% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ILLUSTRATIONS FROM GEOMETRY OF THE}

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\centerline{\Largebf THEORY OF ALGEBRAIC QUATERNIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   3 (1847), Appendix, pp.\ xxxi--xxxvi.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\centerline{\largeit Illustrations from Geometry of the Theory
of Algebraic Quaternions.}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{Communicated February~10, 1845.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), Appendix, pp.\ xxxi--xxxvi.]}

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The spirit of Sir William Hamilton's communication, which was
designed as a further illustration from geometry of the author's
theory of algebraic quaternions, consisted in regarding
operations on such quaternions as admitting of being ultimately
interpreted as {\it operations on straight lines\/}; each line
being considered as having not only a determinate {\it length},
but also a determinate {\it direction}.  The {\it quotient\/}
$\displaystyle {{\rm b} \over {\rm a}}$ obtained by the division
of one such line~(${\rm b}$) by another~(${\rm a}$), is,
generally, in the author's view, a {\it quaternion\/}; it
depends, in general, on {\it four\/} distinct elements, of which
one, namely, the {\it modulus}, is a positive or absolute number
expressing the {\it relative magnitude\/} of the dividend and
divisor lines, while the three other elements serve jointly to
express the {\it relative direction\/} of those two lines.  Of
the three latter, one is the {\it amplitude}, and marks the
{\it inclination\/} of one line to the other, or the magnitude of
the angle which they include; while the two others determine the
{\it plane\/} of that angle, and are what have been called, in a
former communication, the {\it directional coordinate}, such as
the {\it longitude\/} and {\it colatitude\/} of the quaternion.
In this comparatively geometrical view, as in the more
algebraical view which was formerly stated to the Academy, the
consideration of these four elements, modulus, amplitude,
longitude, and colatitude, presents itself, therefore, naturally.
We may also speak of the {\it axis\/} of a quaternion, meaning
thereby the axis perpendicular to the plane of the two straight
lines of which that quaternion is a quotient; and may say, that
such an axis is itself {\it positive\/} or {\it negative}, or
that it is taken in the positive or in the negative direction,
according as it is the axis of a positive or a negative
{\it rotation}, from the divisor to the dividend line.
Quaternions may be said to be {\it coaxal\/} when their axes
coincide, or only differ in sign.  A quaternion is not altered in
value when the two lines of which it is the quotient are
transferred, without altering their directions, to any other
positions in space; or when their lengths are both changed
together in any common ratio; or when they are both made to
revolve together, through any common amount of rotation, round
the axis of the quaternion, without ceasing to be still in (or
parallel to) the same common plane as before.  It is, therefore,
always possible to prepare any two proposed quaternions, or
geometrical quotients or fractions of the kind above described,
so as to have one {\it common denominator\/} or divisor line; and
then the {\it addition\/} or {\it subtraction\/} of those two
quaternions is effected, by retaining that common line as the
denominator or divisor of the new quaternion, and by adding or
subtracting the numerator lines, in order to obtain the new
numerator of the same new quaternion, that is to say, of the sum
or difference of the two old quaternions; addition and
subtraction of straight {\it lines\/} (when those lines are
supposed to have not only lengths but also {\it directions\/})
being performed according to the rules which have already been
proposed by several writers, and which correspond to compositions
and decompositions of rectilinear {\it motions\/} (or of forces).
{\it Multiplication\/} of two quaternions may be effected by
preparing them so that the denominator~(${\rm b}$) of the
multiplier, may be equal to, or the same line with, the
numerator~(${\rm b}$) of the multiplicand (lines being
{\it equal\/} when their directions as well as their lengths are
the same), and by then treating the numerator~(${\rm c}$) of the
multiplier as the numerator of the product, and the
denominator~(${\rm a}$) of the multiplicand as the denominator of
the product: and {\it division\/} may be regarded as the return
to the multiplier, from a given product and multiplicand.

With this view of multiplication, it is evident that the product
of the moduli of the two factors is equal to the modulus of the
product.  It is clear also, that if we construct a spherical
triangle~${\sc a} {\sc b} {\sc c}$, of which the three corners,
or the radii drawn to them from the centre of the sphere,
represent the directions of the three lines
${\rm a}$,~${\rm b}$,~${\rm c}$, then the arc, or side of the
triangle,~${\sc a} {\sc b}$, will represent the amplitude of the
multiplicand quaternion; the arc or side~${\sc b} {\sc c}$ will
represent the amplitude of the multiplier; and the remaining arc
or side~${\sc a} {\sc c}$ the amplitude of the product, so that
the spherical triangle will be constructed with these three
amplitudes for its three sides.  And we see that in the triangle
thus constructed, the spherical angles at ${\sc a}$ and
${\sc c}$, which are respectively opposite to the amplitudes of
the multiplier and multiplicand, are equal to the respective
inclinations of the axes of the multiplicand and multiplier to
the axis of the product of the quaternions; and that the
remaining spherical angle at ${\sc b}$, which is opposite to the
amplitude of the product, is equal to the supplement of the
inclination of the axes of the factors to each other: a form
almost the same with that under which the fundamental connexion
of quaternions with spherical trigonometry was stated by Sir
William Hamilton, in his first letter on the subject, to John T.
Graves, Esq., which was written in October 1843, and has been
printed in the Supplementary number of the {\it Philosophical
Magazine\/} for December 1844.  The other form of the same
fundamental connexion, which was communicated to the Academy in
November, 1843, may be deduced from the foregoing, by the
consideration of that polar or supplementary triangle, of which
the corners mark the directions of the {\it axes\/} of the
factors and the product, and were then called the
{\it representative\/} points of the three quaternions compared.
If the order of the factors be changed, the (positive) axis of
the product falls to the other side of the plane of the axes of
the factors, being always so situated that the rotation round the
axis of the multiplier from the axis of the multiplicand to that
of the product is positive; {\it multiplication of quaternions\/}
is therefore seen, in this as in other ways, to be {\it not in
general a commutative operation}, or the result depends, in
general, essentially on the {\it order\/} in which the factors
are taken.

The same remarkable conclusion follows from the comparison of the
lately mentioned spherical triangle~${\sc a} {\sc b} {\sc c}$
with another triangle ${\sc c}' {\sc b} {\sc a}'$, vertically
opposite and equal thereto, and such that the common
corner~${\sc b}$ bisects each of the two arcs
${\sc c}' {\sc c}$, ${\sc a}' {\sc a}$, joining the two pairs of
corresponding corners; which other triangle may represent the
directions of three lines ${\rm c}'$,~${\rm b}$,~${\rm a}'$,
related to the system of the three former lines
${\rm c}$,~${\rm b}$,~${\rm a}$, by the following equations
between geometrical quotients, or quaternions,
$${{\rm a}' \over {\rm b}} = {{\rm b} \over {\rm a}},\quad
  {{\rm b}' \over {\rm c}} = {{\rm c} \over {\rm b}};$$
for then, by the definition of multiplication of such quotients
here proposed, we have the two different results,
$${{\rm c} \over {\rm b}} \times {{\rm b} \over {\rm a}}
   = {{\rm c} \over {\rm a}};\quad
  {{\rm b} \over {\rm a}} \times {{\rm c} \over {\rm b}}
   = {{\rm a}' \over {\rm c}'};$$
and although these two resulting quaternion products have equal
moduli and equal amplitudes, yet they have in general
{\it different axes}, because the arcs ${\sc a} {\sc c}$ and
${\sc a}' {\sc c}'$, though equally long, are parts of different
great circles, and are therefore situated in different planes.
However, in that particular but useful and often occurring case,
where the two factors have one common axis, the order of those
factors becomes indifferent; and if attention be paid to positive
and negative signs, it may be said that {\it coaxial
quaternions\/} may be multiplied together, in either order, by
adding their amplitudes, multiplying their moduli, and retaining
their common axis.  In general, it may be proved, from the views
here given of multiplication and addition, that, although the
{\it commutative\/} property of ordinary multiplication does not
usually extend to operations on quaternions, yet the
{\it distributive\/} and {\it associative\/} properties of that
operation do always so extend; and that the commutative and
associative properties of addition hold good in like manner for
quaternions: results which were indeed stated to the Academy in
November 1843, as consequences from the algebraical definitions
of a quaternion, and of operations performed thereon, but have
now been mentioned again, as following from more geometrical
definitions also.

Comparing the view here proposed with that which was submitted to
the Academy in November 1844, a quaternion may be said to reduce
itself to a {\it scalar\/} (or ordinary real number), when the
two straight lines, of which it is the quotient, are
{\it parallel\/}; the scalar being positive when those lines are
similar, but negative when they are opposite in direction.  And,
on the other hand, the scalar part vanishes, and the quaternion
becomes a pure {\it vector}, when it is a quotient of two
{\it rectangular\/} lines: and, in this last case, it may be
conveniently constructed by a {\it third line\/} perpendicular to
both of them, namely, by one drawn in the direction of the
positive axis of the quaternion, with a length which bears to an
assumed unit of length the ratio marked by the modulus.  This
third line, which thus represents or constructs the quotient of
two other lines perpendicular to it and to each other, may, by a
suitable choice of those two lines, receive any proposed length,
and any proposed direction; and {\it every straight line having
length and direction in space\/} may, in this view, be regarded
as a particular quaternion, namely, as one of the class above
called {\it vectors}.  It is easy to {\it prove\/} that when
lines are thus treated as quotients, they have the same sums,
differences, and quotients, as those obtained by the processes or
conceptions above described or alluded to; and hence it would be
natural to {\it define}, as we should be at liberty to do, that
the {\it product\/} of two lines is also in general a quaternion,
obtained by multiplying two vector factors together, according to
the rules of multiplication of quaternions.  We should then be
able to establish, in this new way, all the rules, already
communicated to the Academy, for the {\it multiplication of
straight lines in space\/}; and especially should be conducted
anew to those {\it two rules}, or principles, which presented
themselves to the author in his earliest researches on
quaternions (as described in the printed letter already referred
to), and which he still regards as fundamental in their theory:
namely, first, that {\it the product of two straight lines, which
agree in direction, is to be considered as a negative number},
namely, as the product of their two lengths taken negatively;
and, secondly, that {\it the product of two rectangular lines is
to be regarded as a third line perpendicular to both}, of which
the length represents the product of their lengths, and
{\it to\/} which the rotation, {\it from\/} the multiplicand
line, {\it round\/} the multiplier line, is positive.  The
paradoxical, or, at least, unusual appearance of these two
fundamental rules, combined with the variety of the applications
of which the author has found them susceptible, induce him to
hope that he shall be pardoned for thus offering new
confirmations or new illustrations of them, derived from
considerations of the manner in which they present themselves
from various points of view.

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