From An Elementary Treatise On Quaternions by Peter Guthrie Tait.
1. FOR at least two centuries the geometrical representation
of the negative and imaginary algebraic quantities,
and
has been a favourite subject of speculation with mathematicians.
The essence of almost all of the proposed processes consists in
employing such expressions to indicate the DIRECTION, not the
length, of lines.
2. Thus it was long ago seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities of the same kind by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics.
3. Wallis, towards the end of the seventeenth century, proposed
to represent the impossible roots of a quadratic equation
by going
out of the line on which, if real, they would have been laid off.
This construction is equivalent to the consideration of
as a
directed unit-line perpendicular to that on which real quantities
are measured.
4. In the usual notation of Analytical Geometry of two
dimensions, when rectangular axes are employed, this amounts
to reckoning each unit of length along
as
, and on
as
; while on
each unit is
, and on
it is
.
T. Q. I.
If we look at these four lines in circular order, i.e. in the order of positive rotation (that of the northern hemisphere of the earth about its axis, or opposite to that of the hands of a watch), they give
In this series each expression is derived from that which precedes
it by multiplication by the factor
. Hence we may consider
as an operator, analogous to a handle perpendicular to the
plane of
, whose effect on any line in that plane is to make it
rotate (positively) about the origin through an angle of
.
5. In such a system, (which seems to have been first developed,
in 1805, by Buee) a point in the plane of reference is defined by a
single imaginary expression. Thus
may be considered
as a single quantity, denoting the point,
, whose coordinates are
and
. Or, it may be used as an expression for the line
joining that point with the origin. In the latter sense, the expression
implicitly contains the direction, as well as the
length, of this line; since, as we see at once, the direction is
inclined at an angle
to the axis of
, and the length is
. Thus, say we have
the line
considered as that by which we pass from one extremity,
, to the other,
. In this sense it is called a vector.
Considering, in the plane, any other vector,
the addition of these two lines obviously gives
and we see that the sum is the diagonal of the parallelogram on
,
. This is the law of the composition
of simultaneous
velocities; and it contains, of course, the law of subtraction of one
directed line from another.
6. Operating on the first of these symbols by the factor
,
it becomes
; and now, of course, denotes the point
whose
and
coordinates are
and
; or the line joining this
point with the origin. The length is still
, but the angle
the line makes with the axis of x is
; which is
evidently greater by
than before the operation.
7. De Moivre's theorem
tends to lead us still further in the
same direction. In fact, it is easy to see that if we use, instead
of
, the more general factor
, its effect on
any line is to turn it through the (positive) angle
in the plane
of
. [Of course the former factor,
, is merely the particular case of this, when
.]
Thus
by direct multiplication. The reader will at once see that the new
form indicates that a rotation through an angle
has taken place,
if he compares it with the common formulae for turning the coordinate axes through a given angle. Or, in a less simple manner,
thus
Length =
as before.
Inclination to axis of x
8. We see now, as it were, why it happens that
In fact, the first operator produces
successive rotations in the
same direction, each through the angle
; the second, a single
rotation through the angle
.
9. It may be interesting, at this stage, to anticipate so far as to remark that in the theory of Quaternions the analogue of
is
where
Here, however,
is not the algebraic
, but is any directed unit-line whatever in space.
10. In the present century Argand, Warren, Mourey, and
others, extended the results of Wallis and Buee. They attempted to express as a line the product of two lines each represented by a
symbol such
. To a certain extent they succeeded,
but all their results remained confined to two dimensions.
The product, II, of two such lines was defined as the fourth
proportional to unity and the two lines, thus
or
The length of II is obviously the product of the lengths of the factor lines; and its direction makes an angle with the axis of x which is the sum of those made by the factor lines. From this result the quotient of two such lines follows immediately.
11. A very curious speculation, due to Servois and published
in 1813 in Gergonne's Annales, is one of the very few, so far as has
been discovered, in which a well-founded guess at a possible mode
of extension to three dimensions is contained. Endeavouring to
extend to space the form
for the plane, he is guided by
analogy to write for a directed unit-line in space the form
where
are its inclinations to the three axes. He perceives easily that
must be non-reals: but, he asks,
"seraient-elles imaginaires reductibles a la forme generale
?" The
of the Quaternion Calculus furnish an answer to this question. (See Chap. II.) But it may be remarked that, in applying the
idea to lines in a plane, a vector OP will no longer be represented
(as in 5) by
but by
And if, similarly.
the addition of these two lines gives for
(which retains its
previous signification)
12. Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions; and, though many such had been made before 1843, none, with the single exception of Hamilton's, have resulted in simple, practical methods; all, however ingenious, seeming to lead almost at once to processes and results of fearful complexity.
For a lucid, complete, and most impartial statement of the claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions. He there shews how his long protracted investigations of Sets culminated in this unique system of tridimensional-space geometry.
13. It was reserved for Hamilton to discover the use and
properties of a class of symbols which, though all in a certain sense
square
roots of
, may be considered as real unit lines, tied down
to no particular direction in space; the expression for a vector is,
or may be taken to be,
but such vector is considered in connection with an extraspatial
magnitude
, and we have thus the notion of a QUATERNION
This is the fundamental notion in the singularly elegant, and enormously powerful, Calculus of Quaternions.
While the schemes for using the algebraic
to indicate
direction make one direction in space expressible by real numbers,
the remainder being imaginaries of some kind, and thus lead to
expressions which are heterogeneous; Hamilton's system makes all
directions in space equally imaginary, or rather equally real, there
by ensuring to his Calculus the power
of dealing with space
indifferently in all directions.
In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its reference lines solely from the problem it is applied to.
14. But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, with which the student is supposed to be, to some extent at least, acquainted. Such assistance, it will be found, can (as a rule) soon be dispensed with; and Hamilton regarded any apparent necessity for an oc casional recurrence to it, in higher applications, as an indication of imperfect development in the proper methods of the new Calculus.
We commence, therefore, with some very elementary geometrical ideas, relating to the theory of vectors in space. It will subsequently appear how we are thus led to the notion of a Quaternion.
15. Suppose we have two points
and
in space, and suppose A given, on how many numbers does
relative position
depend?
If we refer to Cartesian coordinates (rectangular or not) we find
that the data required are the excesses of
three coordinates
over those of A. Hence three numbers are required.
Or we may take polar coordinates. To define the moon's position with respect to the earth we must have its Geocentric Latitude and Longitude, or its Right Ascension and Declination, and, in addition, its distance or radius-vector. Three again.
16. Here it is to be carefully noticed that nothing has been
said of the actual coordinates of either
or
, or of the earth
and moon, in space; it is only the relative coordinates that are
contemplated.
Hence any expression, as
, denoting a line considered with
reference to direction and currency as well as length, (whatever
may be its actual position in space) contains implicitly three
numbers, and all lines parallel and equal to
, and concurrent
with it, depend in the same way upon the same three. Hence, all
lines which are equal, parallel, and concurrent, may be represented
by a common symbol, and that symbol contains three distinct numbers.
In this sense a line is called a VECTOR, since by it we pass from
the one extremity,
, to the other,
; and it may thus be
considered as an instrument which carries A to B: so that a
vector may be employed to indicate a definite translation in space.
[The term "currency" has been suggested by Cayley for use instead of the somewhat vague suggestion sometimes taken to be involved in the word "direction." Thus parallel lines have the same direction, though they may have similar or opposite currencies. The definition of a vector essentially includes its currency.]
17. We may here remark, once for all, that in establishing a new Calculus, we are at liberty to give any definitions whatever of our symbols, provided that no two of these interfere with, or contradict, each other, and in doing so in Quaternions simplicity and (so to speak) naturalness were the inventor's aim.
18. Let
be represented by
, we know that
involves three separate numbers, and that these depend solely upon the
position of
relatively to
. Now if
be equal in length to
and if these lines be parallel, and have the same currency, we may
evidently write
where it will be seen that the sign of equality between vectors contains implicitly equality in length, parallelism in direction, and concurrency. So far we have extended the meaning of an algebraical symbol. And it is to be noticed that an equation between vectors, as
,
contains three distinct equations between mere numbers.
19. We must now define
(and the meaning of - will follow)
in the new Calculus. Let
be any three points, and (with
the above meaning of = ) let
If we define
(in accordance with the idea (16) that a vector
represents a translation) by the equation
or
we contradict nothing that precedes, but we at once introduce the
idea that vectors are to be compounded, in direction and magnitude,
like simultaneous velocities. A reason for this may be seen in
another way if we remember that by adding the (algebraic) differ
ences of the Cartesian coordinates of
and
, to those of the
coordinates of
and
, we get those of the coordinates of
and
. Hence these coordinates enter linearly into the expression for
a vector. (See, again, 5.)
20. But we also see that if
and
coincide (and
may be
any point)
for no vector is then required to carry
to
. Hence the above
relation
may be written, in this case,
or, introducing, and by the same act defining, the symbol
,
Hence, the symbol
, applied to a vector, simply shews that its
currency is to be reversed.
And this is consistent with all that precedes; for instance,
and
or
are evidently but different expressions of the same truth.
21. In any triangle,
, we have, of course,
and, in any closed polygon, whether plane or gauche,
In the case of the polygon we have also
These are the well-known propositions regarding composition of velocities, which, by Newton's second law of motion, give us the geometrical laws of composition of forces acting at one point.
22. If we compound any number of parallel vectors, the result is obviously a numerical multiple of any one of them.
Thus, if
are in one straight line,
where
is a number, positive when
lies between
and
, otherwise negative: but such that its numerical value, independent
of sign, is the ratio of the length of
to that of
. This is
at once evident if
and
be commensurable; and is easily
extended to incommensurables by the usual reductio ad absurdum.
23. An important, but almost obvious, proposition is that any vector may be resolved, and in one way only, into three components parallel respectively to any three given vectors, no two of which are parallel, and which are not parallel to one plane.
Let
,
,
be the three fixed c
vectors,
any other vector. From
draw
parallel to
, meeting the plane
in
. [There must be a definite point
,
else
, and therefore
, would be parallel
to
, a case specially excepted.] From
draw
parallel to
, meeting
in
.
Then we have
( 21), and these components are respectively parallel to the three given
and these components are respectively parallel to the three given vectors. By 22 we may express
as a numerical multiple
of
,
of
, and
of
. Hence we have, generally, for
any vector in terms of three fixed non-coplanar vectors,
,
which exhibits, in one form, the three numbers on which a vector
depends (16). Here
are perfectly definite, and can have
but single values.
24. Similarly any vector, as
, in the same plane with
and
, can be resolved (in one way only) into components
,
, parallel respectively to
and
; so long, at least, as these two vectors are not parallel to each other.
25. There is particular advantage, in certain cases, in em
ploying a series of three mutually perpendicular unit-vectors as
lines of reference. This system Hamilton denotes by
.
Any other vector is then expressible as
Since
are unit-vectors,
are here the lengths of
conterminous edges of a rectangular parallelepiped of which
is the vector-diagonal; so that the length of
is, in this case,
Let
be any other vector, then (by the proposition of 23) the vector equation
obyiously involves the following three equations among numbers,
Suppose
to be drawn eastwards,
northwards, and
upwards,
this is equivalent merely to saying that if two points coincide, they
are equally to the east (or west) of any third point, equally to the
north (or south) of it, and equally elevated above (or depressed below)
its level.
26. It is to be carefully noticed that it is only when
are not coplanar that a vector equation such as
or
necessitates the three numerical equations
For, if
be coplanar (24), a condition of the following form must hold
Hence
and the equation
now requires only the two numerical conditions
27. The Commutative and Associative Laws hold in the combination of vectors by the signs
and
. It is obvious that, if we
prove this for the sign
, it will be equally proved for
, because
before a vector (20) merely indicates that it is to be reversed
before being considered positive.
Let
be, in order, the corners of a parallelogram; we
have, obviously,
And
Hence the commutative law is true for the addition of any two vectors, and is therefore generally true.
Again, whatever four points are represented by
, we
have
or substituting their values for
respectively, in these three expressions,
And thus the truth of the associative law is evident.
28. The equation
where
is the vector connecting a variable point with the origin,
a definite vector, and
an indefinite number, represents the
straight line drawn from the origin parallel to
(22).
The straight line drawn from
, where
, and parallel
to
, has the equation
In words, we may pass directly from
to
by the vector
or
; or we may pass first to
, by means of
or
, and then to
along a vector parallel to
(16).
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