QUATERNIONS.

1.

CHAPTER I.

VECTORS, AND THEIR COMPOSITION.

FOR

'OR more than a century and a half the geometrical 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.

B

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 Or each unit is +1, and on Or 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

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 ry, 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+b1 may be considered as a single quantity, denoting the point whose cöordinates 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

b

at an angle tan-1 to the axis of x, and the length is √a2+b2.

α

6. Operating on this symbol by the factor

1, it becomes -b+a√1; and now, of course, denotes the point whose x and y cöordinates are - and a; or the line joining this point with the origin. The length is still √a+b2, but the angle the line makes with the axis of x is tan-1

greater than before the operation.

7); which is evidently 90°

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 (cos a+ √1 sin a) (a + b√ −1)

π

= 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—

Length

=

√(a cosa―b sin a)2 + (a sin a + b cos a)2 = √ a2+b2

Inclination to axis of r

as before.

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

(cos a+ √1 sin a)TM

= 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, 0 a real angle, and

This expression for a quaternion bears a very close analogy to the forms employed in De Moivre's Theorem; but there is the essential difference (to which Hamilton's chief invention referred) that is not the algebraic √1, but may be any directed unitline whatever in space.

10. In the present century Argand, Warren, and others, extended the results of Wallis and De Moivre. They attempted to express as a line the product of two lines each represented by a symbol such as a+b√-1. To a certain extent they succeeded, but simplicity was not gained by their methods, as the terrible array of radicals in Warren's Treatise sufficiently proves.

11. A very curious speculation, due to Servois, and published in 1813 in Gergonne's Annales, is the only one, so far as has been discovered, in which the slightest trace of an anticipation of Quaternions is contained. Endeavouring to extend to space the form a+b1 for the plane, he is guided by analogy to write for a directed unit-line in space the form

p cos a+q cos ß +r cos y,

He perceives

where a, ß, y are its inclinations to the three axes. easily that p, q, r must be non-reals: but, he asks, "seraientelles imaginaires réductibles à la forme générale A+B √−1?” This he could not answer. In fact they are the i, j, k of the Quaternion Calculus. (See Chap. II.)

12. Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions; and, though many such have been made within the last forty years, none, with the single exception of Hamilton's, have resulted in simple, practical methods; all, however ingenious, seeming to lead at once to processes and results of fearful complexity.

For a lucid, complete, and most impartial statement of the

claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions.

13. It was reserved for Hamilton to discover the use of √√—1 as a geometric reality, tied down to no particular direction in space, and this use was the foundation of the singularly elegant, yet enormously powerful, Calculus of Quaternions.

While all other schemes for using √1 to indicate direction make one direction in space expressible by real numbers, the remainder being imaginaries of some kind, leading in general to equations which are heterogeneous; Hamilton makes all directions in space equally imaginary, or rather equally real, thereby ensuring to his Calculus the power of dealing with space indifferently in all directions.

In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its reference lines solely from the problem it is applied to.

14. But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, which are in fact a mere particular case of Quaternions in which most of the distinctive features are lost. We shall find in a little that it is capable of soaring above these entirely, after having employed them in its establishment; and, indeed, as the inventor's works amply prove, it can be established, ab initio, in various ways, without even an allusion to Cartesian Geometry. As this work is written for students acquainted with at least the elements of the Cartesian method, we keep to the firstmentioned course of exposition; especially as we thereby avoid some reasoning which, though rigorous and beautiful, might be apt, from its subtlety, to prove repulsive to the beginner.

We commence, therefore, with some very elementary geometrical ideas.