# 1911 Encyclopædia Britannica/Quaternions

**QUATERNIONS,** in mathematics. The word “quaternion” properly means “ a set of four.” In employing such a word to denote a new mathematical method, Sir W. R. Hamilton was probably influenced by the recollection of its Greek equivalent, the Pythagorean Tetractys (Τετρακτύς, the number four), the mystic source of all things. Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space. From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space—or, what comes to the same thing, as the factor, or operator, which changes one directed line into another. Its analytical definition will appear later.

*History.*—The evolution of quaternions belongs in part to
each of two weighty branches of mathematical history—the
interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry. Sir W. R. Hamilton was led to his great invention by keeping geometrical applications constantly before him while he endeavoured to give a real significance to √−1. We will
therefore confine ourselves, so far as his predecessors are concerned,
to attempts at interpretation which had geometrical
applications in view.

One geometrical interpretation of the negative sign of algebra
was early seen to be mere reversal of direction along a line.
Thus, when an image is formed by a plane mirror, the distance
of any point in it from the mirror is simply the negative of that
of the corresponding point of the object. Or if motion in one
direction along a line be treated as positive, motion in the
opposite direction along the same line is negative. In the case
of time, measured from the Christian era, this distinction is at
once given by the letters A.D. or B.C., prefixed to the date. And
to find the position, in time, of one event relatively to another,
we have only to subtract the date of the second (taking account
of its sign) from that of the first. Thus^{[1]} to find the interval
between the battles of Marathon (400 B.C.) and Waterloo
(Ap. 1815) we have

+1815−(−490)＝2305 years.

And it is obvious that the same process applies in all cases in which we deal with quantities which may be regarded as of one directed dimension only, such as distances along a line, rotations about an axis, &c. But it is essential to notice that this is by no means necessarily true of operators. To turn a line through a certain angle in a given plane, a certain operator is required; but when we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle. This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed,—which merely means that−1 is commutative with all other quantities.

John Wallis seems to have been the first to push this idea
further. In his *Treatise of Algebra* (1685) he distinctly proposes
to construct the imaginary roots of a quadratic equation by
going out of the line on which the roots, if real, would have been
constructed.

In 1804 the Abbé Buée (*Phil. Trans.*, 1806), apparently
without any knowledge of Wallis's work, developed this idea so
far as to make it useful in geometrical applications. He gave, in
fact, the theory of what in Hamilton's system is called Composition
of Vectors in one plane—*i.e.* the combination, by + and
−, of complanar directed lines. His constructions are based on
the idea that the imaginaries ± √−1 represent a unit line, and
its reverse, perpendicular to the line on which the real units
± 1 are measured. In this sense the imaginary expression
*a* + *b* √−1 is constructed by measuring a length *a* along the
fundamental line (for real quantities), and from its extremity a
line of length *b* in some direction perpendicular to the fundamental
line. But he did not attack the question of the representation
of products or quotients of directed lines. The step he
took is really nothing more than the kinernatical principle of
the composition of linear velocities, but expressed in terms of the
algebraic imaginary.

In 1806 (the year of publication of Buée’s paper) Jean Robert
Argand published a pamphlet^{[2]} in which precisely the same
ideas are developed, but to af considerably greater extent. For
an interpretation is assigned to the product of two directed lines
in one plane, when each is expressed as the sum of a real and an
imaginary part. This product is interpreted as another directed
line, forming the fourth term of a proportion, of which the first
term is the real (positive) unit-line, and the other two are the
factor-lines. Argand's work remained unnoticed until the
question was again raised in Gergonne’s *Annales*, 1813, by
J. F. Français. This writer stated that he had found the germ
of his remarks among the papers of his deceased brother, and
that they had come from Legendre, who had himself received
them from some one unnamed. This led to a letter from
Argand, in which he stated his communications with Legendre,
and gave a résumé of the contents of his pamphlet. In a further
communication to the Annales, Argand pushed on the applications
of his theory. He has given by means of it a simple proof
of the existence of n roots, and no more, in every rational
algebraic equation of the nth order with real coefficients. About
1828 John Warren (1796–1852) in England, and C. V. Mourey in
France, independently of one another and of Argand, reinvented
these modes of interpretation; and still later, in the writings of
Cauchy, Gauss and others, the properties of the expression
*a*+*b*√−1 were developed into the immense and most important
subject now called the *theory of complex numbers* (see Number).
From the more purely symbolical view it was developed by
Peacock, De Morgan, &c., as *double algebra*.

Argand’s method may be put, for reference, in the following
form. The directed line whose length is *a*, and which makes an
angle θ with the real (positive) unit line, is expressed by *a*(cos θ+*i* sin θ),
where *i* is regarded as + √−1. The sum of two such lines (formed
by adding together the real and the imaginary parts of two such
expressions) can, of course, be expressed as a third directed line)—the
diagonal of the parallelogram of which they are conterminous
sides. The product, P, of two such lines is, as we have seen, given
by

1:*a*(cosθ+*i*sin θ) : : *a*′(cosθ′+*i* sin θ′):P,

or P＝*aa*′{cos (θ+θ′)+*i* sin (θ+θ′)}.

Its length is, therefore, the product of the lengths of the factors,
and its inclination to the real unit is the sum of those of the factors.
If we write the expressions for the two lines in the form A+B*i*,
A′+B′*i*, the product is AA′−BB′+*i*(AB′+BA′); and the fact
that the length of the product line is the product of those of the
factors is seen in the form

(A^{2}+B^{2}) (A′^{2}+B′^{2})＝(AA′−BB′)2+(AB′+BA′)^{2}.

In the modern theory of complex numbers this is expressed by
saying that the *Norm* of a product is equal to the product of the
norms of the factors.

Argand's attempts to extend his method to space generally
were fruitless. The reasons will be obvious later; but we
mention them just now because they called forth from F. T.
Servois (Gergonne’s *Annales*, 1813) a very remarkable comment,
in which was contained the only yet discovered trace of an
anticipation of the method of Hamilton. Argand had been
led to deny that such an expression as *i ^{i}* could be expressed
in the form A+B

*i*,—although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of −π/2. Servois says, with reference to the general representation of a directed line in space:—

“L’analogie semblerait exiger que le trinôme fût de la forme
*p* cos α+*q* cos β + *r* cos γ : α, β, γ étant les angles d’une droite avec
trois axes rectangulaires; et qu'on eût

(*p* cos α + *q* cos β + *r* cos γ)(*p*′ cos α + *q*′ cos β + *r* ′ cos γ)

＝cos^{2}α+cos^{2} β +cos^{2}＝1. Les valeurs de *p*, *g*, *r*, *p*′, *q*′, *r* ′, qui
satisferaient at cette condition seraient absurdes; mais seraient-elles
imaginaires, deductibles at la forme générale A+B √−1? Voila
une question d'analyse fort singuliere que je soumets a vos lumiéres.
La simple proposition que jevous en fais suffit pour vous faire voir
que je ne crois point que toute fonction analytique non réelle soit
vraiment reductible à la forme A+B √−1.”

As will be seen later, the fundamental *i*, *j*, *k* of quaternions,
with their reciprocals, furnish a set of six quantities which
satisfy the conditions imposed by Servois. And it is quite
certain that they cannot be represented by ordinary imaginaries.

Something far more closely analogous to quaternions than
anything in Argand's work ought to have been suggested by
De Moivre's theorem (1730). Instead of regarding, as Buée
and Argand had done, the expression *a*(cos θ+*i* sinθ) as a
directed line, let us suppose it to represent the operator Which,
when applied to any line in the plane in which θ is measured,
turns it in that plane through the angle θ, and at the same
time increases its length in the ratio *a* : 1. From the new
point of view we see at once, as it were, why it is true that

(cos θ+*i* sin θ)^{m}＝cos *m*θ + *i* sin *m*θ.

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,”^{[3]} 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:—

(B_{1}, B_{2})−(A_{1}, A_{2}) ＝(B_{1}−A_{1}, B_{2}−A_{2})＝(α, β);

(a, b) (α, β)＝(aα−bβ, bα+a β).^{[4]}

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:—

*i*^{2}＝*j*^{2}＝k^{2}＝−1, *ij*＝−*ji*＝*k*, *jk*＝−*kj*＝*i*, *ki*＝−*ik*＝*j*.^{[5]}

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+*i*B+*j*C+*k*D,

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
are parallel, or θ=0, the product, which is now the square of any
(unit) line is −1. And when the two factor lines are at right angles
to one another, or θ=π/2, the product is simply *ix*″+*jy*″+*kz*″, the
unit line perpendicular to both. Hence, and in this lies the main
element of the symmetry and simplicity of the quaternion calculus,
all systems of three mutually rectangular unit lines in space have
the same properties as the fundamental system *i*, *j*, *k*. In other
words, if the system (considered as rigid) be made to turn about
till the first factor coincides with i and the second with *j*, the product
will coincide with *k*. This fundamental system, therefore,
becomes unnecessary; and the quaternion method, in every case,
takes its reference lines solely from the problem to which it is
applied. It has therefore, as it were, a unique internal character
of its own.

Hamilton, having gone thus far, proceeded to evolve these results
from a characteristic train of a priori or metaphysical reasoning.
Let it be supposed that the product of two directed lines is something
which has quantity; *i.e.* it may be halved, or doubled, for
instance. Also let us assume (*a*) space to have the same properties
in all directions, and make the convention (*b*) that to change the
sign of any one factor changes the sign of a product. Then the
product of two lines which have the same direction cannot be, even
in part, a directed quantity. For, if the directed gart have the same
direction as the factors, (*b*) shows that it will e reversed by reversing
either, and therefore will recover its original direction when
both are reversed. But this would obviously be inconsistent
with (*a*). If it be perpendicular to the factor lines, (a) shows that
it must have simultaneously every such direction. Hence it must
be a mere number.

Again, the product of two lines at right angles to one another
cannot, even in part, be a number. For the reversal of either factor
must, by (*b*), change its sign. But, if we look at the two factors
in their new position by the light of (*a*), we see that the sign must
not change. But there is nothing to prevent its being represented
by a directed line if, as further applications of (*a*) and (*b*) show we
must do, we take it perpendicular to each of the factor lines. Hamilton
seems never to have been quite satisfied with the apparent heterogeneity
of a quaternion, depending as it does on a numerical and
a directed part. He indulged in a great deal of speculation as to
the existence of an extra-spatial unit, which was to furnish the
*raison d’étre* of the numerical part, and render the quaternion
homogeneous as well as linear. But for this we must refer to his
own works.

Hamilton was not the only worker at the theory of sets. The
year after the first publication of the quaternion method, there
appeared a work of great originality, by Grassmann^{[6]} in which
results closely analogous to some of those of Hamilton were
given. In particular, two species of multiplication (“inner”
and “outer ”) of directed lines in one plane were given. The
results of these two kinds of multiplication correspond respectively
to the numerical and the directed parts of Hamilton’s
quaternion product. But Grassmann distinctly states in his
preface that he had not had leisure to extend his method to
angles in space. Hamilton and Grassmann, while their earlier
work had much in common, had very different objects in view.
Hamilton had geometrical application as his main object; when
he realized the quaternion system, he felt that his object was
gained, and thenceforth confined himself to the development
of his method. Grassmann’s object seems to have been, all
along, of a much more ambitious character, viz. to discover, if
possible, a system or systems in which every conceivable mode
of dealing with sets should be included. That he made very
great advances towards the attainment of this object all will
allow; that his method, even as completed in 1862, fully
attains it is not so certain. But his claims, however great they
may be, can in no way conflict with those of Hamilton, whose
mode of multiplying couples (in which the “inner” and “outer”
multiplication are essentially involved) was produced in 1833,
and whose quaternion system was completed and published
before Grassmann had elaborated for press even the rudimentary
portions of his own system, in which the veritable difficulty of
the whole subject, the application to angles in space, had not
even been attacked. Grassmann made in 1854 a somewhat
savage onslaught on Cauchy and De St Venant, the former of
whom had invented, while the latter had exemplified in application,
the system of “*clefs algébriques*,” which is almost precisely
that of Grassmann. But it is to be observed that Grassmann,
though he virtually accused Cauchy of plagiarism, does not
appear to have preferred any such charge against Hamilton.
He does not allude to Hamilton in the second edition of his
work. But in 1877, in the *Mathematische Annalen*, xii., he
gave a paper “ On the Place of Quaternions in the *Ausdehnungslehre*,”
in which he condemns, as far as he can, the nomenclature
and methods of Hamilton.

There are many other systems, based on various principles, which
have been given for application to geometry of directed lines, but
those which deal with products of lines are all of such complexity
as to be practically useless in application. Others, such as the
*Barycentrische Calcül* of Möbius, and the *Méthode des equipollences*
of Bellavitis, give elegant modes of treating space problems, so
long as we confine ourselves to projective geometry and matters of
that order; but they are limited in their field, and therefore need
not be discussed here. More general systems, having close analogies
to quaternions, have been given since Hamilton's discovery was
published. As instances we may take Goodwin's and O'Brien's
papers in the *Cambridge Philosophical Transactions* for 1849. (See
also Algebra: *special kinds*.)

*Relations to other Branches of Science*.-The above narrative
shows how close is the connexion between quaternions and the
ordinary Cartesian space-geometry. Were this all, the gain by
their introduction would consist mainly in a clearer insight into
the mechanism of co-ordinate systems, rectangular or not—a
very important addition to theory, but little advance so far as
practical application is concerned. But, as yet, we have not
taken advantage of the perfect symmetry of the method.
When that is done, the full value of Hamilton’s grand step
becomes evident, and the gain is quite as extensive from the
practical as from the theoretical point of view. Hamilton, in
fact, remarks,^{[7]} “I regard it as an inelegance and imperfection
in this calculus, or rather in the state to which it has hitherto
been unfolded, whenever it becomes, or *seems* to become,
necessary to have recourse . . . to the resources of ordinary
algebra, for the *solution of equations in quaternions*.” This
refers to the use of the *x*, *y*, *z* co-ordinates,—associated, of course,
with *i*, *j*, *k*. But when, instead of the highly artificial expression
*ix*+*jy*+*kz*, to denote a finite directed line, we employ a single
letter, α. (Hamilton uses the Greek alphabet for this purpose),
and find that we are permitted to deal with it exactly as we
should have dealt with the more complex expression, the
immense gain is at least in part obvious. Any quaternion may
now be expressed in numerous simple forms. Thus we may
regard it as the sum of a number and a line, *a*+α, or as the
product, βγ, or the quotient, δε^{−1}, of two directed lines, &c.,
while, in many cases, we may represent it, so far as it is required,
by a single letter such as *q*, *r*, &c.

Perhaps to the student there is no part of elementary mathematics
so repulsive as is spherical trigonometry. Also, everything
relating to change of systems of axes, as for instance in
the kinematics of a rigid system, where we have constantly to
consider one set of rotations with regard to axes fixed in space,
and another set with regard to axes fixed in the system, is a
matter of troublesome complexity by the usual methods. But
every quaternion formula is a proposition in spherical (sometimes
degrading to plane) trigonometry, and has the full advantage of
the symmetry of the method. And one of Hamilton’s earliest
advances in the study of his system (an advance independently
made, only a few months later, by Arthur Cayley) was the
interpretation of the singular operator *q*()*q*^{−1}, where *q* is a
quaternion. Applied to any directed line, this operator at once
turns it, conically, through a definite angle, about a definite
axis. Thus rotation is now expressed in symbols at least as
simply as it can be exhibited by means of a model. Had
quaternions effected nothing more than this, they would still
have inaugurated one of the most necessary, and apparently
impracticable, of reforms.

The physical properties of a heterogeneous body (provided
they vary continuously from point to point) are known to, depend,
in the neighbourhood of any one point of the body, on a quadric
function of the co-ordinates with reference to that point. The
same is true of physical quantities such as potential, temperature,
&c., throughout small regions in which their variations are
continuous; and also, without restriction of dimensions, of
moments of inertia, &c. Hence, in addition to its geometrical
applications to surfaces of the second order, the theory of quadric
functions of position is of fundamental importance in physics.
Here the symmetry points at once to the selection of the three
principal axes as the directions for *i*, *j*, *k*; and it would appear
at first sight as if quaternions could not simplify, though they
might improve in elegance, the solution of questions of this
kind. But it is not so. Even in Hamilton's earlier work it
was shown that all such questions were reducible to the solution
of linear equations in quaternions; and he proved that this, in
turn, depended on the determination of a certain operator,
which could be represented for purposes of calculation by a
single symbol. The method is essentially the same as that
developed, under the name of “matrices,” by Cayley in 1858;
but it has the peculiar advantage of the simplicity which is
the natural consequence of entire freedom from conventional
reference lines.

Sufficient has already been said to show the close Connexion
between quaternions and the theory of numbers. But one
most important connexion with modern physics must be pointed
out. In the theory of surfaces, in hydro kinetics, heat-conduction,
potentials, &c., we constantly meet with what is called
“Laplace’s operator,” viz. *d ^{2}dx^{2}*+

*d*

^{2}

*dy*

^{2}+

*d*

^{2}

*dz*

^{2}. We know that this is an invariant;

*i.e.*it is independent of the particular directions chosen for the rectangular co-ordinate axes. Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression

*d*

*dx*+

*d*

*dy*+

*d*

*dz*could be, like

*ix*+

*jy*+

*kz*, effectively expressed by a single letter. He chose for this purpose V. And we now see that the square of V is the negative of Laplace's operator; while V itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity. Thus, applied to a potential, it gives the direction and magnitude of the force; to a distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the flux of heat, &c. No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like Clerk-Maxwell (in his Electricity and Magnetism). Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydro kinetics and in electrodynamics into the pregnant language of quaternions. (P. G. T.)

Supplementary Considerations.-There are three fairly well marked
stages of development in quaternions as a geometrical
method. (1) Generation of the concept through imaginaries
and development into a method applicable to Euclidean
geometry. This was the work of Hamilton himself, and the
above account (contributed to the 9th ed. of the *Ency. Brit.* by
Professor P. G. Tait, who was Hamilton's pupil and after him
the leading exponent of the subject) is a brief résumé of this
first, and by far the most important and most difficult, of the
three stages. (2) Physical applications. Tait himself may be
regarded as the chief contributor to this stage. (3) Geometrical
applications, different in kind from, though more or less allied
to, those in connexion with which the method was originated.
These last include (*a*) C. J. Joly's projective geometrical applications
starting from the interpretation of the quaternion as a
point-symbol;^{[8]} these applications may be said to require no
addition to the quaternion algebra; (*b*) W. K. Clifford's bi-quaternions
and G. Combebiac's tri-quaternions, which require
the addition of quasi-scalars, independent of one another and of
true scalars, and analogous to true scalars. As an algebraic
method quaternions have from the beginning received much
attention from mathematicians. An attempt has recently been
made under the name of multenions to systematize this algebra.

We select for description stage (3) above, as the most characteristic
development of quaternions in recent years. For
(3) (*a*) we are constrained to refer the reader to Joly’s own
*Manual of Quaternions* (1905).

The impulse of W. K. Clifford in his paper of 1873 (“Preliminary
Sketch of Bi-Quaternions,” Mathematical Papers,
p. 181) seems to have come from Sir R. S. Ball's paper on the
Theory of Screws, published in 1872. Clifford makes use of a
quasi-scalar w, commutative with quaternions, and such that if
p, q, &c., are quaternions, when *p*+ω*q*=*p*′+ω*q*′, then necessarily
p=p', q=q'. He considers two cases, viz. ω^{2}=1 suitable
for non-Euclidean space, and ω^{2}=0 suitable for Euclidean
space; we confine ourselves to the second, and will call the
indicated bi-quaternion *p*+ω*q* an *octonion*. In octonions the
analogue of Hamilton's vector is localized to the extent of being
confined to an indehnitely long axis parallel to itself, and is
called a rotor; if p is a rotor then wp is parallel and equal to ρ,
and, like Hamilton's vector, ωρ is not localized; ωρ is therefore
called a vector, though it differs from Hamilton's vector in that
the product of any two such vectors ωρ and ωσ is zero because
ω^{2}=0 . ρ+ωσ where ρ, σ are rotors (*i.e.* ρ is a rotor and ωσ a
vector), is called a motor, and has the geometrical significance of
Ball's wrench upon, or twist about, a screw. Clifford considers
an octonion *p*+ω*q* as the quotient of two motors ρ+ωσ, ρ′+ωσ′.
This is the basis of a method parallel throughout to the
quaternion method; in the specification of rotors and motors
it is independent of the origin which for these purposes the
quaternion method, pure and simple, requires.

Combebiac is not content with getting rid of the origin in these limited circumstances. The fundamental geometrical conceptions are the point, line and plane. Lines and complexes thereof are sufficiently treated as rotors and motors, but points and planes cannot be so treated. He glances at Grassmann's methods, but is repelled because he is seeking a unifying principle, and he finds that Grassmann offers him not one but many principles. He arrives at the tri-quaternion as the suitable fundamental concept.

We believe that this tri-quaternion solution of the very
interesting problem proposed by Combebiac is the best one.
But the first thing that strikes one is that it seems unduly
complicated. A point and a plane fix a line or axis, viz.
that of the perpendicular from point to plane, and therefore
a calculus of points and planes is *ipso facto* a calculus of lines
also. To fix a weighted point and a weighted plane in
Euclidean space we require 8 scalars, and not the 12 scalars
of a tri-quaternion. We should expect some species of bi quaternion
to suffice. And this is the case. Let η, ω be two
quasi-scalars such that η^{2}=η, ωη=ω, ηω=ω^{2}=0. Then the bi quaternion
oyq-I-wr suffices. The plane is of vector magnitude
éVq, its equation is 12S*pq*=S*r*, and its expression is the
bi-quaternion nVq+wSr; the point is of scalar magnitude
12S*q*, and its position vector is β, where 12Vβ*q*=V*r* (or what is
the same, β= [V*r*+*q*.V*r*. *q*^{−1}]/S*q*), and its expression is r;Sq+wVr.
(Note that the 12 here occurring is only required to ensure
harmony with tri-quaternions of which our present bi-quaternions,
as also octonions, are particular cases.) The
point whose position vector is V*rq*^{−1} is on the axis and may
be called the centre of the bi-quaternion; it is the centre of a
sphere of radius S*rq*^{−1} with reference to which the point and
plane are in the proper quaternion sense polar reciprocals,
that is, the position vector of the point relative to the centre
is S*rq*^{−1}. Vg/Sq, and that of the foot of perpendicular from
centre on plane is S*rq*^{−1}. S*q*/V*q*, the product being the (radius)',
that is (S*rq*^{−1})^{2}. The axis of the member xQ+x'Q' of the
second-order complex Q, Q′ (where Q=η*q*+ω*r*, Q′=η*q*′+ω*r* ′
and *x*, *x*′ are scalars) is parallel to a fixed plane and intersects
a fixed transversal, viz. the line parallel to q'q'1 which
intersects the axes of Q and Q'; the plane of the member
contains a fixed line; the centre is on a fixed ellipse which intersects' the transversal; the axis is on a fixed ruled surface
to which the plane of the ellipse is a tangent plane, the ellipse
being the section of the ruled surface by the plane; the ruled
surface is a cylindroid deformed by a simple shear parallel
to the transversal. In the third-order complex the centre
locus becomes a finite closed quartic surface, with three (one
always real) intersecting nodal axes, every plane section of
which is a trinodal quartic. The chief defect of the geometrical
properties of these bi-quaternions is that the ordinary algebraic
scalar finds no place among them, and in consequence Q^{−1} is
meaningless.

Putting 1−η=ξ we get Combebiac's tri-quaternion under
the form Q=ξ*p*+η*q*+ω*r*. This has a reciprocal Q^{−1}=ξ*p*^{−1}=η*q*^{−1}
−ω*p*^{−1}*rq*^{−1}, and a conjugate KQ (such that K[QQ′]=
KQ'KQ, K[KQ]=Q) given by KQ=§ Kq+r;Kp+wK1'; the
product QQ' of Q and Q' is Zpp'-l-17qq'+w(p1'-l-rq'); the
quasi-vector § (1-K)Q is Combebiac's linear element and may
be regarded as a point on a line; the quasi-scalar (in a different
sense from the rest of this article) § (1-l-K)Q is Cornbebiac's
scalar (S*p*+S*q*)+Combebiads plane. Combebiac does not use
K; and in place of 5,17 he uses }, L=7]"E, so that /J.Z=I, (.0}.l.= -pw
=w, w2=o. Combebiac's tri-quaternion may be regarded from
many simplifying points of view. Thus, in place of his general
tri-quaternion we might deal with products of an odd number
of point-plane-scalars (of form μ*q*+ω*r*) which are themselves
point-plane»scalars; and products of an even number which
are octonions; the quotient of two point-plane-scalars would
be an octonion, of two octonions an octonion, of an octonion
by a point-plane-scalar or the inverse a point-plane-scalar.
Again a unit point pi may be regarded as by multiplication
changing (*a*) from octonion to point-plane-scalar, (*b*) from
point-plane-scalar to octonion, (*c*) from plane-scalar to linear
element, (*d*) from linear element to plane-scalar.

If Q=ξ*p*+η*q*+ω*r* and we put Q=(1+12ω*t*)(ξ*p*+η*q*)×
(1+12ω*t*)^{−1} we find that the quaternion *t* must be 2*f*(*r*)/*f*(*q*−*p*),
where *f*(*r*)=*rq*−K*pr*. The point ρ=V*t* may be called the
centre of Q and the length St may be called the radius. If
Q and Q' are commutative, that is, if QQ'1=Q'Q, then Q and
Q' have the same centre and the same radius. Thus Q",
Q, Q', Q3, . . . have a common centre and common radius.
Q and KQ have a common centre and equal and opposite
radii; that is, the t of KQ is the negative conjugate of that
of Q. When Su=o, (I'i'%(.|.)1l) () (I-l-%(.0ll) '1 is an operator
which shifts (without further change) the tri-quaternion
operand an amount given by n in direction and distance.

Bibliography.—In 1904 Alexander Macfarlane published a
*Bibliography of Quaternions and allied systems of Mathematics* for
the International Association for promoting the study of Quaternions
and allied systems of Mathematics (Dublin University Press);
the pamphlet contains 86 pages. In 1899 and 1901 Sir W. R.
Hamilton's classical Elements of Quaternions of 1866 was republished
under C. J. Joly's editorship, in two volumes (London). Joly adds
valuable notes and thirteen important appendices. In 1890 the
3rd edition of P. G. Tait's *Elementary Treatise on Quaternions*
appeared (Cambridge). In 1905 C. J. Joly published his *Manual*
*of Quaternions* (London); the valuable contents of this are doubled
by copious so-called examples; every earnest student should take
these as part of the main treatise. The above three treatises may
be regarded as the great storehouses; the handling of the subject
is very different in the three. The following should also be
mentioned: A. McAulay, *Octonions, a development of Clifford's*
*Bi-quaternions* (Cambridge, 1898); G. Combebiac, *Calcul des*
*triquaternions* (Paris, 1902); Don Francisco Pérez de Munoz,
*Introduccion al estudio del calculo de Cuaterniones y otras Algebras*
*especiales* (Madrid, 1905); A. McAulay, *Algebra after Hamilton, or*
*Multenions* (Edinburgh, 1908). (A. McA.)

- ↑ Strictly speaking, this illustration of Tait’s is in error by unity because in our calendar there is no year denominated zero. Thus the interval between June the first of 1 B.C. and June the first of 1 A.D. is one year, and not two years as the text implies. (A.McA.)
- ↑
*Essai sur une manière de représenter les Quantités Imaginaires**dans les Constructions Géométriques*. A second edition was published by J. Hoüel (Paris, 1874). There is added an important Appendix. consisting of the papers from Gergonne's*Annales*which are referred to in the text above. Almost nothing can, it seems. be learned of Argand's private life, except that in all probability he was born at Geneva in 1768. - ↑
*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. - ↑ Die Ausdehnungslehre, Leipsic, 1844; 2nd ed., vollstandig und in strenger Form bearbeitet, Berlin. 1862. See also the collected works of Mobius, and those of Clifford, for a general explanation of Grassmann's method.
- ↑
*Lectures on Quaternions*, § 513. - ↑ It appears from Joly’s and Macfarlane’s references that J. B. Shaw, in America, independently of Joly, has interpreted the quaternion as a point-symbol.