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% Title: On Multiple Algebra                                              %
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% Author: Josiah Willard Gibbs                                            %
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% Release Date: January 20, 2016 [EBook #50977]                           %
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The Project Gutenberg EBook of On Multiple Algebra, by Josiah Willard Gibbs

This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever.  You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
www.gutenberg.org.  If you are not located in the United States, you'll have
to check the laws of the country where you are located before using this ebook.

Title: On Multiple Algebra

Author: Josiah Willard Gibbs

Release Date: January 20, 2016 [EBook #50977]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK ON MULTIPLE ALGEBRA ***
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Produced by Andrew D. Hwang
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\PageSep{1}
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\begin{center}
\textit{ON MULTIPLE ALGEBRA.}
\bigskip
\hrule
\bigskip

\large AN \\
ADDRESS
\bigskip

\footnotesize BEFORE THE SECTION OF \\
\normalsize MATHEMATICS AND ASTRONOMY
\bigskip

\footnotesize OF \\
\LARGE THE AMERICAN ASSOCIATION
\bigskip

\footnotesize FOR THE \\
\large ADVANCEMENT OF SCIENCE
\bigskip

\footnotesize AT THE \\
\normalsize BUFFALO MEETING, AUGUST, 1886.
\vfill\vfill

\Rule
\footnotesize BY \\
\large J. WILLARD GIBBS, \\
\normalsize VICE-PRESIDENT.
\Rule
\vfill\vfill

\footnotesize [From the \textsc{Proceedings of the American Association for the Advancement
of Science}, Vol.~XXXV.]
\vfill\vfill

\large SALEM, MASS. \\
\normalsize PRINTED AT THE SALEM PRESS. \\
1886.
\end{center}
\PageSep{2}
%[Blank page]
\PageSep{3}
\MainMatter
\begin{center}
\large ADDRESS
\medskip

\normalsize BY \\
\medskip

\large J. WILLARD GIBBS,
\medskip

\footnotesize VICE PRESIDENT, SECTION~A, MATHEMATICS~AND~ASTRONOMY.

\Rule
\textit{MULTIPLE ALGEBRA.}
\Rule
\end{center}

\First{It} has been said that ``the human mind has never invented a
labor-saving machine equal to algebra.''\footnote
  {\Title{The Nation}, Vol.~XXXIII, p.~237.}
If this be true, it is
but natural and proper that an age like our own, characterized by
the multiplication of labor-saving machinery, should be distinguished
by an unexampled development of this most refined and
most beautiful of machines. That such has been the case, none
will question. The improvement has been in every part. Even to
enumerate the principal lines of advance would be a task for any
one; for me an impossibility. But if we should ask, in what direction
the advance has been made, which is to characterize the
development of algebra in our day, we may, I think, point to that
broadening of its field and methods, which gives us \emph{multiple algebra}.

Of the importance of this change in the conception of the office
of algebra, it is hardly necessary to speak: that it is really characteristic
of our time will be most evident if we go back some two-
or threescore years, to the time when the seeds were sown which
are now yielding so abundant a harvest. The failure of Möbius,
Hamilton, Grassmann, Saint-Venant to make an immediate impression
upon the course of mathematical thought in any way
commensurate with the importance of their discoveries is the most
conspicuous evidence that the times were not ripe for the methods
which they sought to introduce. A satisfactory theory of the
imaginary quantities of ordinary algebra, which is essentially a
simple case of multiple algebra, with difficulty obtained recognition
\PageSep{4}
in the first third of this century. We must observe that this
\emph{double algebra}, as it has been called, was not sought for or invented;---it
forced itself, unbidden, upon the attention of mathematicians,
and with its rules already formed.

But the idea of double algebra, once received, although as it
were unwillingly, must have suggested to many minds, more or
less distinctly, the possibility of other multiple algebras, of higher
orders, possessing interesting or useful properties.

The application of double algebra to the geometry of the plane
suggested not unnaturally to Hamilton the idea of a triple algebra
which should be capable of a similar application to the geometry
of three dimensions. He was unable to find a satisfactory triple
algebra, but discovered at length a quadruple algebra, \emph{quaternions},
which answered his purpose, thus satisfying, as he says in
one of his letters, an intellectual want which had haunted him at
least fifteen years. So confident was he of the value of this algebra,
that the same hour he obtained permission to lay his discovery
before the Royal Irish Academy, which he did on November~13,
1843.\footnote
  {\Title{Phil.\ Mag.}\ (3), Vol.~XXV, p.~490; \Title{North British Review}, Vol.~XLV (1866), p.~57.}
This system of multiple algebra is far better known
than any other, except the ordinary double algebra of imaginary
quantities,---far too well known to require any especial notice at
my hands. All that here requires our attention is the close historical
connection between the imaginaries of ordinary algebra
and Hamilton's system, a fact emphasized by Hamilton himself
and most writers on quaternions. It was quite otherwise with
Möbius and Grassmann.

The point of departure of the \Title{Barycentrischer Calcul} of Möbius,
published in~1827,---a work of which Clebsch has said that it can
never be admired enough,\footnotemark---is the use of equations in which the
\footnotetext{See his eulogy on Plücker, p.~14, \Title{Gött.\ Abhandl.}, Vol.~XVI\@.}
terms consist of letters representing points with numerical coëfficients,
to express barycentric relations between the points. Thus,
that the point~$S$ is the centre of gravity of weights, $a$,~$b$, $c$,~$d$,
placed at the points $A$,~$B$, $C$,~$D$, respectively, is expressed by
the equation
\[
(a + b + c + d)S = aA + bB + cC + dD.
\]
An equation of the more general form
\[
aA + bB + cC + \text{etc.}, = pP + qQ + rR + \text{etc.}
\]
\PageSep{5}
signifies that the weights $a$,~$b$, $c$,~etc., at the points $A$,~$B$, $C$,~etc.,
have the same sum and the same centre of gravity as the weights
$p$,~$q$, $r$,~etc., at the points $P$,~$Q$, $R$,~etc., or, in other words, that
the former are barycentrically equivalent to the latter. Such
equations, of which each represents four ordinary equations, may
evidently be multiplied or divided by scalars,\footnote
  {I use this term in Hamilton's sense, to denote the ordinary positive and negative
  quantities of algebra. It may, however, be observed that in most cases in which I
  shall have occasion to use it, the proposition would hold without exclusion of imaginary
  quantities,---that this exclusion is generally for simplicity and not from necessity.}
may be added or
subtracted, and may have their terms arranged and transposed,
exactly like the ordinary equations of algebra. It follows that
the elimination of letters representing points from equations of
this kind is performed by the rules of ordinary algebra. This is
evidently the beginning of a quadruple algebra, and is identical,
as far as it goes, with Grassmann's marvellous geometrical algebra.

In the same work we find, also, for the first time, so far as I am
aware, the distinction of positive and negative consistently carried
out on the designation of segments of lines, of triangles and
of tetrahedra, viz., that a change in place of two letters, in such
expressions as $AB$, $ABC$, $ABCD$, is equivalent to prefixing the
negative sign. It is impossible to overestimate the importance
of this step, which gives to designations of this kind the generality
and precision of algebra.

Moreover, if $A$,~$B$,~$C$ are three points in the same straight line,
and $D$~any point outside of that line, the author observes that we
have
\[
AB + BC + CA = 0,
\]
and, also, with $D$~prefixed,
\[
DAB + DBC + DCA = 0.
\]
Again, if $A$,~$B$, $C$,~$D$ are four points in the same plane, and $E$~any
point outside of that plane, we have
\[
ABC - BCD + CDA - DAB = 0,
\]
and also, with $E$~prefixed,
\[
EABC - EBCD + ECDA - EDAB = 0.
\]

The similarity to multiplication in the derivation of these formulæ
cannot have escaped the author's notice. Yet he does not
seem to have been able to generalize these processes. It was reserved
\PageSep{6}
for the genius of Grassmann to see that $AB$~might be regarded
as the product of $A$~and~$B$, $DAB$~as the product of $D$ and~$AB$,
and $EABC$~as the product of $E$ and~$ABC$. That Möbius
could not make this step was evidently due to the fact that he
had not the conception of the addition of other multiple quantities
than such as may be represented by masses situated at
points. Even the addition of vectors (\ie, the fact that the
composition of directed lines could be treated as an addition,)
seems to have been unknown to him at this time, although he
subsequently discovered it, and used it in his \Title{Mechanik des Himmels},
which was published in~1843. This addition of vectors, or
\emph{geometrical addition}, seems to have occurred independently to
many persons.

Seventeen years after the \Title{Barycentrischer Calcul}, in~1844, the year
in which Hamilton's first papers on quaternions appeared in print,
Grassmann published his \Title{Lineale Ausdehnungslehre}, in which he
developed the idea and the properties of the \emph{external} or \emph{combinatorial
product}, a conception which is perhaps to be regarded as the
greatest monument of the author's genius. This volume was to
have been followed by another, of the nature of which some intimation
was given in the preface and in the work itself. We are especially
told that the \emph{internal product},\footnote
  {See the preface.}
which for vectors is identical
except in sign with the scalar part of Hamilton's product (just as
\Typo{Grassman's}{Grassmann's} external product of two vectors is practically identical
with the vector part of Hamilton's product), and the \emph{open product},\footnote
  {See §\;172.}
which in the language of to-day would be called a matrix, were to
be treated in the second volume. But both the internal product
of vectors and the open product are clearly defined, and their fundamental
properties indicated, in this first volume.

This remarkable work remained unnoticed for more than twenty
years, a fact which was doubtless due in part to the very abstract
and philosophical manner in which the subject was presented. In
consequence of this neglect, the author changed his plan, and instead
of a supplementary volume, published in~1862 a single volume
entitled \Title{Ausdehnungslehre}, in which were treated, in an entirely
different style, the same topics as in the first volume, as well
as those which he had reserved for the second.

Deferring for the moment the discussion of these topics in order
to follow the course of events, we find in the year following the
\PageSep{7}
first \Title{Ausdehnungslehre} a remarkable memoir of Saint-Venant,\footnote
  {\Title{C. R.} Vol.~XXI, p.~620.}
in which are clearly described the addition both of vectors and of
oriented areas, the differentiation of these with respect to a scalar
quantity, and a multiplication of two vectors and of a vector and
an oriented area. These multiplications, called by the author
\emph{geometrical}, are entirely identical with Grassmann's external multiplication
of the same quantities.

It is a striking fact in the history of the subject, that the short
period of less than two years was marked by the appearance of
well-developed and valuable systems of multiple algebra by British,
German, and French authors, working apparently entirely independently
of one another. No system of multiple algebra had appeared
before, so far as I know, except such as were confined to
additive processes with multiplication by scalars, or related to the
ordinary double algebra of imaginary quantities. But the appearance
of a single one of these systems would have been sufficient
to mark an epoch, perhaps the most important epoch in the
history of the subject.

In 1853 and~1854, Cauchy published several memoirs on what
he called \emph{clefs algébriques}.\footnote
  {\Title{C. R.} Vols.~XXXVI, ff.}
These were units subject generally
to combinatorial multiplication. His principal application was to
the theory of elimination. In this application, as in the law of
multiplication, he had been anticipated by Grassmann.

We come next to Cayley's celebrated \Title{Memoir on the Theory of
Matrices}\footnote
  {\Title{Phil.\ Trans.}\ Vol.~CXLVIII.}
in~1858, of which Sylvester has said that it seems to him
to have ushered in the reign of Algebra the Second.\footnote
  {\Title{Amer.\ Journ.\ Math.}\ Vol.~VI, p.~271.}
I quote this
dictum of a master as showing his opinion of the importance of
the subject and of the memoir. But the foundations of the theory
of matrices, regarded as multiple quantities, seem to me to have
been already laid in the \Title{Ausdehnungslehre} of~1844. To Grassmann's
treatment of this subject we shall recur later.

After the \Title{Ausdehnungslehre} of~1862, already mentioned, we
come to Hankel's \Title{Vorlesungen über die complexen Zahlen}, 1867.
Under this title the author treats of the imaginary quantities of
ordinary algebra, of what he calls \emph{alternirende Zahlen}, and of
quaternions. These alternate numbers, like Cauchy's \emph{clefs}, are
quantities subject to Grassmann's law of combinatorial multiplication.
This treatise, published twenty-three years after the
\PageSep{8}
first \Title{Ausdehnungslehre}, marks the first impression which we can
discover of Grassmann's ideas upon the course of mathematical
thought. The transcendent importance of these ideas was fully
appreciated by the author, whose very able work seems to have
had considerable influence in calling the attention of mathematicians
to the subject.

In 1870, Professor Benjamin Peirce published his \emph{Linear Associative
Algebra}, subsequently developed and enriched by his son,
Professor C.~S. Peirce. The fact that the edition was lithographed
seems to indicate that even at this late date a work of this kind
could only be regarded as addressed to a limited number of readers.
But the increasing interest in such subjects is shown by the
republication of this memoir in~1881,\footnote
  {\Title{Amer.\ Journ.\ Math.}, Vol.~IV\@.}
as by that of the first \Title{Ausdehnungslehre}
in~1878.

The article on quaternions which has just appeared in the \Title{Encyclopædia
\Typo{Brittanica}{Britannica}} mentions twelve treatises, including second
editions and translations, besides the original treatises of Hamilton.
That all the twelve are later than 1861 and all but two later
than 1872 shows the rapid increase of interest in this subject in the
last years.

Finally, we arrive at the \Title{Lectures on the Principles of Universal
Algebra} by the distinguished foreigner whose sojourn among us
has given such an impulse to mathematical study in this country.
The publication of these lectures, commenced in~1884 in the \Title{American
Journal of Mathematics}, has not as yet been completed,---a
want but imperfectly supplied by the author's somewhat desultory
publication of many remarkable papers on the same subject
(which might be more definitely expressed as the algebra of matrices)
in various foreign journals.
\bigskip

It is not an accident that this century has seen the rise of multiple
algebra. The course of the development of ideas in algebra
and in geometry, although in the main independent of any aid
from this source, has nevertheless to a very large extent been of
a character which can only find its natural expression in multiple
algebra.

Our Modern Higher Algebra is especially occupied with the
theory of linear transformations. Now what are the first notions
which we meet in this theory? We have a set of $n$~variables, say
\PageSep{9}
$x$,~$y$,~$z$, and another set, say $x'$,~$y'$,~$z'$, which are homogeneous linear
functions of the first, and therefore expressible in terms of
them by means of a block of $n^{2}$~coëfficients. Here the quantities
occur by sets, and invite the notations of multiple algebra. It
was in fact shown by Grassmann in his first \Title{Ausdehnungslehre} and
by Cauchy nine years later, that the notations of multiple algebra
afford a natural key to the subject of elimination.

Now I do not merely mean that we may save a little time or
space by writing perhaps~$\rho$ for $x$,~$y$ and~$z$; $\rho'$~for $x'$,~$y'$ and~$z'$; and $\Phi$~for
a block of $n^{2}$~quantities. But I mean that the subject as usually
treated under the title of determinants has a stunted and misdirected
development on account of the limitations of single algebra.
This will appear from a very simple illustration. After a little
preliminary matter, the student comes generally to a chapter entitled
``Multiplication of Determinants,'' in which he is taught that
the product of the determinants of two matrices may be found by
performing a somewhat lengthy operation on the two matrices, by
which he obtains a third matrix, and then taking the determinant
of this. But what significance, what value has this theorem? For
aught that appears in the majority of treatises which I have seen, we
have only a complicated and lengthy way of performing a simple
operation. The real facts of the case may be stated as follows:

Suppose the set of $n$~quantities~$\rho'$ to be derived from the set~$\rho$
by the matrix~$\Phi$, which we may express by
\[
\rho' = \Phi · \rho;
\]
and suppose the set~$\rho''$ to be derived from the set~$\rho'$ by the matrix~$\Psi$, \ie,
\[
\rho'' = \Psi · \rho',
\]
and
\[
\rho'' = \Psi · \Phi · \rho;
\]
it is evident that $\rho''$~can be derived from~$\rho$ by the operation of a
single matrix, say~$\Theta$, \ie,
\[
\rho'' = \Theta · \rho,
\]
so that
\[
\Theta = \Psi · \Phi.
\]

In the language of multiple algebra $\Theta$~is called the product of
$\Psi$~and~$\Phi$. It is of course interesting to see how it is derived from
the latter, and it is little more than a schoolboy's exercise to determine
this. Now this matrix~$\Theta$ has the property that its determinant
is equal to the products of the determinants of $\Psi$ and~$\Phi$.
And this property is all that is generally stated in the books, and
\PageSep{10}
the fundamental property, which is all that gives the subject its
interest, that $\Theta$~is itself the product of $\Psi$ and~$\Phi$ in the language of
multiple algebra, \ie, that operating by~$\Theta$ is equivalent to operating
successively by $\Phi$ and~$\Psi$, is generally omitted. The chapter
on this subject, in most treatises which I have seen, reads very like
the play of Hamlet with Hamlet's part left out.

And what is the cause of this omission? Certainly not ignorance
of the property in question. The fact that it is occasionally
given would be a sufficient bar to this answer. It is because the
author fails to see that his real subject is matrices and not determinants.
Of course, in a certain sense, the author has a right to
choose his subject. But this does not mean that the choice is unimportant,
or that it should be determined by chance or by caprice.
The problem well put is half solved, as we all know. If one chooses
the subject ill, it will develop itself in a cramped manner.

But the case is really much worse than I have stated it. Not
only is the true significance of the formation of~$\Theta$ from $\Psi$ and~$\Phi$ not
given, but the student is often not taught to form the matrix
which is the product of $\Psi$ and~$\Phi$, but one which is the product of one
of these matrices and the conjugate of the other. Thus the proposition
which is proved loses all its simplicity and significance,
and must be recast before the instructor can explain its true bearings
to the student. This fault has been denounced by Sylvester,
and if anyone thinks I make too much of the standpoint from which
the subject is viewed, I will refer him to the opening paragraphs
of the ``Lectures on Universal Algebra'' in the sixth volume of the
\Title{American Journal of Mathematics}, where, with a wealth of illustration
and an energy of diction which I cannot emulate, the most
eloquent of mathematicians expresses his sense of the importance
of the substitution of the idea of the matrix for that of the determinant.
If then so important, why was the idea of the matrix let
slip? Of course the writers on this subject had it to commence
with. One cannot even define a determinant without the idea of
a matrix. The simple fact is that in general the writers on this
subject have especially developed those ideas, which are naturally
expressed in simple algebra, and have postponed or slurred over
or omitted altogether those ideas which find their natural expression
in multiple algebra. But in this subject the latter happen
to be the fundamental ideas, and those which ought to direct the
whole course of thought.

I have taken a very simple illustration, perhaps the very first
\PageSep{11}
theorem which meets the student after those immediately connected
with the introductory definitions, both because the simplest illustration
is really the best, and because I am here most at home.
But the principles of multiple algebra seem to me to shed a flood
of light into every corner of the subjects usually treated under the
title of determinants, the subject gaining as much in breadth from
the new notions as in simplicity from the new notations; and in
the more intricate subjects of invariants, covariants, etc., I believe
that the principles of multiple algebra are ready to perform
an equal service. Certainly they make many things seem very
simple to me, which I should otherwise find difficult of comprehension.

Let us turn to geometry.

If we were asked to characterize in a single word our modern
geometry, we would perhaps say that it is a geometry of position.
Now position is essentially a multiple quantity, or if you prefer,
is naturally represented in algebra by a multiple quantity. And
the growth in this century of the so-called synthetic as opposed to
analytical geometry seems due to the fact that by the ordinary
analysis geometers could not easily express, except in a cumbersome
and unnatural manner, the sort of relations in which they
were particularly interested. With the introduction of the notations
of multiple algebra, this difficulty falls away, and with it the
opposition between synthetic and analytical geometry.

It is, however, interesting and very instructive to observe how
the ingenuity of mathematicians has often triumphed over the limitations
of ordinary algebra. A conspicuous example and one of
the simplest is seen in the \Title{Mécanique Analytique}, where the author,
by the use of what are sometimes called indeterminate equations,
is able to write in one equation the equivalent of an indefinite
number. Thus the equation
\[
X\, dx+ Y\, dy + Z\, dz = 0,
\]
by the indeterminateness of the values of $dx$, $dy$, $dz$, is made equivalent
to the three equations
\[
X = 0,\quad Y =0,\quad Z = 0.
\]
It is instructive to compare this with
\[
Xi + Yj + Zk = 0,
\]
which is the form that Hamilton or Grassmann would have used.
\PageSep{12}
The use of this analytical artifice, if such it can be called, runs all
through the work and is fairly characteristic of it.

Again, the introduction of the potential in the theory of gravity,
or electricity, or magnetism, gives us a scalar quantity instead of
a vector as the subject of study; and in mechanics generally the
use of the force-function substitutes a simple quantity for a complex.
This method is in reality not different from that just mentioned,
since Lagrange's indeterminate equation expresses, at least
in its origin, the variation of the force-function. It is indeed the
real beauty of Lagrange's method that it is not so much an analytical
artifice, as the natural development of the subject.

In modern analytical geometry we find methods in use which are
exceedingly ingenious, and give forms curiously like those of multiple
algebra, but which, at least if logically carried out very far,
are excessively artificial, and that for the expression of the simplest
things. The simplest conceptions of the geometry of three dimensions
are points and planes, and the simplest relation between
these is that a point lies in a plane. Let us see how these notions
have been handled by means of ordinary algebra, and by multiple
algebra. It will illustrate the characteristic difference of the
methods, perhaps as well as the reading of an elaborate treatise.

In multiple algebra a point is designated by a single letter, just
as it is in what is called synthetic geometry, and as it generally is
by the ordinary analyst, when he is not writing equations. But in
his equations, instead of a single letter the analyst introduces several
letters (coördinates) to represent the point.

A plane may be represented in multiple algebra as in synthetic
geometry by a single letter; in the ordinary algebra it is sometimes
represented by three coördinates, for which it is most convenient
to take the reciprocals of the segments cut off by the plane
on three axes. But the modern analyst has a more ingenious method
of representing the plane. He observes that the equation of the
plane may be written
\[
\xi x + \eta y + \zeta z = 1,
\Tag{(1)}
\]
where $\xi$,~$\eta$,~$\zeta$ are the reciprocals of the segments, and $x$,~$y$,~$z$ are
the coördinates of any point in the plane. Now if we set
\[
p = \xi x + \eta y + \zeta z,
\Tag{(2)}
\]
this letter will represent an expression which represents the plane.
In fact, we may say that $p$~implicitly contains $\xi$,~$\eta$, and~$\zeta$, which are
\PageSep{13}
the coördinates of the plane. We may therefore speak of the plane~$p$,
and for many purposes can introduce the letter~$p$ into our equations
instead of $\xi$,~$\eta$,~$\zeta$. For example, the equation
\[
p''' = \frac{p' + p''}{2}
\Tag{(3)}
\]
is equivalent to the three equations
\[
\xi''' = \frac{\xi' + \xi''}{2},\qquad
\eta''' = \frac{\eta' + \eta''}{2},\qquad
\zeta''' = \frac{\zeta' + \zeta''}{2}.
\Tag{(4)}
\]

It is to be noticed that on account of the indeterminateness of the
$x$,~$y$, and~$z$, this method, regarded as an analytical artifice, is identical
with that of Lagrange, also that in multiple algebra we should
have an equation of precisely the same form as~\Eq{(3)} to express
the same relation between the planes, but that the equation would
be explained to the student in a totally different manner. This we
shall see more particularly hereafter.

It is curious that we have thus a simpler notation for a plane
than for a point. This however may be reversed. If we commence
with the notion of the coördinates of a plane, $\xi$,~$\eta$,~$\zeta$, the equation
of a point (\ie, the equation between $\xi$,~$\eta$,~$\zeta$ which will hold for
every plane passing through the point) will be
\[
x\xi + y\eta + z\zeta = 1,
\Tag{(5)}
\]
where $x$,~$y$,~$z$ are the coördinates of the point. Now if we set
\[
q = x\xi + y\eta + z\zeta,
\Tag{(6)}
\]
we may regard the single letter~$q$ as representing the point, and
use it, in many cases, instead of the coördinates $x$,~$y$,~$z$, which indeed
it implicitly contains. Thus we may write
\[
q''' = \frac{q' + q''}{2}
\Tag{(7)}
\]
for the three equations
\[
x''' = \frac{x' + x''}{2},\qquad
y''' = \frac{y' + y''}{2},\qquad
z''' = \frac{z' + z''}{2}.
\Tag{(8)}
\]

Here, by an analytical artifice, we come to equations identical in
form and meaning to those used by Hamilton, Grassmann, and
even by Möbius in~1827. But the explanations of the formulæ
would differ widely. The methods of the founders of multiple algebra
are characterized by a bold simplicity, that of the modern
geometry by a somewhat bewildering ingenuity. That $p$~and~$q$
represent the same expression (in one case $x$,~$y$,~$z$, and in the other
$\xi$,~$\eta$,~$\zeta$ being indeterminate) is a circumstance which may easily become
perplexing. I am not quite certain that it would be convenient
\PageSep{14}
to use both of these abridged notations at the same time. In fact,
if the geometer using these methods were asked to express by an
equation in $p$~and~$q$ that the point~$q$ lies in the plane~$p$, he might
find himself somewhat entangled in the meshes of his own ingenuity,
and need some new artifice to extricate himself. I do not
mean that his genius might not possibly be equal to the occasion,
but I do mean very seriously that it is a vicious method which
requires any ingenuity or any artifice to express so simple a relation.

If we use the methods of multiple algebra which are most comparable
to those just described, a point is naturally represented by
a vector~($\rho$) drawn to it from the origin, a plane by a vector~($\sigma$)
drawn from the origin perpendicularly toward the plane and in
length equal to the reciprocal of the distance of the plane from the
origin. The equation
\[
\sigma''' = \frac{\sigma' + \sigma''}{2}
\Tag{(9)}
\]
will have precisely the same meaning as equation~\Eq{(3)}, and
\[
\rho''' = \frac{\rho' + \rho''}{2}
\Tag{(10)}
\]
will have precisely the same meaning as equation~\Eq{(7)}, viz., that
the point~$\rho'''$ is in the middle between $\rho'$~and~$\rho''$. That the point~$\rho$
lies in the plane~$\sigma$ is expressed by equating to unity the product
of $\rho$~and~$\sigma$ called by Grassmann internal, or by Hamilton called
the scalar part of the product taken negatively. By whatever
name called, the quantity in question is the product of the lengths
of the vectors and the cosine of the included angle. It is of course
immaterial what particular sign we use to express this product, as
whether we write
\[
\rho · \sigma = 1,\qquad\text{or}\qquad
S\rho\sigma = -1.
\Tag{(11)}
\]
I should myself prefer the simplest possible sign for so simple a
relation. It may be observed that $\rho$~and~$\sigma$ may be expressed as
the geometrical sum of their components parallel to a set of perpendicular
axes, viz.,
\[
\rho = xi + yj + zk,\qquad
\sigma = \xi i + \eta j + \zeta k.
\Tag{(12)}
\]
By substitution of these values, equation~\Eq{(11)} becomes by the laws
of this kind of multiplication
\[
x\xi + y\eta + z\zeta = 1.
\Tag{(13)}
\]

My object in going over these elementary matters is to call attention
to the very roundabout way in which the ordinary analysis
\PageSep{15}
makes out to represent a point or a plane by a single letter, as distinguished
from the directness and simplicity of the notations of
multiple algebra, and also to the fact that the representations of
points and planes by single letters in the ordinary analysis are not,
when obtained, as amenable to analytical treatment as are the notations
of multiple algebra.

I have compared that form of the ordinary analysis which relates
to Cartesian axes with a vector analysis. But the case is essentially
the same, if we compare the form of ordinary analysis which
relates to a fundamental tetrahedron with Grassmann's geometrical
analysis, founded on the point as the elementary quantity.

In the method of ordinary analysis, a point is represented by
four coördinates, of which each represents the distance of the point
from a plane of the tetrahedron divided by the distance of the opposite
vertex from the same plane. The equation of a plane may
be put in the form
\[
\xi x + \eta y + \zeta z + \omega w = 0,
\Tag{(14)}
\]
where $\xi$,~$\eta$, $\zeta$,~$\omega$ are the distances of the plane from the four points,
and $x$,~$y$, $z$,~$w$ are the coördinates of any point in the plane. Here
we may set
\[
p = \xi x + \eta y + \zeta z + \omega w,
\Tag{(15)}
\]
and say that $p$~represents the plane. To some extent we can introduce
this letter into equations instead of $\xi$,~$\eta$, $\zeta$,~$\omega$. Thus the
equation
\[
lp' + mp'' + np''' = 0
\Tag{(16)}
\]
(which denotes that the planes $p'$,~$p''$,~$p'''$, meet in a common line,
making angles of which the sines are proportional to $l$,~$m$, and~$n$)
is equivalent to the four equations
\[
l\xi' + m\xi'' + n\xi''' = 0,\qquad
l\eta' + m\eta'' + n\eta''' = 0,\qquad
\text{etc.}
\Tag{(17)}
\]

Again, we may regard $\xi$,~$\eta$, $\zeta$,~$\omega$ as the coördinates of a plane.
The equation of a point will then be
\[
x\xi + y\eta + z\zeta + w\omega = 0.
\Tag{(18)}
\]
If we set
\[
q = x\xi + y\eta + z\zeta + w\omega,
\Tag{(19)}
\]
we may say that $q$~represents the point. The equation
\[
q''' = \frac{q' + q''}{2},
\Tag{(20)}
\]
which indicates that the point~$q'''$ bisects the line between $q'$~and~$q''$,
is equivalent to the four equations
\PageSep{16}
\[
\Typo{\xi' = \frac{\xi'' + \xi'''}{2}}{\xi''' = \frac{\xi' + \xi''}{2}},
\qquad
\Typo{\eta' = \frac{\eta'' + \eta'''}{2}}{\eta''' = \frac{\eta' + \eta''}{2}}, \quad\text{etc.}
\Tag{(21)}
\]

To express that the point~$q$ lies in the plane~$p$ does not seem
easy, without going back to the use of coördinates.

The form of multiple algebra which is to be compared to this is
the geometrical algebra of Möbius and Grassmann, in which points
without reference to any origin are represented by single letters,
say by Italic capitals, and planes may also be represented by
single letters, say by Greek capitals. An equation like
\[
Q''' = \frac{Q' + Q''}{2},
\Tag{(22)}
\]
has exactly the same meaning as equation~\Eq{(20)} of ordinary algebra.
So
\[
l\Pi' + m\Pi'' + n\Pi''' = 0
\Tag{(23)}
\]
has precisely the same meaning as equation~\Eq{(16)} of ordinary algebra.
That the point~$Q$ lies in the plane~$\Pi$ is expressed by equating
to zero the product of $Q$ and~$\Pi$ which is called by Grassmann external
and which might be defined as the distance of the point from
the plane. We may write this
\[
Q × \Pi = 0.
\Tag{(24)}
\]
To show that so simple an expression is really amenable to analytical
treatment, I observe that $Q$~may be expressed in terms of any
four points (not in the same plane) on the barycentric principle
explained above, viz.,
\[
Q = xA + yB + zC + wD,
\Tag{(25)}
\]
and $\Pi$~may be expressed in terms of combinatorial products of $A$,
$B$, $C$, and~$D$, viz.,
\[
\Pi = \xi B × C × D + \eta C × A × D + \zeta D × A × B + \omega A × C × B\Typo{.}{,}
\Tag{(26)}
\]
and by these substitutions, by the laws of the combinatorial product
to be mentioned hereafter, equation~\Eq{(24)} is transformed into
\[
w\omega + x\xi + y\eta + z\zeta = 0,
\Tag{(27)}
\]
which is identical with the formula of ordinary analysis.\footnote
  {The letters $\xi$,~$\eta$, $\zeta$,~$\omega$, here denote the distances of the plane~$\Pi$ from the points $A$,~$B$,
  $C$,~$D$, divided by six times the volume of the tetrahedron \Typo{$A$,~$B$, $C$,~$D$}{$ABCD$}. The letters $x$,~$y$,
  $z$,~$w$, denote the tetrahedral coördinates as above.}

I have gone at length into this very simple point, in order to illustrate
the fact which I think is a general one, that the modern
geometry is not only tending to results which are appropriately
expressed in multiple algebra, but that it is actually striving to
\PageSep{17}
clothe itself in forms which are remarkably similar to the notations
of multiple algebra, only less simple and general, and far less amenable
to analytical treatment, and therefore, that a certain logical
necessity calls for throwing off the yoke under which analytical
geometry has so long labored. And lest this should seem to be
the utterance of an uninformed enthusiasm, or the echoing of the
possibly exaggerated claims of the devotees of a particular branch
of mathematical study, I will quote a sentence from Clebsch and
from Clifford, relating to the past and to the future of multiple algebra.
The former in his eulogy on Plücker,\footnote
  {\Title{Gött.\ Abhandl.}, Vol.~16, p.~28.}
in~1871, speaking
of recent advances in geometry, says that ``in a certain sense the
coördinates of a straight line, and in general a great part of the
fundamental conceptions of the newer algebra, are contained in the
\Title{Ausdehnungslehre} of~1844,'' and Clifford\footnote
  {\Title{Amer.\ Journ.\ Math.}, Vol.~I, p.~350.}
in the last year of his life,
speaking of the \Title{Ausdehnungslehre}, with which he had but recently
become acquainted, expresses ``his profound admiration of that extraordinary
work, and his conviction that its principles will exercise
a vast influence upon the future of mathematical science.''

Another subject in which we find a tendency toward the forms
and methods of multiple algebra, is the calculus of operations.
Our ordinary analysis introduces operators; and the successive
operations $A$~and~$B$ may be equivalent to the operation~$C$. To
express this in an equation we may write
\[
BA(x) = C(x),
\]
where $x$~is any quantity or function. We may also have occasion
to write
\[
A(x) + B(x) = D(x),\qquad\text{or}\qquad
(A + B)(x) = D(x).
\]
But it is almost impossible to resist the tendency to express these
relations in the form
\begin{align*}
BA &= C, \\
A + B &= D,
\end{align*}
in which the operators appear in a sense as quantities, \ie, as subjects
of functional operation. Now since these operators are often
of such nature that they cannot be perfectly specified by a single
numerical quantity, when we treat them as quantities they must
be regarded as multiple quantities. In this way certain formulæ
which essentially belong to multiple algebra get a precarious footing
where they are only allowed because they are regarded as
\PageSep{18}
abridged notations for equations in ordinary algebra. Yet the
logical development of such notations would lead a good way in
multiple algebra, and doubtless many investigators have entered
the field from this side.

One might also notice, to show how the ordinary algebra is becoming
saturated with the notions and notations which seem destined
to turn it into a multiple algebra, the notation so common
in the higher algebra
\[
(a, b, c)\! (x, y, z)
\]
for
\[
ax + by + cz.
\]
This is evidently the same as Grassmann's internal product of the
multiple quantities $(a, b, c)$ and $(x, y, z)$, or, in the language of
quaternions, the scalar part, taken negatively, of the product of
the vectors of which $a$,~$b$,~$c$ and $x$,~$y$,~$z$ are the components. A
similar correspondence with Grassmann's methods might, I think,
be shown in such notations as, for example,
\[
(a, b, c, d)\! (x, y)^{3}.
\]
The free admission of such notations is doubtless due to the fact
that they are regarded simply as abridged notations.

The author of the celebrated ``Memoir on the Theory of Matrices,''
goes much farther than this in his use of the forms of multiple algebra.
Thus he writes explicitly one equation to stand for several,
without the use of any of the analytical artifices which have been
mentioned. This work has indeed, as we have seen, been characterized
as marking the commencement of multiple algebra,---a
view to which we can only take exception as not doing justice to
earlier writers.

But the significance of this memoir with regard to the point
which I am now considering is that it shows that the chasm so
marked in the second quarter of this century is destined to be
closed up. Notions and notations for which a Cayley is sponsor
will not be excluded from good society among mathematicians.
And if we admit as suitable the notations used in this memoir
(where it is noticeable that the author rather avoids multiple algebra,
and only uses it very sparingly), we shall logically be brought
to use a great deal more. For example, if it is a good thing to
write in our equations a single letter to represent a matrix of $n^{2}$~numerical
quantities, why not use a single letter to represent the
$n$~quantities operated upon, as Grassmann and Hamilton have done?
\PageSep{19}
Logical consistency seems to demand it. And if we may use the
sign~$)\!($ to denote an operation by which two sets of quantities are
combined to form a third set, as is the case in this memoir, why not
use other signs to denote other functional operations of which the
result is a multiple quantity? If it be conceded that this is the
proper method to follow where simplicity of conception, or brevity
of expression, or ease of transformation is served thereby, our algebra
will become in large part a multiple algebra.
\bigskip

We have considered the subject a good while from the outside;
we have glanced at the principal events in the history of multiple
algebra; we have seen how the course of modern thought seems to
demand its aid, how it is actually leaning toward it, and beginning
to adopt its methods. It may be worth while to direct our
attention more critically to multiple algebra itself, and inquire into
its essential character and its most important principles.

I do not know that anything useful or interesting, which relates
to multiple quantity, and can be symbolically expressed, falls outside
of the domain of multiple algebra. But if it is asked, what
notions are to be regarded as fundamental, we must answer, here
as elsewhere, those which are most simple and fruitful. Unquestionably,
no relations are more so, than those which are known by
the names of addition and multiplication.

Perhaps I should here notice the essentially different manner in
which the multiplication of multiple quantities has been viewed
by different writers. Some, as Hamilton, or De~Morgan, or Peirce,
speak of the product of two multiple quantities, as if only one
product could exist, at least in the same algebra. Others, as
Grassmann, speak of various kinds of products for the same multiple
quantities. Thus Hamilton seems for many years to have
agitated the question, what he should regard as the product of
each pair of a set of triplets, or in the geometrical application of
the subject, what he should regard as the product of each pair of a
system of perpendicular directed lines.\footnote
  {\Title{Phil.\ Mag.}, (3), \Chg{\textsc{xxv}}{XXV}, p.~490; \Title{North British Review}, \Chg{\textsc{xlv}}{XLV} (1866), p.~57.}
Grassmann asks, What
products, \ie, what distributive functions of the multiple quantities
are most important?

It may be that in some cases the fact that only one kind of product
is known in ordinary algebra has led those to whom the problem
presented itself in the form of finding a new algebra to adopt
this characteristic derived from the old. Perhaps the reason lies
\PageSep{20}
deeper in a distinction like that in arithmetic between concrete
and abstract numbers or quantities. The multiple quantities corresponding
to concrete quantities such as ten apples or three miles,
are evidently such combinations as $\text{ten apples} + \text{seven oranges}$,
$\text{three miles northward} + \text{five miles eastward}$, or six miles in a
direction fifty degrees east of north. Such are the fundamental
multiple quantities from Grassmann's point of view. But if we
ask what it is in multiple algebra which corresponds to an abstract
number like twelve, which is essentially an operator, which changes
one mile into twelve miles, and \$$1,000$ into \$$12,000$, the most general
answer would evidently be, an operator which will work such
changes as, for example, that of $\text{ten apples} + \text{seven oranges}$ into
$\text{fifty apples} + 100 \text{ oranges}$, or that of one vector into another.

Now an operator has, of course, one characteristic relation, viz.,
its relation to the operand. This needs no especial definition,
since it is contained in the definition of the operator. If the operation
is distributive, it may not inappropriately be called multiplication,
and the result is \textit{par excellence} the product of the operator
and operand. The sum of operators \textit{quâ} operators, is an operator
which gives for the product the sum of the products given by the operators
to be added. The product of two operators is an operator
which is equivalent to the successive operations of the factors.
This multiplication is necessarily associative, and its definition is
not really different from that of the operators themselves. And
here I may observe that Professor C.~S. Peirce has shown that
his father's associative algebras may be regarded as operational
and matricular.\footnote
  {\Title{Amer.\ Journ.\ Math.}, Vol.~IV, p.~221.}

Now, the calculus of distributive operators is a subject of great
extent and importance, but Grassmann's view is the more comprehensive,
since it embraces the other with something besides. For
every quantitative operator may be regarded as a quantity, \ie,
as the subject of mathematical operation, but every quantity cannot
be regarded as an operator; precisely as in grammar every
verb may be taken as substantive, as in the infinitive, while every
substantive does not give us a verb.

Grassmann's view seems also the most practical and convenient.
For we often use many functions of the same pair of multiple
quantities, which are distributive with respect to both, and we
need some simple designation to indicate a property of such fundamental
importance in the algebra of such functions, and no advantage
\PageSep{21}
appears in singling out a particular function to be alone
called the product. Even in quaternions, where Hamilton speaks
of only one product of two vectors (regarding it as a special case
of the product of quaternions, \ie, of operators), he nevertheless
comes to use the scalar part of this product and the vector part
separately. Now the distributive law is satisfied by each of these,
which, therefore, may conveniently be called products. In this
sense we have three kinds of products of vectors in Hamilton's
analysis.

Let us then adopt the more general view of multiplication, and
call any function of two or more multiple quantities, which is distributive
with respect to all, a product, with only this limitation,
that when one of the factors is simply an ordinary algebraic quantity
its effect is to be taken in the ordinary sense.

It is to be observed that this definition of multiplication implies
that we have an addition both of the kind of quantity to which the
product belongs, and of the kinds to which the factors belong. Of
course, these must be subject to the general formal laws of addition.
I do not know that it is necessary for the purposes of a
general discussion to stop to define these operations more particularly,
either on their own account or to complete the definition
of multiplication. Algebra, as a formal science, may rest on a
purely formal foundation. To take our illustration again from
mechanics, we may say that if a man is inventing a particular
machine,---a sewing machine,---a reaper,---nothing is more important
than that he should have a precise idea of the operation which
his machine is to perform, yet when he is treating the general
principles of mechanics he may discuss the lever, or the form of
the teeth of wheels which will transmit uniform motion, without
inquiring the purpose to which the apparatus is to be applied; and
in like manner that if we were forming a particular algebra,---a
geometrical algebra,---a mechanical algebra,---an algebra for the
theory of elimination and substitution,---an algebra for the study
of quantics,---we should commence by asking, What are the multiple
quantities, or sets of quantities, which we have to consider?
What are the additive relations between them? What are the
multiplicative relations between them? etc., forming a perfectly
defined and complete idea of these relations as we go along; but
in the development of a general algebra no such definiteness of
conception is requisite. Given only the purely formal law of the
distributive character of multiplication,---this is sufficient for the
\PageSep{22}
foundation of a science. Nor will such a science be merely a pastime
for an ingenious mind. It will serve a thousand purposes in
the formation of particular algebras. Perhaps we shall find that
in the most important cases, the particular algebra is little more
than an application or interpretation of the general.

Grassmann observes that any kind of multiplication of $n$-fold
quantities is characterized by the relations which hold between
the products of $n$~independent units. In certain kinds of multiplication
these characteristic relations will hold true of the products
of any of the quantities.

Thus if the value of a product is independent of the order of the
factors when these belong to the system of units, it will always
be independent of the order of the factors. The kind of multiplication
characterized by this relation and no other between the
products is called by Grassmann \emph{algebraic}, because its rules coincide
with those of ordinary algebra. It is to be observed, however,
that it gives rise to multiple quantities of higher orders. If $n$~independent
units are required to express the original quantities, $n\, \frac{n+1}{2}$~units
will be required for the products of two factors, $n\, \frac{(n+1)(n+2)}{2 · 3}$
for the products of three factors,~etc.

Again, if the value of a product of factors belonging to a system
of units is multiplied by~$-1$ when two factors change places, the
same will be true of the product of any factors obtained by addition
of the units. The kind of multiplication characterized by
this relation and no other is called by Grassmann \emph{external} or \emph{combinatorial}.
For our present purpose we may denote it by the sign~$×$.
It gives rise to multiple quantities of higher orders, $n\, \frac{n-1}{2}$~units
being required to express the products of two factors,
$n\, \frac{(n-1)(n-2)}{2 · 3}$ units for products of three factors,~etc. All products
of more than $n$~factors are zero. The products of $n$~factors may
be expressed by a single unit, viz., the product of the $n$~original
units taken in a specified order, which is generally set equal to~$1$.
The products of $n - 1$ factors are expressed in terms of $n$~units, those
of $n - 2$ factors in terms of $n\, \frac{n-1}{2}$~units, etc. This kind of multiplication
is associative, like the algebraic.

Grassmann observes, with respect to binary products, that these
two kinds of multiplication are the only kinds characterized by
laws which are the same for any factors as for particular units, except
indeed that characterized by no special laws, and that for
which all products are zero.\footnote
  {Crelle's \Title{Journ.\ f.\ Math.}, Vol.~XLIX, p.~138.}
The last we may evidently reject as
\PageSep{23}
nugatory. That for which there are no special laws, \ie, in which
no equations subsist between the products of a system of independent
units, is also rejected by Grassmann, as not appearing to
afford important applications. I shall, however, have occasion to
speak of it, and shall call it the indeterminate product. In this
kind of multiplication, $n^{2}$~units are required to express the products
of two factors, and $n^{3}$~units for products of three factors, etc. It
evidently may be regarded as associative.

Another very important kind of multiplication is that called by
Grassmann \emph{internal}. In the form in which I shall give it, which is
less general than Grassmann's, it is in one respect the most simple
of all, since its only result is a numerical quantity. It is essentially
binary and characterized by laws of the form
\begin{gather*}
i · i = 1,\qquad j · j = 1,\qquad k · k = 1,\quad\text{etc.,} \\
i · j = 0,\qquad j · i = 0,\quad\text{etc.,}
\end{gather*}
where $i$,~$j$,~$k$, etc., represent a system of independent units. I use
the dot as significant of this kind of multiplication.

Grassmann derives this kind of multiplication from the combinatorial
by the following process. He defines the complement
(Ergänzung) of a unit as the combinatorial product of all the other
units, taken with such a sign that the combinatorial product of
the unit and its complement shall be positive. The combinatorial
product of a unit and its complement is therefore unity, and that
of a unit and the complement of any other unit is zero. The internal
product of two units is the combinatorial product of the first
and the complement of the second.

It is important to observe that any scalar product of two factors
of the same kind of multiple quantities, which is positive when the
factors are identical, may be regarded as an internal product, \ie,
we may always find such a system of units, that the characteristic
equations of the product will reduce to the above form. The nature
of the subject may afford a definition of the product independent
of any reference to a system of units. Such a definition
will then have obvious advantages. An important case of this
kind occurs in geometry in that product of two vectors which is
obtained by multiplying the products of their lengths by the cosine
of the angle which they include. This is an internal product
in Grassmann's sense.

Let us now return to the indeterminate product, which I am
inclined to regard as the most important of all, since we may derive
from it the algebraic and the combinatorial. For this end,
\PageSep{24}
we will prefix~$\sum$ to an indeterminate product to denote the sum of
all the terms obtained by taking the factors in every possible order.
Then,
\[
\tsum \alpha \mid \beta \mid \gamma,
\]
for instance, where the vertical line is used to denote the indeterminate
product,\footnote
  {This notation must not be confounded with Grassmann's use of the vertical line.}
is a distributive function of $\alpha$,~$\beta$ and~$\gamma$. It is
evidently not affected by changing the order of the letters. It
is, therefore, an algebraic product in the sense in which the term
has been defined.

So, again, if we prefix~$\sum_{\pm}$ to an indeterminate product to denote
the sum of all terms obtained by giving the factors every
possible order, those terms being taken negatively which are obtained
by an odd number of simple permutations,
\[
\tsum_{\pm} \alpha \mid \beta \mid \gamma,
\]
for instance, will be a distributive function of $\alpha$,~$\beta$,~$\gamma$, which is
multiplied by~$-1$ when two of these letters change places. It
will therefore be a combinatorial product.

It is a characteristic and very important property of an indeterminate
product that every product of all its factors with any
other quantities is also a product of the indeterminate product and
the other quantities. We need not stop for a formal proof of this
proposition, which indeed is an immediate consequence of the
definitions of the terms.

These considerations bring us naturally to what Grassmann calls
\emph{regressive multiplication}, which I will first illustrate by a very
simple example. If $n$, the degree of multiplicity of our original
quantities, is~$4$, the combinatorial product of $\alpha × \beta × \gamma$ and $\delta × \eps$,
viz.,
\[
\alpha × \beta × \gamma × \delta × \eps,
\]
is necessarily zero, since the number of factors exceeds four. But
if for~$\delta × \eps$ we set its equivalent
\[
\delta \mid \eps - \eps \mid \delta,
\]
we may multiply the first factor in each of these indeterminate
products combinatorially by  $\alpha × \beta × \gamma$, and prefix the result,
which is a numerical quantity, as coëfficient to the second factor.
This will give
\[
(\alpha × \beta × \gamma × \delta) \eps - (\alpha × \beta × \gamma × \eps) \delta.
\]
\PageSep{25}
Now, the first term of this expression is a product of $\alpha × \beta × \gamma$,
$\delta$, and~$\eps$, and therefore, by the principle just stated, a product of
$\alpha × \beta × \gamma$ and~$\delta \mid \eps$. The second term is a similar product of
$\alpha × \beta × \gamma$ and~$\eps \mid \delta$. Therefore the whole expression is a product
of $\alpha × \beta × \gamma$ and $\delta \mid \eps - \eps \mid \delta$, that is, of $\alpha × \beta × \gamma$ and~$\delta × \eps$.
This is, except in sign, what Grassmann calls the \emph{regressive product}
of $\alpha × \beta × \gamma$ and~$\delta × \eps$.

To generalize this process, we first observe that an expression
of the form
\[
\tsum_{\pm} \alpha × \beta \mid \gamma × \delta,
\]
in which each term is an indeterminate product of two combinatorial
products, and in which $\sum_{\pm}$ denotes the sum of all terms obtained
by putting every different pair of the letters before the
dividing line, the negative sign being used for any terms which
may be obtained by an odd number of simple permutations of the
letters,---in other words, the expression
\begin{multline*}
\alpha × \beta \mid \gamma × \delta
  - \alpha × \gamma \mid \beta × \delta
  - \alpha × \delta \mid \gamma × \beta \\
  + \beta × \gamma \mid \alpha × \delta
  - \beta × \delta \mid \alpha × \gamma
  + \gamma × \delta \mid \alpha × \beta,
\end{multline*}
is a distributive function of $\alpha$,~$\beta$, $\gamma$, and~$\delta$, which is multiplied by~$-1$
when two of these letters change places, and may, therefore,
be regarded as equivalent to the combinatorial product $\alpha × \beta × \gamma × \delta$. Now, if $n = 5$, the combinatorial product of
\[
\rho × \sigma × \tau\qquad\text{and}\qquad
\alpha × \beta × \gamma × \delta
\]
is zero. But if we multiply the first member of each of the above
indeterminate products by $\rho × \sigma × \tau$, and prefix the result as
coëfficient to the second member, we obtain
\[
(\rho × \sigma × \tau × \alpha × \beta) \gamma × \delta - (\rho × \sigma × \tau × \alpha × \gamma) \beta × \delta + \text{etc.,}
\]
which is what Grassmann calls the regressive product of $\rho × \sigma × \tau$
and $\alpha × \beta × \gamma × \delta$. It is easy to see that the principle may be
extended so as to give a regressive product in any case in which
the total number of factors of two combinatorial products is
greater than~$n$. Also, that we might form a regressive product by
treating the first of the given combinatorials as we have treated
the second. It may easily be shown that this would give the same
result, except in some cases with a difference of sign. To avoid this
inconvenience, we may make the rule, that whenever in the substitution
of a sum of indeterminate products for a combinatorial,
both factors of the indeterminate products are of odd degree, we
\PageSep{26}
change the sign of the whole expression. With this understanding,
the results which we obtain will be identical with Grassmann's
regressive product. The propriety of the name consists
in the fact that the product is of less degree than either of the
factors. For the contrary reason, the ordinary external or combinatorial
multiplication is sometimes called by Grassmann \emph{progressive}.

Regressive multiplication is associative and exhibits a very
remarkable analogy with the progressive. This analogy I have
not time here to develop, but will only remark that in this analogy
lies in its most general form that celebrated \emph{principle of duality},
which appears in various forms in geometry and certain branches
of analysis.

To fix our ideas, I may observe that in geometry the progressive
multiplication of points gives successively lines, planes and
volumes; the regressive multiplication of planes gives successively
lines, points and scalar quantities.

The indeterminate product affords a natural key to the subject
of matrices. In fact, a sum of indeterminate products of the second
degree represents $n^{2}$~scalars, which constitute an ordinary or
quadratic matrix; a sum of indeterminate products of the third
degree represents $n^{3}$~scalars, which constitute a cubic matrix, etc.
I shall confine myself to the simplest and most important case,
that of quadratic matrices.

An expression of the form
\[
\alpha (\lambda · \rho)
\]
being a product of $\alpha$,~$\lambda$, and~$\rho$, may be regarded as a product of
$\alpha \mid \lambda$ and~$\rho$, by a principle already stated. Now if $\Phi$~denotes a
sum of indeterminate products, of second degree, say $\alpha \mid \lambda + \beta \mid \mu + \text{etc.}$, we may write
\[
\Phi · \rho
\]
for
\[
\alpha (\lambda · \rho) + \beta (\mu · \rho) + \text{etc.}
\]
This is\Add{,} like~$\rho$, a quantity of the first degree, and it is a homogeneous
linear function of~$\rho$. It is easy to see that the most general
form of such a function may be expressed in this way. An equation
like
\[
\sigma = \Phi · \rho
\]
represents $n$~equations in ordinary algebra, in which $n$~variables are
expressed as linear functions of $n$~others by means of $n^{2}$~coëfficients.
\PageSep{27}

The internal product of two indeterminate products may be defined
by the equation
\[
(\alpha \mid \beta) · (\gamma \mid \delta) = (\beta · \gamma) \alpha \mid \delta.
\]
This defines the internal product of matrices, as
\[
\Psi · \Phi.
\]
This product evidently gives a matrix, the operation of which is
equivalent to the successive operations of~$\Phi$ and~$\Psi$; \ie,
\[
(\Psi · \Phi) · \rho = \Psi · (\Phi · \rho).
\]
We may express this a little more generally by saying that internal
multiplication is associative when performed on a series of
matrices, or on such a series terminated by a quantity of the first
degree.

Another kind of multiplication of binary indeterminate products
is that in which the preceding factors are multiplied combinatorially,
and also the following. It may be defined by the equation
\[
(\alpha \mid \lambda) \ttimes (\beta \mid \mu) \ttimes (\gamma \mid \nu) = \alpha × \beta × \gamma \mid \lambda × \mu × \nu.
\]
This defines a multiplication of matrices denoted by the same
symbol, as
\[
\Phi \ttimes \Psi \ttimes \Omega,\qquad \Phi \ttimes \Psi \ttimes \Theta.
\]
This multiplication, which is associative and commutative, is of
great importance in the theory of determinants. In fact,
\[
\frac{1}{\Fact{n}} \Phi \ttimes^{n}
\]
is the determinant of the matrix~$\Phi$. A lower power, as the~$m$\Chg{\textsuperscript{th}}{th},
with the divisor $n(n - 1) \dots (n - m + 1)$ would express as multiple
quantity all the subdeterminants of order~$m$.\footnote
  {Quadratic matrices may also be represented by a sum of indeterminate products of
  a quantity of the first degree with a combinatorial product of $(n - 1)$st~degree, as, for
  example, when $n = 4$, by a sum of products of the form
  \[
  \alpha \mid \beta × \gamma × \delta.
  \]

  The theory of such matrices is almost identical with that of those of the other form,
  except that the external multiplication takes the place of the internal, in the multiplication
  of the matrices with each other and with quantities of the first degree.}

It is evident that by the combination of the operations of indeterminate,
algebraic, and combinatorial multiplication, we obtain
multiple quantities of a more complicated nature than by the use
of only one of these kinds of multiplication. The indeterminate
product of combinatorial products we have already mentioned. The
\PageSep{28}
combinatorial product of algebraic products, and the indeterminate
product of algebraic products, are also of great importance,
especially in the theory of quantics. These three multiplications,
with the internal, especially in connection with the general property
of the indeterminate product given above, and the derivation
of the algebraic and combinatorial products from the indeterminate,
which affords a generalization of that property, give rise to
a great wealth of multiplicative relations between these multiple
quantities. I say ``\emph{wealth} of multiplicative relations'' designedly, for
there is hardly any kind of relations between things which are the
objects of mathematical study, which add so much to the resources
of the student as those which we call multiplicative, except, perhaps
the simpler class, which we call additive, and which are presupposed
in the multiplicative. This is a truth quite independent
of our using any of the notations of multiple algebra, although a
suitable notation for such relations will of course increase their
value.
\bigskip

Perhaps, before closing, I ought to say a few words on the applications
of multiple algebra.

First of all, geometry, and the geometrical sciences, which treat
of things having position in space, kinematics, mechanics, astronomy,
physics, crystallography, seem to demand a method of this
kind, for position in space is essentially a multiple quantity, and
can only be represented by simple quantities in an arbitrary and
cumbersome manner. For this reason, and because our spatial intuitions
are more developed than those of any other class of mathematical
relations, these subjects are especially adapted to introduce
the student to the methods of multiple algebra. Here,
Nature herself takes us by the hand, and leads us along by easy
steps, as a mother teaches her child to walk. In the contemplation
of such subjects, Möbius, Hamilton, and Grassmann formed their
algebras, although the philosophical mind of the last was not satisfied
until he had produced a system unfettered by any spatial
relations. It is probably in connection with some of these subjects
that the notions of multiple algebra are most widely disseminated.

Maxwell's \Title{Treatise on Electricity and Magnetism} has done so
much to familiarize students of physics with quaternion notations,
that it seems impossible that this subject should ever again be entirely
divorced from the methods of multiple algebra.

I wish that I could say as much of astronomy. It is, I think,
\PageSep{29}
to be regretted, that the oldest of the scientific applications of
mathematics, the most dignified, the most conservative, should
keep so far aloof from the youngest of mathematical methods; and
standing as I do to-day, by some chance, among astronomers,
although not of the guild, I cannot but endeavor to improve the
opportunity by expressing my conviction of the advantages which
astronomers might gain by employing some of the methods of
multiple algebra. A very few of the fundamental notions of a vector
analysis, the addition of vectors and what quaternionists would
call the scalar part and the vector part of the product of two vectors
(which may be defined without the notion of the quaternion),--- these
three notions with some four fundamental properties relating
to them are sufficient to reduce enormously the labor of
mastering such subjects as the elementary theory of orbits, the
determination of an orbit from three observations, the differential
equations which are used in determining the best orbit from an indefinite
number of observations by the method of least squares, or
those which give the perturbations when the elements are treated
as variable. In all these subjects the analytical work is greatly
simplified, and it is far easier to find the best form for numerical
calculation than by the use of the ordinary analysis.

I may here remark that in its geometrical applications multiple
algebra will naturally take one of two principal forms, according
as vectors or points are taken as elementary quantities, \ie, according
as something having magnitude and direction, or something
having magnitude and position at a point, is the fundamental
conception. These forms of multiple algebra may be
distinguished as \emph{vector analysis} and \emph{point analysis}. The former
we may call a triple, the latter a quadruple algebra, if we determine
the degree of the algebra from the degree of multiplicity of
the fundamental conception. The former is included in the latter,
since the subtraction of points gives us vectors, and in this way
Grassmann's vector analysis is included in his point analysis.
Hamilton's system, in which the vector is the fundamental idea, is
\Typo{nevetherless}{nevertheless} made a quadruple algebra by the addition of ordinary
numerical quantities. For practical purposes, we may regard Hamilton's
system as equivalent to Grassmann's algebra of vectors.
Such practical equivalence is of course consistent with great differences
of notation, and of the point of view from which the subject
is regarded.

Perhaps I should add a word in regard to the nature of the problems
\PageSep{30}
which require a vector analysis, or the more general form of
Grassmann's point analysis. The distinction of the problems is
very marked, and corresponds precisely to the distinction familiar
to all analysts between problems which are suitable for Cartesian
coördinates, and those which are suitable for the use of tetrahedral,
or, in plane geometry, triangular coördinates. Thus, in
mechanics, kinematics, astronomy, physics, or crystallography,
Grassmann's point analysis will rarely be wanted. One might
teach these subjects for years by a vector analysis, and never perhaps
feel the need of any of the notions or notations which are peculiar
to the point analysis, precisely as in ordinary algebra one
might use the Cartesian coördinates in teaching these subjects,
without any occasion for the use of tetrahedral coördinates. I think
of one exception, which, however, confirms the rule. The very
important theory of forces acting on a rigid body is much better
treated by point analysis than by vector analysis, exactly as in
ordinary algebra it is much better treated by tetrahedral coördinates
than by Cartesian,---I mean for the purpose of the elegant development
of general propositions. A sufficient theory for the purposes
of numerical calculations can easily enough be given by
any method, and the most familiar to the student is for such practical
purposes of course the best. On the other hand, the projective
properties of bodies, the relations of collinearity, and similar
subjects, seem to demand the point analysis for their adequate
treatment.

If I have said that the algebra of vectors is contained in the
algebra of points, it does not follow that in a certain sense the
algebra of points is not deducible from the algebra of vectors. In
mathematics, a part often contains the whole. If we represent
points by vectors drawn from a common origin, and then develop
those relations between such vectors representing points, which
are independent of the position of the origin,---by this simple
process we may obtain a large part, possibly all, of an algebra of
points. In this way the vector analysis may be made to serve very
conveniently for many of those subjects which I have mentioned
as suitable for point analysis. The vector analysis, thus enlarged,
is hardly to be distinguished from a point analysis, but the treatment
of the subject in this way has somewhat of a makeshift
character, as distinguished from the unity and simplicity of the
subject when developed directly from the idea of something situated
at a point.
\PageSep{31}

Of those subjects which have no relations to space, the elementary
theory of eliminations and substitutions, including the theory
of matrices and determinants, seems to afford the most simple
application of multiple algebra. I have already indicated what
seems to me the appropriate foundation for the theory of matrices.
The method is essentially that which Grassmann has sketched in
his first \Title{Ausdehnungslehre}, under the name of the \emph{open product} and
has developed at length in the second.

In the theory of quantics, Grassmann's algebraic product finds
an application, the quantic appearing as a sum of algebraic products
in Grassmann's sense of the term. As it has been stated
that these products are subject to the same laws as the ordinary
products of algebra, it may seem that we have here a distinction
without an important difference. If the quantics were to be subject
to no farther multiplications, except the algebraic in Grassmann's
sense, such an objection would be valid. But quantics
regarded as sums of algebraic products, in Grassmann's sense,
are multiple quantities and subject to a great variety of other multiplications
than the algebraic, by which they were formed. Of
these, the most important are doubtless the combinatorial, the internal,
and the indeterminate. The combinatorial and the internal
may be applied, not only to the quantic as a whole or to the algebraic
products of which it consists, but also to the individual factors
in each term, in accordance with the general principle which
has been stated with respect to the indeterminate product and
which will apply also to the algebraic, since the algebraic may be
regarded as a sum of indeterminate products.

In the differential and integral calculus it is often advantageous
to regard as multiple quantities various sets of variables, especially
the independent variables, or those which may be taken as
such. It is often convenient to represent in the form of a single
differential coëfficient, as
\[
\frac{d\tau}{d\rho},
\]
a block or matrix of ordinary differential coëfficients. In this
expression, $\rho$~may be a multiple quantity representing say $n$~independent
variables, and $\tau$~another representing perhaps the same
number of dependent variables. Then $d\rho$~represents the $n$~differentials
of the former, and $d\tau$~the $n$~differentials of the latter.
The whole expression represents an operator which turns~$d\rho$ into~$d\tau$,
so that we may write identically
\PageSep{32}
\[
d\tau = \frac{d\tau}{d\rho}\, d\rho.
\]
Here we see a matrix of $n^{2}$~differential coëfficients represented
by a quotient. This conception is due to Grassmann, as well as
the representation of the matrix by a sum of products, which we
have already considered. It is to be observed that these multiple
differential coëfficients are subject to algebraic laws very similar
to those which relate to ordinary differential coëfficients when there
is a single independent variable, \eg,
\begin{align*}
\frac{d\sigma}{d\tau}\, \frac{d\tau}{d\rho} &= \frac{d\sigma}{d\rho},\displaybreak[0] \\
\frac{d\rho}{d\tau}\, \frac{d\tau}{d\rho} &= 1.
\end{align*}

In the integral calculus, the transformation of multiple integrals
by change of variables is made very simple and clear by the methods
of multiple algebra.

In the geometrical applications of the calculus, there is a certain
class of theorems, of which Green's and Poisson's are the
most notable examples, which seem to have been first noticed in
connection with certain physical theories, especially those of electricity
and magnetism, and which have only recently begun to find
their way into treatises on the calculus. These not only find simplicity
of expression and demonstration in the infinitesimal calculus
of multiple quantities, but also their natural position, which
they hardly seem to find in the ordinary treatises.

But I do not so much desire to call your attention to the diversity
of the applications of multiple algebra, as to the simplicity
and unity of its principles. The student of multiple algebra suddenly
finds himself freed from various restrictions to which he has
been accustomed. To many, doubtless, this liberty seems like an
invitation to license. Here is a boundless field in which caprice
may riot. It is not strange if some look with distrust for the result
of such an experiment. But the farther we advance, the more
evident it becomes that this too is a realm subject to law. The
more we study the subject, the more we find all that is most useful
and beautiful attaching itself to a few central principles. We begin
by studying \emph{multiple algebras}: we end, I think, by studying
\textsc{multiple algebra}.

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