left of A, say at Q, we have AC = AQ = AB + BQ = AB − QB, hence $AC⁄CB$ = $AB⁄QB$ − 1.

Here AB < QB, hence the ratio AB : QB is positive and always less than one, so that the whole is negative and < 1. If C is at the point at infinity it is −1, and then increases as C moves to the right, till for C at A we get the ratio = 0. Hence—

“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 value −1, reaches 0 at A, +1 at M, ∞ at B, now changes sign to −∞, and increases till at an infinite distance it reaches again the value −1. It assumes therefore all possible values from −∞ to +∞, 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 one single point in the line AB.

[Relations between segments of lines are interesting as showing an application of algebra to geometry. The genesis of such relations from algebraic identities is very simple. For example, if a, b, c, x be any four quantities, then 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 (a − b) = AB − AC = −BC, paying regard to signs; (a − c) = AB − AD = 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:—

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 Clearing of fractions by multiplying throughout by AB · BC · CA, we have to prove −AD · AX · BC − BD · BX · CA − CD · 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

a2b − ab2 = −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

Clearing of fractions by multiplying throughout by AB · BC · CD · DA, and reducing to a common origin O (calling OA = a, OB = b, &c.), an equation containing the second and lower powers 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 infinity, and the second by taking F at infinity, and then making the obvious permutations of the points.]

§ 12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane is called the projection of A on the plane from S as centre of projection. If we have two planes and ′ and a point S, we may project every point A in to the other plane. If A′ is the projection of A, then A is also the projection of A′, so that the relations are reciprocal. To every figure in we get as its projection a corresponding figure in ′.

We shall determine such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be sufficient to consider at first only constructions in one plane.

Let us suppose we have given in a plane two lines p and p′ and a centre S (fig. 4); we may then project the points in p from S to p′. Let A′, B′ ... be the projections of A, B ..., the point at infinity in p which we shall denote by I will be projected into a finite point