By Sir William R. Hamilton

[*Proceedings of the Royal Irish Academy, Nov. 13, 1843,
vol. 2, 424-434*]

It is known to all students of algebra that an imaginary equation of
the form has been employed so as to conduct to very varied
and important results. Sir Wm. Hamilton proposes to consider some of
the consequences which result from the following system of imaginary
equations, or equations between a *system of three different
imaginary quantities*:

no linear relation between *i*, *j*, *k* being supposed to exist, so
that the equation

in which

and *w*, *x*, *y*, *z*, *w*', *x*', *y*', *z*' are real, is equivalent
to the four separate equations

Sir W. Hamilton calls an expression of the form a
*quaternion*; and the four real quantities *w*, *x*, *y*, *z* he
calls the *constituents* thereof. Quaternions are added or
subtracted by adding or subtracting their constituents, so that

Their multiplication is, in virtue of the definitions (A) (B) (C), effected by the formulae

which give

and therefore

if we call the positive quantity

the *modulus* of the quaternion . The modulus of the
product of any two quaternions is therefore equal to the product of
the moduli. Let

then, because the equations (D) give

we have

Consider *x*, *y*, *z* as the rectangular coordinates of a point of
space, and let be the point where the radius vector of
*x*, *y*, *z* (prolonged if necessary) intersects the spheric surface
described about the origin with a radius equal to unity; call
the *representative point* of the
quaternion , and let the polar coordinates and ,
which determine upon the sphere, be called the
*co-latitude* and the *longitude* of the representative
point , or of the quaternion itself; let also the
other angle be called the *amplitude* of the
quaternion; so that a quaternion is completely determined by its
modulus, amplitude, co-latitude and longitude. Construct the
representative points and , of the other
factor , and of the product ; and complete the
spherical triangle by drawing the arcs
, , . Then, the
equations (G) become

and consequently shew that the angles of the triangle are

these angles are therefore respectively equal to the amplitudes of the
factors, and the supplement (to two right angles) of the amplitude of
the product. The equations (D) show, further, that the
*product-point* is to the right or left of the
*multiplicand-point* , with respect to the
*multipler-point* , according as the semiaxis of +*z*
(or its intersection with the spheric surface) is to the right or left
of the semiaxis of +*y*, with respect to the semiaxis of +*x*: that
is, according as the positive direction of rotation in longitude is
towards the right or left. A change in the order of the two
quaternion-factors would throw the product-point from the
right to the left, or from the left to the right of .

It results from these principles, that if
be any spherical triangle; if, also,
be the rectangular coordinates of ,
those of , and
of ,
the centre of the sphere being origin, and the radius being unity; and
if the rotation round +*x* from +*y* to +*z* be of the same
(right-handed or left-handed) character as that round from
to ; then the following formula of
multiplication, according to the rules of quaternions, will hold good:

Developing and decomposing this imaginary or symbolic formula (I), we find that it is equivalent to the system of the four following real equations, or equations between real quantities:

Of these equations (K), the first is only an expression of the
well-known theorem, already employed in these remarks, which serves to
connect a side of any spherical triangle with the three angles
thereof. The three other equations (K) are an expression of another
theorem (which possibly is new), namely that a force
, directed from the centre of the sphere to the
point , is statically equivalent to the system of three
other forces, one directed to , and equal to
, another directed to , and
equal to , and the third equal to
, and directed
towards that pole of the arc , which lies at the same
side of this arc as . It is not difficult to prove this
theorem otherwise; but it may be regarded as interesting to see that
the four equations (K) are included so simply in the one formula (I)
of multiplication of quaternions, and are obtained so easily by
developing and decomposing that formula, according to the fundamental
definitions (A) (B) (C). A new sort of algorithm, or calculus, for
spherical trigonometry, appears to be thus given, or indicated. And
by supposing the three corners of the spherical triangle
to tend indefinitely to close up in that
one point which is the intersection of the spheric surface with the
positive semiaxis of *x*, each coordinate will tend to become
= 1, and each and to vanish, while the sum of the
three angles will tend to become ; so that the following well
known and important equations in the usual calculus of imaginaries, as
connected with plane trigonometry, namely,

(in which ), is found to result, as a limiting case, from the more general formula (I).

In the ordinary theory there are only two different square roots
of negative unity (+*i* and -*i*), and they differ only in their signs.
In the present theory, in order that a quaternion,
*w* + *ix* + *jy* + *kz*, should have its square = -1, it is necessary
and sufficient that we should have

we are conducted, therefore, to the extended expression

which may be called an *imaginary unit*, because its modulus is
= 1, and its square is negative unity. To distinguish one such
imaginary unit from another, we may adopt the notation,

being still that point on the spheric surface which has , , (or , , ) for its rectangular coordinates; and then the formula of multiplication (I) becomes, for any spherical triangle, in which the rotation round , from to , is positive,

If be the *positive pole* of the arc
, or the pole to which the least rotation
from round is positive, then the product
of the two imaginary units in the first member of this formula
(which may be any two such units), is the following:

we have also, for the product of the same two factors, taken in the opposite order, the expression

which differs only in the sign of the imaginary part; and the product of these two products is unity, because, in general,

we have, therefore,

and the products
and may
be said to be *reciprocals* of each other.

In general, in virtue of the fundamental equations of definition,
(A), (B), (C), although the *distributive* character of the
multiplication of ordinary algebraic quantities (real or imaginary)
extends to the operation of the same name in the theory of quaternions,
so that

yet the *commutative* character is lost, and we cannot generally
write for the new as for the old imaginaries,

since we have, for example, *ji* = - *ij*. However, in virtue of
the same definitions, it will be found that another important
property of the old multiplication is preserved, or extended
to the new, namely, that which may be called the *associative*
character of the operation, and which may have for its type the formula

thus we have, generally,

and so on for any number of factors; the notation being employed to express that one determined quaternion, which, in virtue of the theorem (Q), is obtained, whether we first multiply as a multiplicand by as a multiplier, and then multiply the product as a multiplicand by as a multiplier; or multiply first by , and then by . With the help of this principle, we might easily prove the equation (P), by observing that its first member .

In the same manner it is seen at once that

whatever *n* points upon the spheric surface may be denoted by
, , , ,... :
and by combining this principle with that expressed by (M),
it is not difficult to prove that for any spherical polygon
, the following formula
holds good:

which includes the theorem (I') for the case of a spherical
triangle, and in which the arrangement of the *n* points may be supposed,
for simplicity, to be such that the rotations round from
to , round from
to , and so on, are all positive, and each less than
two right angles, though it is easy to interpret the expression
so as to include also the cases where any or all of these conditions
are violated. When the polygon becomes infinitely small, and therefore
plane, the imaginary units become all equal to each other, and may be
denoted by the common symbol *i*; and the formula (R) agrees then with
the known relation, that

Again, let , , be, respectively, the representative points of any three quaternions , , , and let , , be the representative points of the three other quaternions, , , , derived by multiplication from the former; then the algebraical principle expressed by the formula (Q) may be geometrically enunciated by saying that the two points and are the foci of a spherical conic which touches the four sides of the spherical quadrilateral ; and analogous theorems respecting spherical pentagons and other polygons may be deduced, by constructing similarly the formulæ (Q'),&c.

In general, a quaternion , like an ordinary imaginary quantity, may be put under the form,

provided that we assign to , or , the extended meaning (L), which involves two arbitrary angles; and the same general quaternion may be considered as a root of a quadratic equation, with real coefficients, namely

which easily conducts to the following expression for a quotient, or formula for the division of quaternions,

if we define or to mean that quaternion which gives the product , when it is multiplied as a multiplicand by as a multiplier. The same general formula (S'') of division may easily be deduced from the equation (O), by writing that equation as follows,

or it may be obtained from the four general equations of
multiplication (D), by treating the four constituents
of the multiplicand, namely *w*', *x*', *y*', *z*', as the
four sought quantities, while *w*, *x*, *y*, *z*, and
*w*'', *x*'', *y*'', *z*'', are given; or from a construction
of spherical trigonometry, on principles already laid down.

The general expression (S) for a quaternion may be raised
to any power with a real exponent *q*, in the same manner
as an ordinary imaginary expression, by treating the square
root of -1 which it involves as an imaginary unit
having (in general) a fixed direction; raising
the modulus to the proposed real power; and multiplying
the amplitude , increased or diminished by any whole
number of circumferences, by the exponent *q*: thus

if *q* be real, and if *n* be any whole number. For example,
a quaternion has in general two, and only two, different square
roots, and they differ only in their signs, being both
included in the formula,

in which it is useless to assign to *n* any other values than
0 and 1; although, in the particular case where the
original quaternion reduces itself to a real and negative quantity,
so that , this formula (T') becomes

the direction of remaining here entirely undetermined; a result agreeing with the expression (L) or (L') for . In like manner the quaternions, which are cube roots of unity, are included in the expression

denoting here again an imaginary unit, with a direction altogether arbitrary.

If we make, for abridgment

the series here indicated will be always convergent, whatever quaternion may be; and we can always separate its real and imaginary parts by the formula

which gives, reciprocally, for the inverse function , the expression

*u* being any whole number, and being the natural,
or Napierian, logarithm of , or, in other words, that
real quantity, positive or negative, of which the function *f*
is equal to the given real and positive modulus . And
although the ordinary property of exponential functions, namely

does not in general hold good, in the present theory, unless
the two quaternions and be codirectional,
yet we may raise the function *f* to any real power by the formula

which it is natural to extend, by definition, to the case where
the exponent *q* becomes itself a quaternion. The general
equation,

when put under the form

will then give

and thus the general expression for a quaternion *q*, which is
one of the logarithms of a given quaternion
to a given base , is found to involve two independent
whole numbers *n* and *n*', as in the theories of Graves and Ohm,
respecting the general logarithms of ordinary imaginary quantities
to ordinary imaginary bases.

For other developments and applications of the new theory, it is necessary to refer to the original paper from which this abstract is taken, and which will probably appear in the twenty-first volume of the Transactions of the Academy.

Links:

D.R. Wilkins(

School of Mathematics

Trinity College, Dublin