A SYLLABUS

OF

MODERN PLANE GEOMETRY.

A. I. G. T.

It is requested that any observations or criticisms upon this Syllabus
may be addressed to the Secretary, the REV. J. J. MILNE, Invermark,
Alleyn Park, West Dulwich, S.E.

London:

MACMILLAN AND CO.

AND NEW YORK.

RICHARD CLAY AND SONS, LIMITED,

LONDON AND BUNGAY,

A SYLLABUS OF

MODERN PLANE GEOMETRY.

I. INTRODUCTION.

(1) In Elementary Geometry a straight line of definite length, or a SEGMENT, is considered with reference to absolute length only, and the question of whether it is to be added to or subtracted from another line is a matter of explicit statement.

In the Modern Plane Geometry account is taken of the direction of measurement along a straight line, and this is expressed with the help of the signs + and

The fundamental rules with regard to these signs are that (i) if A and B be any two points on a straight line,

(ii) if A, B, P be any three points on a straight line,

AP + PB AB.

(2) Two fixed points (A and B) being taken on a straight line, the position of any other point P (on the same straight line) is given when the ratio of the segments AP : PB is given.

If AP: PB is positive, AB is internally divided in P.

If AP: PB is negative, AB is externally divided in P.

In either case P is nearer to A or to B, according as AP : PB numerically < or > unity.

The point at an infinite distance beyond A is identically the same point as that at an infinite distance beyond B; for

when P is at A, AP : PB is zero;

when P is at the middle point of AB then AP : PB is + 1, as P approaches B the ratio continually increases in value, becoming equal in the limit to +∞;

as P passes through B the sign of the ratio changes and gradually decreases numerically from ∞ (in the limit) to

- 1, which is the value it assumes when P is at an infinite distance beyond B.

Up to this point the value of the ratio has undergone a continuous change, and the motion of P has also been continuous. As P moves from the point at an infinite distance beyond A to A, the ratio gradually decreases numerically in value from I till in the limit it assumes the value zero. The changes in the value of the ratio have been continuous throughout. This, then, points to the conclusion that the motion of P has also been continuous; that is, that the point at an infinite distance beyond B may be considered as identically the same as that at an infinite distance beyond A, and that the straight line may be considered as a continuous locus.

NOTE.— When a straight line AB is mentioned, a confusion is likely to arise in the mind of the student as to whether the Finite straight line AB is meant, or the continuous Locus of which AB