% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 1st June 1999.

\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
   \voffset=-0.4 true mm \hoffset=-5.4 true mm

\def\folio{\ifnum\pageno>0 \number\pageno \else\fi}

\font\Largebf=cmbx10  scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\largesc=cmcsc10  scaled \magstep1

\font\tensc=cmcsc10
\font\sevensc=cmcsc10 scaled 700
\newfam\scfam \def\sc{\fam\scfam\tensc}
\textfont\scfam=\tensc \scriptfont\scfam=\sevensc

\pageno=0

\null\vskip72pt

\centerline{\Largebf EXERCISES IN QUATERNIONS}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (The Cambridge and Dublin Mathematical Journal,
4 (1849), pp.\ 161--168.)}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 1999}

\vskip36pt\eject

\null\vskip36pt

\centerline{\it EXERCISES IN QUATERNIONS.}

\vskip12pt

\centerline{By {\sc Sir William Rowan Hamilton}.}

\bigbreak

\centerline{[{\it The Cambridge and Dublin Mathematical Journal},
   vol.~iv (1849), pp.\ 161--168.]}

\nobreak\bigskip

1.
Although the following paper, or series, will be founded on the
same {\it principles\/} as the communications on Symbolical
Geometry in the present {\it Journal}, and on Quaternions in the
{\it Philosophical Magazine}, yet its {\it plan\/} will be in
many respects different.  And the writer hopes that without
either, on the one hand, too much repeating from those papers,
or, on the other hand, interfering with their continuation, he
may be able, by remarks, rules, formul{\ae}, and examples which
will be submitted in the present Exercises, to give some
acceptable assistance to those mathematicians, or to those
mathematical students, who may do him the honour of desiring to
familiarise themselves with the Calculus of Quaternions; or who
may even be disposed to give that Calculus a trial, as a branch
of symbolical science, and as an instrument of geometrical and
physical research.

\bigbreak

2.
The general {\it conception\/} of {\it directed lines\/} has
occurred to several authors; and many have also perceived the
existence of an important, and indeed fundamental {\it analogy\/}
between the geometrical composition and decomposition of {\it
motions}, and the algebraical addition and subtraction of
positive and negative {\it numbers\/}: which analogy has been
felt to be so close and strong, as to invite and justify the
application of the {\it names\/} and {\it marks\/} of addition
and subtraction to the operations of constructing the
intermediate and transverse diagonals of a parallelogram, when
two coinitial sides of that parallelogram are given, as two
directed lines, of which (what thus come to be called) the
{\it sum\/} and {\it difference\/} are to be taken.  With these,
as {\it preliminary conceptions}, to the introduction of which
into science the present writer is aware that he cannot in any
manner pretend, he wishes to be allowed to regard his readers as
being {\it already\/} familiar, {\it before\/} entering on the
study of the quaternions: although that study will certainly tend
to impress them still more on the mind.  They are, indeed,
{\it common\/} to his own and to several other systems, to which
systems, in {\it other\/} respects, the theory of the quaternions
offers rather a {\it contrast\/} than a resemblance.

\bigbreak

3.
Yet, at this stage, he desires to invite attention to an unusual
mode of {\it notation}, which appears to him to embody under a
convenient and simple form, and under one which adapts itself as
readily to {\it lines in space\/} as to lines in a single plane,
those preliminary and fundamental conceptions.  He denotes, by
the symbol
$${\sc b} - {\sc a}
   \eqno (1),$$
that finite straight line which is drawn {\it from\/} the
point~${\sc a}$ {\it to\/} the point~${\sc b}$; and which line,
when {\it thus\/} denoted, is understood to have a determined
length, direction, and situation in space, as soon as the two
points ${\sc a}$ and ${\sc b}$ themselves are supposed to
receive determined poitions.  And then, by merely writing the two
formul{\ae},
$$({\sc c} - {\sc b}) + ({\sc b} - {\sc a})
   =  {\sc c} - {\sc a}
   \eqno (2),$$
$$({\sc c} - {\sc a}) - ({\sc b} - {\sc a})
   =  {\sc c} - {\sc b}
   \eqno (3),$$
he {\it expresses}, under the form of two {\it identities}, those
conceptions of the geometrical addition and subtraction of
directed lines, as analogous to the composition and decomposition
of motions, and as performed according to the same rules, in
which conceptions themselves he has already carefully disclaimed
all private or personal property.  For, with the use above
explained of the notation (1), the {\it geometrical identity\/}
(2) expresses that if the directed line to ${\sc c}$ from
${\sc b}$ be {\it geometrically added\/} to the directed line to
${\sc b}$ from ${\sc a}$, according to the rules of the
composition of motions, the {\it geometrical sum}, or the {\it
resultant line}, will be that {\it third\/} directed line which
is drawn to ${\sc c}$ to ${\sc a}$; whatever may be the positions
in space of the three points ${\sc a} \, {\sc b} \, {\sc c}$.
And the {\it other\/} geometrical identity (3) expresses in like
manner this converse proposition, that if the directed line from
${\sc a}$ to ${\sc b}$, which is, by the notation (1), denoted by
the symbol ${\sc b} - {\sc a}$, be {\it geometrically
subtracted\/} from the directed line ${\sc c} - {\sc a}$, which
is drawn from ${\sc a}$ to ${\sc c}$, according to the laws of
decomposition of motion, the {\it geometrical difference\/}
obtained by this subtraction will be that directed line
${\sc c} - {\sc b}$, which is drawn from the final
point~${\sc b}$ of the subtrahend line ${\sc b} - {\sc a}$ to the
final point~${\sc c}$ of that other given and coinitial directed
line ${\sc c} - {\sc a}$, from which the subtraction is to be
performed.

\bigbreak

4.
Instead of compounding only {\it two\/} successive motions, or
{\it rectilinear steps\/} in space, we may compound
{\it any number\/} of such steps; or, in other words, instead of
considering a triangle ${\sc a} \, {\sc b} \, {\sc c}$, we
consider a {\it polygon\/}
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e} \ldots$:
and the known results for such more complex cases may still be
expressed with great simplicity, and under the form of
{\it geometrical identities}, by adhering to the same method of
notation.  Thus, for a rectilinear quadrilateral,
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d}$,
whether this be or be not in one plane, we shall always have the
formula
$$({\sc d} - {\sc c}) + ({\sc c} - {\sc b}) + ({\sc b} - {\sc a})
   =  {\sc d} - {\sc a}
   \eqno (4);$$
for a pentagon, whether plane or gauche,
$$    ({\sc e} - {\sc d})
    + ({\sc d} - {\sc c})
    + ({\sc c} - {\sc b})
    + ({\sc b} - {\sc a})
   =  {\sc e} - {\sc a}
   \eqno (5);$$
and similarly for other polygons, in space or in one plane: the
line which is drawn from the initial to the final point of any
{\it unclosed\/} polygon being regarded (in this as in many other
systems) as the {\it geometrical sum of all the successive
sides\/} of the figure which it thus serves to {\it close\/};
exactly as in the formula (2) the {\it directed base\/}
${\sc c} - {\sc a}$ of the triangle ${\sc a} {\sc b} {\sc c}$ was
the geometrical {\it sum\/} of the two successive {\it sides\/}
${\sc b} - {\sc a}$ and ${\sc c} - {\sc b}$, obtained by
{\it adding\/} the second of those two sides to the first.

\bigbreak

5.
If the closing line be drawn in the order of succession of the
sides, or in the order of the motion along the polygon which has
been above supposed to be performed; or if the polygon be given
as closed; then the {\it sum of all\/} the successive lines,
including the closing line, will be a {\it null line}, because
the motion thus conceived would simply bring a moving point
{\it back\/} to its original or initial position.  Accordingly,
in the notation above proposed, we shall have the following
formul{\ae} of identity, which {\it express\/} this conception of
{\it return\/}:
$${\sc a} - {\sc a}
   =  ({\sc a} - {\sc b}) + ({\sc b} - {\sc a})
   =  ({\sc a} - {\sc c}) + ({\sc c} - {\sc b}) + ({\sc b} - {\sc a})
   =  \hbox{\&c.}
   \eqno (6).$$

\bigbreak

6.
We may agree to {\it suppress the symbol of a null line}, when it
occurs as written to the {\it left-hand\/} of any complex symbol
denoting the result of any geometrical addition or subtraction;
and then, by changing ${\sc c}$ to ${\sc b}$ in the identity (2),
and to ${\sc a}$ in the identity (3), we shall obtain the
formul{\ae},
$$\eqalignno{
+ ({\sc b} - {\sc a})
   &= ({\sc b} - {\sc b}) + ({\sc b} - {\sc a})
    = {\sc b} - {\sc a}
   &(7);\cr
- ({\sc b} - {\sc a})
   &= ({\sc a} - {\sc a}) - ({\sc b} - {\sc a})
    = {\sc a} - {\sc b}
   &(8);\cr}$$
which allow of our interpreting the two isolated but
{\it affected\/} symbols of lines,
$$+ ({\sc b} - {\sc a})
   \quad\hbox{and}\quad
  - ({\sc b} - {\sc a})
   \eqno (9),$$
as denoting respectively the directed line ${\sc b} - {\sc a}$
{\it itself}, and the {\it opposite\/} of that line, namely the
directed line ${\sc a} - {\sc b}$: two lines being said to be
mutually opposites, when the beginning and end of the first line
coincide respectively with the end and the beginning of the
second.  A null line is {\it its own\/} opposite,
$$+ ({\sc a} - {\sc a}) = - ({\sc a} - {\sc a})
   \eqno (10);$$
but any {\it actual line}, that is, a line ${\sc b} - {\sc a}$
with any finite length, is distinguished form the opposite line
${\sc a} - {\sc b}$ by the contrast between their {\it
directions}.  A line ${\sc b} - {\sc a}$ may be {\it
subtracted\/} from another line ${\sc c} - {\sc a}$, by
{\it adding\/} the second line ${\sc c} - {\sc a}$ to the
{\it opposite\/} ${\sc a} - {\sc b}$ of the first; for we have,
by the identities (2) and (3),
$$({\sc c} - {\sc a}) - ({\sc b} - {\sc a})
   =  {\sc c} - {\sc b}
   =  ({\sc c} - {\sc a}) + ({\sc a} - {\sc b})
   \eqno (11).$$

\bigbreak

7.
Two directed lines being regarded as {\it equal\/} to each other,
when, and only when, their {\it directions\/} as well as their
{\it lengths\/} are identical, although their {\it situations\/}
will generally be different, the {\it equation\/}
$${\sc d} - {\sc c} = {\sc b} - {\sc a}
   \eqno (12) $$
will express, generally, that the four points
${\sc a} \, {\sc b} \, {\sc d} \, {\sc c}$
are the four successive corners of a {\it parallelogram\/}; the
corner~${\sc d}$ being opposite to ${\sc a}$, and ${\sc c}$ to
${\sc b}$: and we may still retain this mode of speaking, even
when the altitude or the area or the parallelogram vanishes, by
the four points ${\sc a} \, {\sc b} \, {\sc d} \, {\sc c}$ coming
to range themselves on one right line.  From this signification
of the equation (12) it is evident that this equation admits of
{\it inversion}, and of {\it alternation}, so that it may be
written thus (inversely),
$${\sc c} - {\sc d} = {\sc a} - {\sc b}
   \eqno (13),$$
because the opposites of equal lines are equal; or thus
(alternately),
$${\sc d} - {\sc b} = {\sc c} - {\sc a}
   \eqno (14),$$
by Euclid {\sc i}.~33, or because the two {\it paths of
transport}, from ${\sc a}$ and ${\sc b}$ to ${\sc c}$ and ${\sc
d}$ respectively, must be {\it themselves\/} equal directed
lines, in order to allow of the first given directed line
${\sc b} - {\sc a}$ being carried, {\it without rotation}, by the
simultaneous motion of its two extreme points along those two
paths of transport, so as to come to {\it coincide\/} with the
second given directed line ${\sc d} - {\sc c}$; which second line
would not be (in the foregoing sense) {\it equal\/} to the first,
unless this perfect coincidence could be effected by such
transport without rotation.  (The writer may remark, in passing,
that he agrees with those who hold that all such considerations
as these, of {\it motions abstracted from causes of motion}, do
not vitiate, in any degree however small, the {\it purity\/} of
geometrical science: to think otherwise would be indeed, as he
conceives, to condemn, so far, those ancient geometers, including
Euclid, who generated surfaces, for example the sphere, by
motion.)  Directed lines which are {\it equal\/} to the same
directed line are also equal to each other; and the sums and
differences of equal directed lines, similarly taken, are equal
directed lines.  Lines which are opposites of equal directed
lines may be said, by an extension of the former definition of
opposites, to be themselves also opposite lines.

\bigbreak

8.
Since, under the condition expressed by the equation (12), the
line ${\sc d} - {\sc a}$ is the {\it directed diagonal\/} of the
parallelogram ${\sc a} \, {\sc b} \, {\sc d} \, {\sc c}$,
intermediate between the two directed sides ${\sc b} - {\sc a}$
and ${\sc c} - {\sc a}$, and coinitial with them, it ought (by a
known principle above mentioned) to be found to be their
geometrical sum: and, accordingly,
$${\sc d} - {\sc a}
   =  ({\sc d} - {\sc c}) + ({\sc c} - {\sc a})
   =  ({\sc b} - {\sc a}) + ({\sc c} - {\sc a})
   \eqno (15);$$
or, adding the sides in a different order, and employing the
principle of {\it alternation}, whereby we pass from the equation
(12) to (14),
$${\sc d} - {\sc a}
   =  ({\sc d} - {\sc b}) + ({\sc b} - {\sc a})
   =  ({\sc c} - {\sc a}) + ({\sc b} - {\sc a})
   \eqno (16).$$
It is therefore allowed to {\it change the order\/} of the
summands in the addition of any two directed lines; a conclusion
which is easily extended to any number of such lines, in space or
in one plane, so as to shew that geometrical {\it addition\/} is
a {\it commutative operation}, or that the {\it sum\/} of any
given system of directed lines is always equal to the same given
directed line, in whatever {\it order\/} the summation is
effected.  Addition of directed lines is also an {\it
associative\/} operation, in the sense that any number of
successive summands may be collected of {\it associated\/}
together (as is done in calculation by enclosing their symbols
within brackets) into one partial group, and their sum then added
as a single summand to the rest: for this comes merely (when its
geometrical signification is examined) to drawing a {\it diagonal
of a polygon}, plane or gauche.  It is understood that in order
to avail ourselves of the identity (2), for the purpose of adding
an {\it arbitrary\/} but given line ${\sc b}' - {\sc a}'$ to
another line ${\sc b} - {\sc a}$, when the beginning ${\sc a}'$
of the proposed {\it addend\/} line does not {\it already\/}
coincide with the end~${\sc b}$ of the line to which the addition
is to be performed, we are to {\it make\/} it coincide, by a
transport without rotation; this process of construction being
symbolically expressed by by the formula
$$({\sc b}' - {\sc a}') + ({\sc b} - {\sc a})
   =  {\sc c} - {\sc a},
   \quad\hbox{if }
      {\sc b}' - {\sc a}' = {\sc c} - {\sc b}
   \eqno (17).$$

\bigbreak

9.
When three points ${\sc a} \, {\sc b} \, {\sc c}$ are so related
as to satisfy the equation
$${\sc c} - {\sc b} = {\sc b} - {\sc a}
   \eqno (18),$$
which gives, by principles and notations already explained,
$${\sc c} - {\sc a} = ({\sc b} - {\sc a}) + ({\sc b} - {\sc a})
   \eqno (19);$$
then, by a natural and obvious use of numerical coefficients, we
may write also, as other expressions for the same relation of
position between the three points (namely that the
point~${\sc b}$ bisects the straight line connecting ${\sc a}$
and ${\sc c}$), the two following equations, of which each
includes the other:
$${\sc c} - {\sc a} = 2 ({\sc b} - {\sc a});\quad
  {\sc b} - {\sc a} = {\textstyle {1 \over 2}} ({\sc c} - {\sc a})
   \eqno (20).$$
And generally, if $a$ denote any positive or negative number,
whether integral or fractional, and whether commensurable or
incommensurable, the notation
$${\sc c} - {\sc a} = a ({\sc b} - {\sc a})
   \eqno (21) $$
may conveniently be employed to express that the point~${\sc c}$
is situated on the same indefinite right line as the points
${\sc a}$ and ${\sc b}$, being at the same side of ${\sc a}$ as
${\sc b}$ if the coefficient~$a$ be positive, but at the opposite
side of ${\sc a}$ if $a$ be negative, and at a distance from
${\sc a}$ which bears to the distance of ${\sc b}$ to ${\sc a}$
the ratio of $\pm a$ to $1$.  When the coefficient~$a$ becomes
zero, then both members of (21) become null lines, and ${\sc c}$
coincides with ${\sc a}$.  With such an use of coefficients we
shall have, as in ordinary algebra, the two identities
$$(a' \pm a) ({\sc b} - {\sc a})
   = a' ({\sc b} - {\sc a}) \pm a ({\sc b} - {\sc a})
   \eqno (22);$$
$$a \{ ({\sc b}' - {\sc a}') \pm ({\sc b} - {\sc a}) \}
   = a ({\sc b}' - {\sc a}') \pm a ({\sc b} - {\sc a})
   \eqno (23);$$
which we may express in words by saying that the operation of
{\it multiplication\/} of a directed line by a numerical
coefficient is a {\it distributive operation}, whether relatively
to the multiplying number, or relatively to the multiplied line;
this {\it distributive\/} property of such {\it multiplication\/}
of lines by numbers depending mainly on the {\it commutative\/}
property of the {\it addition\/} of lines among themselves.

\bigbreak

10.
The equation (21) expresses sufficiently that the point~${\sc c}$
is situated {\it somewhere\/} upon the indefinite straight line
which passes through the two points ${\sc a}$ and ${\sc b}$, or
that the {\it three\/} points ${\sc a} \, {\sc b} \, {\sc c}$ are
{\it collinear\/}; and it expresses {\it nothing more\/} than
this relation of collinearity, if we conceive the number~$a$ to
remain undetermined.  The formula (21) may therefore, with this
use of an indeterminate numerical coefficient~$a$, be regarded as
the {\it equation of an indefinite right line in space\/};
{\it one such equation\/} being {\it sufficient\/} in this mode
of dealing with the subject.  This equation (21) may however be
made to assume a more symmetric form, by introducing the
consideration of an arbitrary fourth point~${\sc d}$, supposed to
be situated anywhere in space, with which the three collinear
points ${\sc a} \, {\sc b} \, {\sc c}$ shall be compared.  For
thus, by writing the equation successively under the following
forms,
$$({\sc c} - {\sc d}) - ({\sc a} - {\sc d})
   =  a ({\sc b} - {\sc d}) - a ({\sc a} - {\sc d})
   \eqno (24),$$
$${\sc c} - {\sc d} 
   =  a ({\sc b} - {\sc d}) + (1 - a) ({\sc a} - {\sc d})
   \eqno (25),$$
$$(b - ba) ({\sc a} - {\sc d}) + ba ({\sc b} - {\sc d})
       - b ({\sc c} - {\sc d})
   = 0
   \eqno (26),$$
$$l ({\sc a} - {\sc d}) + m ({\sc b} - {\sc d})
       + n({\sc c} - {\sc d})
   = 0
   \eqno (27),$$
in each of the two last of which forms the symbol for zero is
understood to denote a null line, we see, by comparing these two
forms, that the coefficients $l$, $m$, $n$, in the form (27), are
connected among themselves by the equation of condition
$$l + m + n = 0
   \eqno (28);$$
and, conversely, that under this last condition the formula (27)
expresses that the three points ${\sc a} \, {\sc b} \, {\sc c}$
are ranged upon one common right line.  In fact, when the
condition (28) is satisfied, we can eliminate the coefficient~$l$
by it from (27), and so obtain the equation
$$m ({\sc b} - {\sc a}) + n ({\sc c} - {\sc a}) = 0
   \eqno (29),$$
which evidently agrees with the form (21), and conducts to
similar consequences.

\bigbreak

11.
Let ${\sc e}$ be a new point situated anywhere upon the
indefinite straight line ${\sc c} {\sc d}$; and therefore
satisfying an equation of the form
$$p ({\sc e} - {\sc d}) = n ({\sc c} - {\sc d})
   \eqno (30),$$
where $n$ may denote the same coefficient as in (27).
Eliminating this coefficient~$n$, the symbol~${\sc c}$ for the
point of intersection of the two indefinite straight lines
${\sc a} {\sc b}$, ${\sc d} {\sc e}$, disappears; and there
results, as the expression of the condition that {\it some\/}
such point~${\sc c}$ shall exist, or that the {\it four points\/}
${\sc a} \, {\sc b} \, {\sc d} \, {\sc e}$ shall be situated in
{\it one common plane}, an equation of the following form,
$$l ({\sc a} - {\sc d}) + m ({\sc b} - {\sc d})
       + p ({\sc e} - {\sc d})
   = 0
   \eqno (31);$$
which seems to resemble the equation (27), but differs from it in
this important respect, that the sum of the three new
coefficients $l \, m \, p$ does not now generally vanish, as the
sum of the three old coefficients $l \, m \, n$ did vanish, in
virtue of the condition (28).  And by comparing the {\it equation
of coplanarity\/} (31) with the {\it condition of collinearity\/}
(28), we may now see that this last mentioned condition (28), in
combination with the equation (27), expressed that the three
given points ${\sc a} \, {\sc b} \, {\sc c}$ are {\it coplanar
with any arbitrary fourth point\/}~${\sc d}$; which can only be
by those {\it three\/} points ${\sc a} \, {\sc b} \, {\sc c}$
being {\it collinear\/} with each other.  In fact, under the
condition (28), we have
$$l ({\sc d} - {\sc d}') + m ({\sc d} - {\sc d}')
       + n ({\sc d} - {\sc d}')
   = 0
   \eqno (32),$$
by adding which to (27) we obtain the same result, namely
$$l ({\sc a} - {\sc d}') + m ({\sc b} - {\sc d}')
       + n ({\sc c} - {\sc d}')
   = 0
   \eqno (33),$$
as if we had simply changed the symbol of the fourth point~${\sc
d}$ to that of any arbitrary fifth point~${\sc d}'$.  By
introducing the symbol~${\sc o}$ of a new and arbitrary point of
space, with which the four coplanar points
${\sc a} \, {\sc b} \, {\sc d} \, {\sc e}$
may be compared, through drawing lines from it to them, the
equation of coplanarity (31) assumes the form
$$l ({\sc a} - {\sc o}) + m ({\sc b} - {\sc o})
      + n ({\sc d} - {\sc o}) + p ({\sc e} - {\sc o})
   = 0
   \eqno (34);$$
where $n$ is a new coefficient, connected with the others by the
condition
$$l + m + n + p = 0
   \eqno (35).$$

These remarks, and a few others which shall be offered in some
following articles, may be of use, as serving to illustrate and
exemplify an unusual mode of {\it notation\/}\footnote*{The
notation ${\sc b} - {\sc a}$ for a {\it directed right line in
space}, was proposed in a note to the first article of the paper
on Symbolical Geometry, printed in the present {\it Journal\/}
(towards the end of 1845): and it had been long familiar to the
writer, as an extension to {\it space\/} of a similar notation
relatively to {\it time}, which had been published by him in the
year 1835, to express a {\it time-step}, or directed interval in
time, from any {\it one moment\/} (not number) denoted by
${\sc a}$, to any {\it other moment\/} of time denoted by
${\sc b}$.  (See the Essay on Algebra as the Science of Pure
Time, in the 1${}^{\rm st}$ part of the {\sc xvii}${}^{\rm th}$
volume of the {\it Transactions of the Royal Irish Academy\/}).
With respect to the mere fact of {\it distinguishing\/} between
two elementary geometrical symbols, ${\sc a} {\sc b}$ and
${\sc b} {\sc a}$, as denoting two {\it opposite lines}, the
present author cheerfully acknowledges that this simple and
natural distinction has often been noticed and employed by other
writers on Geometry.}
in geometry; but they can only be regarded as {\it preparatory\/}
to the theory of the {\it quaternions}, because that theory, in
its {\it geometrical\/} aspect, depends essentially on the
conception of the {\it multiplication\/} and {\it division\/} of
{\it one\/} directed line {\it by another line\/} of the same
kind, and not merely by a numerical {\it coefficient\/}: a
{\sc quaternion} (in the author's system) being always equal to
such a {\it product or quotient of two directed lines in space}.

\nobreak\bigskip

\centerline{[{\it To be continued.}]}

\bye
