For this equation merely states that m turnings of a line
through successive equal angles, in one plane, give the same
result as a single turning through rn times the common angle.
To make this process applicable to any plane in space, it is
clear that we must have a special value of i for each such plane.
In other words, a unit line, drawn in any direction whatever,
must have −1 for its square. In such a system there will be
no line in space specially distinguished as the real unit line:
all will be alike imaginary, or rather alike real. We may
state, in passing, that every quaternion can be represented as
a (cos θ+ π sin θ), where a is a real number, θ a real angle,
and π a directed unit line whose square is −1. Hamilton
took this grand step, but, as we have already said, without
any help from the previous work of De Moivre. The course
of his investigations is minutely described in the preface to
his first great work (Lectures on Quaternions, 1853) on the
subject. Hamilton, like most of the many inquirers who
endeavoured to give a real interpretation to the imaginary of
common algebra, found that at least two kinds, orders or
ranks of quantities were necessary for the purpose. But,
instead of dealing with points on a line, and then wandering
out at right angles to it, as Buée and Argand had done, he
chose to look on algebra as the science of “pure time,”[1] and
to investigate the properties of “ sets ” of time-steps. In its
essential nature a set is a linear function of any number of
“distinct” units of the same species. Hence the simplest
form of a set is a “couple”; and it was to the possible laws
of combination of couples that Hamilton first directed his
attention. It is obvious that the way in which the two
separate time-steps are involved in the couple will determine
these laws of combination. But Hamilton's special object
required that these laws should be such as to lead to certain
assumed results; and he therefore commenced by assuming
these, and from the assumption determined how the separate
time-steps must be involved in the couple. It we use Roman
letters for mere numbers, capitals for instants of time, Greek
letters for time-steps, and a parenthesis to denote a couple,
the laws assumed by Hamilton as the basis of a system were as
follows:—
(B1, B2)−(A1, A2) ＝(B1−A1, B2−A2)＝(α, β);
(a, b) (α, β)＝(aα−bβ, bα+a β).[2]
To show how we give, by such assumptions, a real interpretation to the ordinary algebraic imaginary, take the simple case a＝0, b＝1, and the second of the above formulae gives
(0, 1)(α, β)＝(β, α)
Multiply once more by the number-couple (o, 1), and we have (0,1)(0,1)(α,β)=(0, 1)(−α,β)=(−α,−β)=(−1,0)(α,β)=−(α,β)
Thus the number-couple (0, 1), when twice applied to a step-couple, simply changes its sign. That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets the vast possessions of his vizier and of his barber. He determines to rob them both (an operation which may be very satisfactorily expressed by −1); but, being a wag, he chooses his own way of doing it. He degrades his vizier to the office of barber, taking all his goods in the process; and makes the barber his vizier. Next day he repeats the operation. Each of the victims has been restored to his former rank, but the operator −1 has been applied to both.
Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy+jz, by which he proposed to represent the directed line in space whose projections on the co-ordinate axes are x, y, z. The composition of two such lines by the algebraic addition of their several projections agreed with the assumption of Buée and Argand for the case of coplanar lines. But, assuming the distributive principle, the product of two lines appeared to give the expression
xx′ −yy′ −zz′ +i(yx′ +xy′) +j(xz′ +zx′) +ij(yz′ +zy′).
For the square of j, like that of i, was assumed to be negative unity. But the interpretation of ij presented a difficult yin fact the main difficulty of the whole investigation-and it is specially interesting to see how Hamilton attacked it. He saw that he could get a hint from the simpler case, already thoroughly discussed, provided the two factor lines were in one plane through the real unit line. This requires merely that
y : z :: y′ : z′ ; or yz′−zy′＝0;
but then the product should be of the same form as the separate factors. Thus, in this special case, the term in ij ought to vanish. But the numerical factor appears to be yz′+zy′, while it is the quantity yz′−zy′ which really vanishes. Hence Hamilton was at first inclined to think that ij must be treated as nil. But he soon saw that “a less harsh supposition” would suit the simple case. For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyzi′ and jizy′, the term would vanish of itself when the factor lines are coplanar provided ij＝−ji, for it would then assume the form ij(yz′ − zy′). He had now the following expression for the product of any two directed lines:—
xx′ −yy′ −zz′ +i(yx′ +xy′) +j(xz′ +zx′) +ij(yz′ +zy′).
But his result had to be submitted to another test, the Law of the Norms. As soon as he found, by trial, that this law was satisfied, he took the final step. “This led me,” he says, “to conceive that perhaps, instead of seeking to confine ourselves to triplets, . . . we ought to regard these as only imperfect forms of Quaternions, . . . and that thus my old conception of sets might receive a new and useful application.” In a very short time he settled his fundamental assumptions. He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication:—
i2＝j2＝k2＝−1, ij＝−ji＝k, jk＝−kj＝i, ki＝−ik＝j.[3]
And now the product of two quaternions could be at once expressed as a third quaternion, thus—
(a+ib+jc+kd)(a′+ib′+jc′+kd′)＝A+iB+jC+kD,
where
A＝aa′ − bb′ − cc′ − dd′,
B＝ab′+ba′+cd′ − dc′,
C＝ac′+ca′+db′ − bd,
D＝ad′+da′+bc′ − cb.
Hamilton at once found that the Law of the Norms holds,—not being aware that Euler had long before decomposed the product of two sums of four squares into this very set of four squares. And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion
- (xx'+yy' +zz') +i(yz' -zy') +j(zx' -xz') +k(xy' -yx')
To any one acquainted, even to a slight extent, with the elements of Cartesian geometry of three dimensions, a glance at the extremely suggestive constituents of this expression shows how justly Hamilton was entitled to say: “When the conception had been so far unfolded and fixed in my mind, I felt that the new instrument for applying calculation to geometry, for which I had so long sought, was now, at least in part, attained.” The date of this memorable discovery is October 16, 1843.
Suppose, for simplicity, the factor-lines to be each of unit length. Then x, y, z, x', y', z' express their direction-cosines. Also, if 6 be the angle between them, and x", y”, z” the direction-cosines of a line perpendicular to each of them, we have xx'+yy'+zz'=cos θ, yz'-zy”=x" sin θ, &c., so that the product of two unit lines is now expressed as −cos θ+(ix″+jy″+kz″) sin θ. Thus, when the factors
- ↑ Theory of Conjugate Functions, or Algebraic Couples, with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, read in 1833 and 1835. and published in Trans. R. I. A. xvii. ii. (1835).
- ↑ Compare these with the long-subsequent ideas of Grassmann.
- ↑ It will be easy to see that, instead of the last three of these, we may write the single one ijk = −1.
