QUATERNIONS.

CHAPTER I.

VECTORS, AND THEIR COMPOSITION.

OR more than a century and a half the geometrical

1. FOR

representation of the negative and imaginary algebraic quantities, -1 and √-1, or, as some prefer to write them,

and, has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in employing such quantities to indicate the direction, not the length, of lines.

2. Thus it was soon seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics.

3. Wallis, in the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going out of the line on which, if real, they would have been laid off.

His construction is equivalent to the consideration of √-1 as a directed unit-line perpendicular to that on which real quantities are measured.

4. In the usual notation of Analytical Geometry of two dimensions, when rectangular axes are employed, this amounts to reckoning each unit of length along Oy as + √1, and on Oy' as -√1; while on Ox each unit is +1, and on Ox' it is 1. If we look at these four lines in circular order, i. e. in the order of positive rotation (opposite to that of the hands of a watch), they give

17

1, √−1, −1, -√-1.

In this series each expression is derived from that which precedes it by multiplication by the factor 1. Hence we may consider 1 as an operator, analogous to a handle perpendicular to the plane of xy, whose effect on any line is to make it rotate (positively) about the origin through an angle of 90°.

5. In such a system, a point is defined by a single imaginary expression. Thus a+b√1 may be considered as a single quantity, denoting the point whose coordinates are a and b. Or, it may be used as an expression for the line joining that point with the origin. In the latter sense, the expression a+b√-1 implicitly contains the direction, as well as the length, of this line; since, as we see at once, the direction is inclined

at an angle tan-1 to the axis of x, and the length is

a

a+b2.

6. Operating on this symbol by the factor 1, it becomes -b+a√−1; and now, of course, denotes the point whose a and y cöordinates are -b and a; or the line joining this point with the origin. The length is still a2+b2, but the angle the line

7. De Moivre's Theorem tends to lead us still farther in the same direction. In fact, it is easy to see that if we use, instead of √1, the more general factor cos a+ √1 sin a, its effect on any line is to turn it through the (positive) angle a in the plane of x, y. [Of course the former factor, √1, is merely the particular case of this, when a =

Thus (cosa + √1 sin a) (a+b√1)

π

2

]

= a cos a−b sin a+ √-1 (a sin a + b cos a), by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle a has taken place, if he compares it with the common formulæ for turning the coordinate axes through a given angle. Or, in a less simple manner, thus—

8. We see now, as it were, why it happens that

(cos a+ √1 sin a)m = cos ma+ √1 sin ma.

In fact, the first operator produces m successive rotations in the same direction, each through the angle a; the second, a single rotation through the angle ma.

9. It may be interesting, at this stage, to anticipate so far as to state that a Quaternion can, in general, be put under the form N(cos + sin 0),

where N is a numerical quantity, O a real angle, and

1.