which depends solely on its argument 0. Hence we may interpret the step

AP = r. (cose + √−1 sinė)

as obtained in the following fashion: Rotate unit length from A B through an angle 0, and then stretch it in the ratio of r: 1. The latter part of this operation

will be signified by the modulus r, the former by the operator (cose + √−1 sině). Thus if A D be of unit length and lying in A B, we may read—

A Pr. (cose + √-1 sine). A D,

and we look upon our complex number as a symbol denoting the combination of two operations performed on a unit step a D.

Starting then from the idea of a complex number as denoting position, we have been led to a new operation represented by the symbol cose + √−1 sinė. This is obviously a generalised form of our old symbol √-1. The operator cose + √−1 sine applied to any step bids us turn the step through an angle 0. We shall see that this new conception has important results.

§ 13. On the Operation which turns a Step through a given Angle.

Suppose we apply the operator (cos✪ + √−1 sinė) twice to a unit step. Then the symbolic expression for this operation will be

or

(cose +1 sinė) (cose +√-1 sinė),

(cose + √1 sinė)2.

But to turn a step first through an angle 0 and then through another angle is clearly the same operation as turning it by one rotation through an angle 20, or as applying the operator cos20 +√-1 sin20. Hence we are able to assert the equivalence of the operations expressed by the equation

(cose +1 sine)2= cos20+ √-1 sin20.

In like manner the result of turning a step by n operations through successive angles equal to must be identical with the result of turning it at once through an angle equal to n times 0, or we may write

(cose +1 sine)" cosne + √-1 sinne.

=

This important equivalence of operations was first expressed in the above symbolical forn by De Moivre, and it is usually called after him De Moivre's Theorem.

We are now able to consider the operation by means of which a step A P can be transformed into another a Q. We must obviously turn AP about A counter-clockwise till it coincides in position with AQ; in this case P will fall on P', so that A P’ = A P. Then we must stretch A P' into AQ; this will be a process of multiplying it by some quantity p, which is equal to the ratio of A Q to a P'.

Expressing this symbolically, if be the angle

PAQ, we have

(cos +1 sing). A P = A P'.

p. (cos + √-1 sino). a P = p. a P′ = A Q. This last equation we can interpret in various ways: (i) p. (cos +-1 sino) is a complex number of which p is the modulus and the argument. Hence we may say that to multiply a step by a complex number is to turn the step through an angle equal to the argument and to alter its length by a stretch represented by the modulus.

(ii) Or, again, we may consider the step A P as itself representing a complex number, x+√−1 y, or if r be

B

FIG. 77.

the scalar value of AP and the angle B A P, we may put A Pr(cose+V-1 sine). Similarly AQ will be a complex number, and its scalar magnitude (= p. AP′

=

pr) will be its modulus, while the angle B A Q = 0 + $ will be its argument. We have then the following identity

p (cos +√1 sino) . r (cos +√ −1 sin0) =

pr. (cos@+&+ √−1 cosp+0).

This may be read in two ways:

=

First, the product of two complex numbers is itself a complex number, and has the product of the moduli for its modulus, the sum of the arguments for its argument.

Or secondly, if we turn unit step through an angle and give a stretch r, and then turn the result obtained through an angle and give it a stretch p, the result will be the same as turning unit step through an angle + and giving it a stretch equal to pr.

Thus we see that any relation between complex numbers may be treated either as an algebraical fact relating to such numbers, or as a theorem concerning operations of turning and stretching unit steps.

(iii) We may consider what answer the above identity gives to the question: What is the ratio of two directed steps A Q and A P? Or, using the notation suggested on p. 45, we ask: What is the meaning of the AQ | symbol ? A step like AP (or AQ) which has | AP magnitude, direction, and sense is, as we have noted, termed a vector. We therefore ask: What is the ratio of two vectors, or what operation will convert one into the other? The answer is: An operation which is the product of a turning (or spin) and a stretch. Now the stretch is a scalar quantity, a numerical ratio by which the scalar magnitude of AP is connected with that of AQ. The stretch therefore is a scalar operation. Further, the turning or spin converts the direction of AP into that of AQ, and it obviously takes place by spinning A P round an axis perpendicular to the plane of the paper in which both AP and AQ lie. Thus the second part of the operation by which we convert AP into AQ denotes a spin (counter-clockwise) through a definite angle about a certain axis. The amount of the spin might be measured by a step taken along that axis. Thus, for instance, if the spin were through 6 units of angle, we might measure 6 units of length along the axis to

denote its amount. We may also agree to take this length along one direction of the axis ('out from the face of the clock') if the spin be counter-clockwise, and in the opposite direction (' behind the face of the clock') if the spin be clockwise. Thus we see that our spinning operation may be denoted by a line or step having magnitude, direction, and sense; that is, by a vector. We are now able to understand the nature of the ratio

of two vectors; it is an operation consisting of the pro- X

duct of a scalar and a vector. This product was termed by Sir William Hamilton a quaternion, and made the foundation of a very powerful calculus.

Thus a quaternion is primarily the operation which converts one vector step into another. It does this by means of a spin and a stretch. If we have three points in plane space, the reader will now understand how the position of the third with regard to the first can be made identical with that of the second by means of a spin and a stretch of the step joining the first to the third, that is, by means of a quaternion.1

§ 14. Relation of the Spin to the Logarithmic Growth of Unit Step.

Let us take a circle of unit radius and endeavour to find how its radius grows in describing unit angle about the centre. Hitherto we have treated of growth only in the direction of length; and hence it might be supposed that the radius of a circle does not 'grow' at all as it revolves about the centre. But our method of adding vector steps suggests at once an obvious extension of our conception of growth. Let a step AP become

1 The term 'stretch' must be considered to include a squeeze or a stretch denoted by a scalar quantity p less than unity.