In the same manner we have for any number of steps

AB+BC + CD + ... LM +MN=AN.

As a special case let N coincide with A, then the sum is AA which we denote by zero and thus

AB+BC+... LM+MA = 0.

The Associative and Commutative laws are easily proved to be true if the definition of equal steps be remembered.

21. Subtraction of Steps. By the difference of two steps we have to understand that step which added to the second gives the first, that is

(The sign here denotes the operation of subtraction, not the sense.)

Therefore instead of subtracting AB we may add BA. Hence the rule:-In order to subtract a step, change its sign and add, just as in Algebra.

This completely justifies the use of the signs + and − to denote sense.

It is often convenient to express all steps in a line as steps from a common origin, thus if O be the origin in the line, then the step AB is the difference of the two steps OB and OA or

P

22. Areas. A straight line OP moving in a plane will, in general, describe an area. If the point O be fixed whilst P moves along a given curve (the length of OP varying), then OP starting from the position OA will sweep over an area OAP. If OP turns further the area will increase, if it turns back the area will diminish. We may say therefore that the area is positive or negative according as OP turns in one or the opposite sense. Which sense is taken as positive is a matter of choice; we take the counter-clockwise turning as positive.

Fig. 8.

Suppose P to describe a closed curve returning to its original position A, then OP will have swept over or have generated the area enclosed by the curve. This will be true wherever the point O is taken. The sense of turning is known if the sense in which P describes the boundary is given. Hence the sense in which an area is to be taken is fixed by giving the boundary a sense, which sense is indicated by an arrowhead.

R.

A

The point O may be taken outside the figure; in this case, as P moves from A round the boundary in the sense indicated, OP will sweep over every point Q within the boundary once and once only and that in the positive sense; but it will sweep over a point outside the boundary either not at all (as R) or once in the positive sense and once in the negative sense (as S), the one cancelling the other. Here again, the area generated by the moving line of variable length is that enclosed by the curve described by its end point, the sense of the area being given by the sense of the boundary.

Fig. 9.

-

The justification for the use of the signs and — to

denote the sense of areas can be shewn just as in the case

B

0

Fig. 10.

But

of steps in a line.

In the figure, by the difference of two areas OAC and OAB we mean that area OBC which added to OAB gives the area OAC or

ОАС - ОАВ = ОВС if OBC+0AB=OAC.

(The minus sign denoting the operation of subtraction.)

ОАВ =- ОВА

(changing the sense of the boundary);

.. OAC-(— OBA) = OBC.

Now OBA+0AC=OBC, for this says-start with P at B, move to A and then from A to C and the total area swept out will be OBC. Hence the sign for subtraction together with the sign for sense are equivalent to the sign, just as in ordinary Algebra.

23. Aspect of Planes. The rule given about the sense of turning and of areas in a plane is perfectly definite as long as we have to deal with one plane only at which we look from one side only. But suppose a closed curve, say a circle, drawn on transparent paper and a sense given to the boundary, say counterclockwise, then on turning the paper over, the sense would appear clockwise. For figures in space, we must therefore also specify the side of the plane from which we have to look at the area.

Just as parallel lines are said to have the same direction, so parallel planes are said to have the same aspect. Parallel planes having parallel perpendiculars, the aspect of a plane is determined by the direction of its

normal.

If we give the normal to a plane a definite sense, then the arrowhead indicating it will at one side of the plane point towards the plane, and at the other away from it.

Normal

Positive Face

The latter we shall call the positive side, or face, of the plane, the former its negative face.

24. Turning about an axis. The turning of a line or figure in a plane about a point can be considered as taking place about the normal (at that point) as axis.

If now we connect the sense of this turning with the sense of the axis, we shall have a short and accurate way of describing the aspect of areas and the sense of turning by means of the direction and sense of the normal.

Fig. 11.

Definition. If a line has a definite sense, then that turning shall be taken as positive which appears as clockwise to a person looking along the line in its positive sense.

Negative Face

Positive Face

Fig. 12.

Looked at in the opposite sense the turning will appear counter-clockwise. Hence to a person looking on the positive side of a plane the positive turning will appear counter-clockwise. This agrees with our definition in § 22.

Suppose an ordinary right-handed screw thread on a shaft with a nut on it; give, whilst looking along the shaft, a positive turning to the nut, then the latter will move forwards along the shaft in the positive sense.

The above system of connecting the sense of turning with the sense of an axis is therefore called right-handed.