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Gr 587
ON INFINITY;
75318
AND ON THE SIGN OF EQUALITY.
FROM THE TRANSACTIONS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY,
VOL. XI. PART I.
BY AUGUSTUS DE MORGAN, F.R.A.S.
OF TRINITY COLLEGE, PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.
T T
TTT F
CAMBRIDGE:
PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.
M. DCCC. LXV.
ATHEMATICS
Ga A
9
D386

VII. On Infinity; and on the Sign of Equality. By AUGUSTUS DE MORGAN,
F.R.A.S., of Trinity College, Professor of Mathematics in University
College, London.
If
[Read May 16, 1864.]
SECTION I. On Infinity.
I AM well aware of the position occupied by the subject I am about to treat. I know the
positive way in which opinions are held upon it. Those who teach that we know nothing
are quite sure we know nothing: those who teach that we know a certain something are as
sure both of the character and extent of the known region: those who halt between these
opinions are perfectly satisfied that such halting is the only true position for a rational being.
you wish to oppose a mathematician, confront him on some question in the parts of his
subject which are supposed to be perfectly demonstrated. Doubt about Taylor's theorem or
the binomial theorem, and you may be listened to with attention. But do not dare to
question his notions of infinitely great or infinitely small, be they positive or negative. You
may be simply wrong about Taylor or the binomial: but about infinity you are not merely
wrong, but absurd; and not merely absurd, but manifestly and palpably absurd.
It is quite
right it should be so, for the principles of the subject are immediately founded on first con-
sciousness; and no one can easily help being very certain both of the justice of his own con-
clusions, and of the prudence of his own hesitations. I am as positive about my own views as
any one can be: but with a qualification which enables me to avoid collision with others, and
an excuse which many others do not possess.
I am perfectly certain that in my own mind there is subjective¹ reality both of the in-
finite and the infinitesimal. But first, I have not that certainty which most persons seem to
¹ Most mathematical writers on first principles seem hardly
to know the distinction of subjective and objective. Many
apply the word metaphysics to everything psychological: D'A-
lembert uses this word, as opposed to physical. And meta-
physics being a word of fear-see schoolmen, fetters, Aristotle,
&c.-all that relates to psychology is often rejected with dis-
dain. In no subject is the great distinction-the fundamental
antithesis of thought-more needed than in the treatment of
infinity.
The writers of whom I speak are thorough materialists:
and the notion which they give of mathematics shuts up their
geometry in its literal name, land-surveying. Their pupils,
when they come to be critics, will describe this or that teacher
as giving metaphysics at the beginning of geometry, if he only
impress on his pupils that their diagrams are to be suggestive
of that ideal perfection which pencil and paper cannot realise.
I once had a lecture which I shall never forget, from an itin-
erant vendor of black-lead pencils, upon the fineness of the
points which his tools would take, and the consequent exact-
ness with which geometry could be taught by them. Dis-
dain of metaphysics should be relegated to the pencil trade,
as a legitimate source of puff.
1
2
MR DE MORGAN, ON INFINITY;
own.
shew of the uniform structure of all intellects. I am not prepared to say that the bases of
thought are the same in all minds. As little am I prepared to say that they are not. When
therefore I express myself in the usual way, presuming an absolute right and wrong in this
matter, and very confidently laying it down, I desire to be understood as admitting that in
the mind of another person there may be different conclusions, as well grounded as my own:
I do not assume that our grounds must be the same. What any patient thinker on psychology
tells me about his consciousness I believe, because I know the truth of what I say about my
Provided always that what I am told do not contradict those great forms of thought
about which there is no difference in common life. When a writer or speaker, puzzled with
the existence of infinitesimals, and also puzzled with their non-existence, shows some hanker-
ing after a tertium quid, wants subjective existence in a certain sense, and subjective non-
existence in another certain sense, I leave him to construct his own uncertain nonsense. I
cannot give up the excluded middle of the logicians: so I go on the principle of the excluded
mediator; I have nothing to say to him. I stand up for the maxim that the infinitesimal
either is or is not a conception of any one given mind.
Secondly, I have the advantage of a long period of deliberation, the third of a century,
a generation, about a sixtieth of the time elapsed since Euclid wrote. One of the first things
which impressed my mind, in my earliest studies at Cambridge, was the very definite character
of the youthful conclusions which were formed upon this difficult question; I mean among all
who could think upon first principles, and did think. Some could not, of whom some knew
they could not; and, as the fashion is, these last disdained what they had no tools to dig for,
as much as the fox despised the grapes which he could not reach. I cannot imagine, said a
good mathematician, what people mean by writing about first principles; it seems to me like
poking into nothing: I have no doubt he truly described his own state. But it was other-
wise with many. In my own day of elementary study (1823—1827) an old system had fallen,
and the new system was so fully established that there was no halting between two in the
student's reading, no cross-purposes in the teaching of different tutors, at least in my college.
But the old system was still remembered and discussed, and excited much thought about
fundamental principles, to the great advantage of many. I observed that, in the class which
poked into nothing¹, almost every hopeful undergraduate, almost every mature bachelor of
arts, had his own true and immutable foundation of the differential calculus. The effect on
¹ There existed then, as now, the undergraduate with whom
mathematics was but a subject of examination, and the bache-
lor with whom it was but an instrument of inquiry into nature.
The French books then studied abounded in those faulty
generalisations which existed in the last century. It must be
noted, to the credit of the University, that the tendency towards
mere examination problems, which has now existed for many
years, has not fostered that habit of imperfect demonstration
which it would have been judged beforehand so likely to foster.
Our books are of a much higher standard of rigour: so much
higher, that I may put an incident on record without being
supposed to satirise existing habits. I chanced to be in com-
pany, among others, with an undergraduate of great (tripos)
expectations, and a high wrangler who had thrown himself
with zeal into some mechanical inquiries. I produced an in-
stance in contradiction of some universal theorem, some Soit-
px-une-fonction-quelconque performance. The contradiction
was palpable enough, but neither the soph nor the bachelor
would surrender the theorem: both had reasons of power to
give. "Such instances are never set in examinations," said
the undergraduate: "such instances never occur in physical
investigations," said the graduate. I have given it as it hap-
pened, without one single brush of colour: I have a shade of
doubt as to whether the two remarks were made in one com-
pany, or in two companies on the same day.
!
AND ON THE SIGN OF EQUALITY.
3
For
my mind was that it would be prudent to defer arriving at any positive conclusion.
more than thirty years I allowed myself to suppose it possible that I might finally and ex-
clusively adopt any one of the systems on which infinity is explained. Accordingly, though
now as positive as if I had only meditated for a week, I may claim the attention due to a
long deliberation, and to conclusions which have formed themselves.
Before I proceed to my own views, I shall begin by shewing that the usual substitutes
for infinity are not always safe; and that the absolute infinitesimal has difficulties which
demand its admission, as much as other difficulties demand its rejection.
The absolute infinite is avoided by recourse to increase without limit. If, for instance,
we have to choose points in space which shall satisfy certain conditions, and if we first choose
them within a given sphere, and then increase the radius of the sphere without limit, do we
not finally allow unlimited choice? What point of space is omitted out of a sphere of infinite.
radius? Certainly not any assignable point. But on the other hand, we know well that we
dare not deny of an infinite sphere anything which is true of any sphere however great: if
there be anything which is true, and is equally true of all the increasing spheres, that truth
is, with obvious reason and invariable success, predicated of the sphere increased ad infinitum.
Now it is certainly true of any sphere, however great, that there is infinitely more space
outside than inside. If this be also true of an infinite sphere, that sphere does not include all
space if it be false, where does the sphere begin to include what it does not include? Where
does the ever remaining external space lose that character? Let us see which of the two
assertions will a problem justify.
Three points are taken at hazard in space, all points being equally likely: what is the
chance of the triangle having three acute angles? I saw this problem solved in a mathe-
matical journal, by a sound use of the integral calculus, on the supposition of the points
being taken within a given sphere. The answer gave a finite chance for an acute-angled
triangle, which remained finite where the sphere was enlarged ad infinitum. But it is
But it is very
easily shewn that the chance of an acute-angled triangle must be infinitely small.
Take any
one base, at its end draw two planes perpendicular to it, and infinitely extended: also,
on the base as a diameter, describe a sphere.
a diameter, describe a sphere. An acute-angled triangle must have its
vertex within the infinite strip between the parallel planes, and outside the sphere. Now
it is clear that the possible vertices outside the strip infinitely outnumber those within
the strip, if points be equally distributed through space. For any given base, and con-
sequently for any number of bases, there is no appreciable chance of an acute-angled
triangle: the same then for all bases and unlimited choice, if any one base be as likely
as any other.
This question about the distinction between space with infinitely distant boundary,
and infinite space, will always raise discussion. Any triangle, however great-and there-
fore an infinite triangle-has six external spaces, each infinitely greater than the triangle.
Three, say P, Q, R, are the angular spaces of the opposite angles; and, S being the
triangle, the other three are angular spaces with the triangle cut off, or P – S, Q – S,
R-S. If four points be taken quite at hazard, all points of the plane being equally
1-2
4
MR DE MORGAN, ON INFINITY;
likely, all triangles are equally likely to be formed by three of the points. Any one-
triangle being taken, the chance that the fourth point so falls that one of the four shall
be in the triangle of the other three is to the chance against it as P+Q+R+ S to
P + Q + R − 3S; that is, S being infinitely small compared with P, or Q, or R, the chance.
is an even chance. Hence, by repeating all the equally probable triangles on common
principles, it is an even chance, four points being taken at hazard, that one shall be in
the triangle of the other three. I am aware of a very ingenious proposal of solution.
which gives 1 to 3 instead of 1 to 1, and I am prepared to discuss it when it shall be
published. I will not at present pronounce decidedly for the above solution; but I am
myself utterly unable to see how it can be questioned.
If we should throw any reasonable doubt upon the security of the increase without
limit, and this by comparison of its results with the absolute infinite, it follows of course
that this same absolute infinite must be made an object of reasoning, were it only to
keep watch upon the increase without limit, which, like fire, is a good servant but a
bad master.
I now proceed to consider the notion of an infinitesimal. Let there be a rectangle
of which two opposite sides are produced without end in one direction: let the name
of the rectangle be a, and that of the infinite strip A. Observe that I utterly exclude
numerical comparison, which has been made a teacher of infinity, but which I think I
shall shew to need teaching itself. Certainly A is infinite as compared with a, and a is
an infinitesimal part of A. The finite is certainly infinitesimal of the infinite. In this
point of view, then, we have a clear conception of an infinitesimal. Assuredly, also, a
is a part of the whole A; and, if we take away a, what remains is less than A.
Many
readers may be prepared to oppose this with the Leibnitian notion that A + a = A, be-
cause a is, relatively, infinitely small: I contend that when any part whatsoever is re-
moved, the remaining part is less than the whole. But I do not know how to prove
this, if any one should deny it. I see then a relation of magnitude, the indefinite cha-
racter of which is perceptible. But I am sure, whatever this relation may be, a + a is
twice as large a part of A as a is of A; and so on. I now take a line PQ, a finite
line, and I suppose a point Z to move from Q to P. So long as PZ is finite and
numerically assignable, it is clear that PZ is a larger fraction of PQ than a is of A. If,
when PZ absolutely vanishes, it be a smaller fraction of PQ than a is of A, then either
continuity must be abandoned, or the position has been past through by Z at which PZ
is the fraction of PQ which a is of A. I do not now say that this reasoning is sound:
I only contend for its introduction of difficulty.
as
I bring forward this case that the objections may appear in force: they take all their
strength when arrayed against the downright language in which I have asserted that there
is-in the mind-the concept of the necessary existence of a part of PQ, the same
that which a is of A. Of the objective infinite I shall afterwards speak: at present I
shall merely say that, except subjectively, there is not even such a thing as the half of
Looking at the object as we must do, according to our notions of what is exter-
PQ.
*
AND ON THE SIGN OF EQUALITY.
nal-all objects being really what Hamilton calls subject-objects-we can object, or put
forward as an object, a universe of space, time, and matter, without any mind to think
of either. In that universe may be P and Q, and M, the bisecting point, as a mind
would think it: but the relation half is one of thought; it ceased to exist when the
mind was put out of existence. It is hard enough to admit an objective universe at
all, mind being eliminated: but all relations are subjective.
one to
This point is of such importance, that it is worth while to dwell on it. A room is
vacant of all human or rational occupants: chairs are in it, and tables, but no
see or think of them. We all say that the furniture is in the room; though some of
those to whom esse is percipi ought not to admit it. Right or wrong, we know that
we can say the chairs are in the room, in a sense in which we can deny that the identity
That a chair is
of a chair with itself, or its difference from a table, is in the room.
a chair, and is not a table, are truths of the reason, and have no residence in the room,
except when a mind is in it. But is not the chair in the room a chair? Surely it is,
to the mind without. Does not each piece of furniture carry with it, as part of itself,
its own identity? No, it does not, if without consciousness: its identity with itself is a
mental relation. But might you not say the same of the form or shape? I think not,
but it matters nothing: what I say is not less true because there is more of which it
may be true.
There is a something in the room which is ready to give perceptions:
A mind which had
but that something does not give the identity of a chair with itself.
not that notion-if we may dare to call such a thing a mind-when its body entered the
room would probably not acquire it by looking at the chair. All relations, all bringing
together of externals by thought of comparison-whether of things with other things, or
of things with themselves—is of the mind: the concept of comparison is not an objective
tie. Thus finite space and infinite space may both be external: but the relation of finite
to infinite, whether logical or numerical, is, with all its difficulties, of the mind which has
the two spaces in thought.
With objects I have nothing to do: my affair is with objections. I return to the
infinitesimal. First, the thing is inconceivable: of this I shall speak when I come to dis-
tinguish between imaged and unimaged concepts. Secondly, an infinite number is really
not an existence: this also I defer, observing that I have not asserted a numerical re-
lation. Thirdly, that I have spoken of relative magnitude, which is in truth a syno-
nyme of ratio, a thing not reducible to clearness, except by number. Not reducible to
definiteness, I grant: but I contend that the notion of relative magnitude is independent
of, and prior to, that of enumeration. A child of two years old, who cannot count three,
sees the joke in a picture of a house, with a horse tall enough to feed out of the chimney.
And what is the joke? That the horse is very much too large, or else the house very
much too small; the child understands both modes of correction. But the horse too large?
Never was so small a horse seen in this world. Here again, the child knows that the
horse is too large for the house. Accordingly, the infant has the idea of relative magni-
tude, without definiteness, without abstraction, but with perfect clearness, perfect appre-
hension of sameness and difference, instant perception of any large amount of error.
5
er
6
MR DE MORGAN, ON INFINITY;
}
But we cannot compare o with PQ, or a with A. In this objection the word com-
pare takes the technical sense of numerical comparison, and I answer that I have not
compared them: whether I can or no is therefore immaterial. In the logical use of the
word I can and do compare them; and I can because I do. I see clearly that the evan-
escence of length leaves a point, οὗ μέρος οὐθέν. I see that the abstraction of the point,
no matter how many times it may be made, is no more abstraction of length than took
place in the first process, that is, none at all. I see also that perpetual and successive
abstraction of a from the end of A does actually leave less each time than there was
before. I therefore see that I must either affirm discontinuity, or admit the conception
of PZ, an infinitesimal of PQ, in which, Z not yet coinciding with P, PZ is to PQ as
a to A. All I mean hitherto is to shew that the method of unlimited increase, as a kind
of graduated approach to the infinite, may not be so safe as is commonly supposed: and
that the notion of the infinitesimal, though difficult, may possibly compel absurdity, if it
be denied. I now proceed to the main points of the subject: I will, in the sequel, put
a greater difficulty in the way than either of those here produced.
I presume the notion of infinite, as distinguished from finite, to have place in the
mind with the ideas of space and time. The two words, finite and infinite, each of which
is the simple contrary or privative of the other, do not admit of definition independently
of each other: this I think is seldom perceived. What is a finite solid? Is it one which
by no multiplication can be made an assignable part of all space?
Then the infinite space
between two parallel planes is finite. Is it a solid contained within a finite boundary?
The definition here assumes the defined word: to which may be added that, definitions
being convertible, we must not pronounce the solid se¯(*²+v") dady to be infinite.
may indeed define a finite straight line as that of which the ends are assignable: the
definition is here convertible. But if we define a finite area as that of which the whole
مر
boundary is assignable, we throw f
α
€
We
x-də among the infinites, when n> 1. Thus, though
no two words are better known than finite and infinite, we have in truth no meaning of
finite other than not infinite. If, then, we have no conception of infinity, neither have
we a conception of finite, as finite: as the schoolmen said, Negatio sine re negata con-
cipi non potest.
But we can know nothing about infinity. I have always noticed that those who, after
consideration¹, argue for the incomprehensibility of infinity, shew a perfect comprehension
1 There are those who reflect, and upon reflection reject:
with them I have no quarrel. There are others who, finding
at the outset that they do know nothing about infinity except
that it is not finite, frame the conclusion that they can know
nothing. This conclusion serves them for life: and it is
perfectly true, so long as they follow their course. They do
know nothing, and they can know nothing. The plan is
equally successful when applied to the distinctive parts of
algebra. Dickens's schoolmaster defines botany to his pupil,
a knowledge of plants, and then sends him to weed the garden :
he goes and knows them. A mathematician begins by spelling
nothing as his knowledge of infinity, and then goes and knows
it. What I say in the text is matter of careful observation,
both of writers and speakers. I have met many whose argu-
ments in favour of no knowledge of infinity are of just the same
force as that of the man who answered the question how long
he had been deaf and dumb.
AND ON THE SIGN OF EQUALITY.
7
ness.
of it: they bring a candle to get light enough to prove that there must be total dark-
Those who disdain the infinite in their writings use it in conversation, and talk
much more clearly than they write. The difficulties seem to me to shew that the final
condition of human knowledge, when the subjects are space and time, is nearly initial.
Nobody can start a new difficulty about infinity, any more than about freewill: turn the
subject as we may, the first perplexities are the last. Every kind of human knowledge
has unanswerable questions at its end: and until the insoluble problems are revealed,
we plume ourselves upon our progress. We knew a great deal about matter so long as
it was a congeries of solid stuff, stuck together anyhow or nohow: but the gradual ac-
quirement of the notions of attraction, crystalline structure, and atomic constitution, have
shewn us our mistake. We have fought our way through much discussion to the alter-
native of realism or idealism, and have thus learnt where our intellect must stop. In
the great concepts of space and time, on which mathematics are founded, the insoluble
problems present themselves nearly at the outset; which seems to me to prove that we
advance more rapidly in this subject than in others. In all matters we have learnt to
say that we do not know what things are, we only know something about them: that
is, we have subjects with attributes, and therefore propositions which can be affirmed.
Thus we can know something about infinity, as about mind, matter, causation, &c.
finity, like other things, has its attributes deniable and undeniable. In my treatment of
the subject I do not pretend to remove difficulties, but only to lessen their conflict.
In-
The history of the notion of infinite will shew that infinite magnitude, abstracted
from all else, is a modern notion. We all know how the Greek geometers evaded it:
and it must be added that Aristotle did not treat of it.
n
ǹ
او
The depov of Aristotle clearly means what we call indefinite, without determinate
boundary: the indefiniteness of infinity is but one case; another is such indefiniteness as
that of marshy land, which passes into solid ground no one can say exactly where. Aristotle
(Metaph. x. or xi. ch. 10) speaks as follows: To daπeipov ǹ tò ádóvatov dieλleiv tập µn
πεφυκέναι διιέναι καθάπερ ἡ φωνὴ ἀόρατος ἢ τὸ διέξοδον ἔχον ἀτελεύτητον, ἢ ὃ μόλις,
ἢ ὁ πεφυκός ἔχειν, μὴ ἔχει διέξοδον ἢ πέρας· ἔτι προσθέσει ἢ ἀφαιρέσει ἢ ἄμφω. This
has been translated as follows:-But the infinite is either that which it is impossible to
pass through, in respect of its not being adapted by nature to be permeated, in the
same way as the voice is invisible; or it is that which possesses a passage without an
end, or that which is scarcely so, or that which by nature is adapted to have, but has
not, a passage or termination. Further, a thing is infinite from subsisting by addition,
or subtraction, or both.'
The
It needs no acumen to discover, and no argument to maintain, that for aóparos we
must read dópio Tos, throughout the chapter: and no apology is wanted for meddling with
a text so corrupt, both by depravation and by interpolation, as that of Aristotle.
following is my paraphrase:-The indefinite is that to which we cannot, of its own nature,
attach the idea of beginning here and ending there, so as to have a notion of going through
it: as the voice, which is inaudible at distance enough on either side of the speaker, but
8
MR DE MORGAN, ON INFINITY;
does not give us a terminus of audibility on either side. Or else it is that which has
a definite boundary on one side, but not on the other, so that we have a passage of
no definite end, or the next thing to it [these words perhaps interpolated]. Or it is that
which by nature is capable of definite passage from boundary to boundary, but under
circumstances has it not. And this either by indefinite addition to, or subtraction from,
a case of definite boundary. I should not like to attempt the whole chapter: the corrup-
tions are not all so easily amended as in the case of dóparos.
The Tо aπερov is described in nearly the same words in book III. of the Physics: the
voice is still aópaтos. The chapters on space and time which follow in the next book have
ἀόρατος.
no discussion about the matter; and the very obscure expressions which are used have a sort
of meaning as to indefinitude which they clearly want as to infinity. For instance, that the
in(de?)finite, when indivisible, [an indivisible infinite?] is not so, except in like manner as the
voice is invisible; εἰ δὲ ἀδιαίρετον οὐκ ἄπειρον εἰ μὴ ὡς ἡ φωνὴ ἀόρατος (ch. 6). Also that it
ei ei w's ʼn
is an addition, and also a subtraction: καὶ τὸ ἄπειρον ἔστι μὲν προσθέσις ἔστι δὲ καὶ ἀφαίρεσις
(ch. 8) [an infinite obtained by subtraction ?].
That Aristotle was treating of the indefinite is clear: and the infinite in magnitude, abs-
tracted from all other concept, was hardly a separate subject of speculation in ancient or
middle time. When we first meet with it, we find it connected with the unbounded in quality,
the unconditioned, the absolute, and all those attempts to describe the Creator by impersonal
negations which have played so large a part in modern philosophy. An acute metaphysician
of our own day has expressed a doubt whether infinity be a lawful subject for the mathe-
matician: what he can mean, if not that, as of old, infinite quantity is not to be abstracted
from the absolute, I cannot understand. Leibnitz himself, the organizer of the calculus of
infinites, was not wholly free from the same notion' (Dutens, Vol. II. p. 220). In our own
day a collection has been made of the merest crudities, things which the lowest mathematical
student learns to laugh at; and this collection was intended to be produced in support of a
'theory of the unconditioned.' To some of these absurdities the name of Leibnitz is attached;
as that a foot is an inch, since both have an infinite number of parts, and one infinite is not
greater than another. No reference was given; and I cannot find the original passage; but
I can very easily believe that, in opposing the notion of a composite infinite, Leibnitz may
have used some such illustration.
In the schools at Cambridge, a century and a half ago, the mathematical infinite, modus
quantitatis, was discussed as to whether a negative notion or not, with inclination to the
negative. The metaphysical infinite was discussed as a positive notion, synonymous with
perfection.
"Je crois avec M. Locke qu'à proprement parler on peut
dire qu'il n'y a point d'espace, de tems, ni de nombre, qui soit
infini, mais qu'il est seulement vrai que pour grand que soit un
espace, un tems, ou un nombre, il y en a toujours un autre plus
grand que lui sans fin: et qu'ainsi le véritable infini ne se
trouve point dans un tout composé de parties. Cependant, il
ne laisse pas de se trouver ailleurs, savoir dans l'absolu, qui est
sans parties, et qui a influence sur les choses composées, parce-
qu'elles résultent de la limitation de l'absolu. Donc l'infini
positif n'étant autre chose que l'absolu, on peut dire qu'il y a
en le sens une idée positif de l'infini, et qu'elle est antérieure a
celle de fini. Au reste, en rejettant un infini composé, on ne
nie point ce que les géomètres démontrent de Seriebus infinitis,
et particulièrement l'excellent Mr Newton."
AND ON THE SIGN OF EQUALITY.
9
The notion of one infinite being inconceivable as greater than another seems to have been
the stumblingblock which prevented Leibnitz from accepting the composite infinite. Cava-
lieri acknowledges the difficulty, but believes himself to have conquered it: he holds the
difference of infinites "pro firmissimo geometriæ fundamento," (Geom. Indivis. Book VII. pref.).
It is difficult to see how those who thought at all of infinity could fail to recognise two
infinites, one infinitely greater than the other, in the strip between two parallels unbounded in
one direction only, and the angular space which contains it.
The above passage of Leibnitz forms part of his remarks on Locke, which were given to
the younger Burnet a "year or two" before 1697. Not far from that time (1693) Leibnitz
wrote as follows:-"Je suis tellement pour l'infini actuel, qu'au lieu d'admettre que la
nature l'abhorre, comme l'on dit vulgairement, je tiens qu'elle l'affecte partout, pour mieux
marquer les perfections de son Auteur. Ainsi je crois qu'il n'y a aucune partie de la matière
que ne soit, je ne dis pas divisible, mais actuellement divisée, et par conséquent la moindre
particelle doit être considerée comme un monde plein d'une infinité de créatures differentes."
(Dutens, 11. 243). This is the boldest announcement of the objective infinite that ever was
made, we may think we must suppose, putting the two passages together, that what Leibnitz
refused was the subjective infinite: unless, which may be the explanation, the infinitely great
was rejected, the infinitely small received.
The objective infinite was the sole matter of discussion, down to the time of D'Alembert;
hardly any, I believe, made the distinction before him; and he denies the existence¹ of the
objective infinite.
It will now be admitted that finite and infinite enter the mind together,
for most admit that no concept can exist except with distinction between it and its contrary or
primitive. D'Alembert (art. FINI) speaks with scorn of those who maintain that there is a
direct and primitive idea of infinite: but he refers especially to those who draw an innate notion
of infinite from an innate notion of a God.
Euler, who expressly lays it down that the infinitesimal is a real 0, makes out his case by a
direct assumption of the conclusion. He postulates that all quantity is either assignable-
expressible in terms of a finite unit—or 0. On this postulate he easily proves that less than
anything assignable is 0. The chapter cited, and the remarks in the preface, are good illus-
trations of discussion on the objective infinite.
D'Alembert elsewhere (Elémens de la Philosophie, Eclaircissement 14) insists upon it that
the infinite is properly only the indefinite, obtained by abstraction of boundary; and also that
it is a limit of the finite. I should suppose that this was written before the article in which
the infinite and indefinite are so clearly and justly distinguished.
1 La quantité infinie est proprement celle qui est plus grande
que toute grandeur assignable; et, comme il n'existe pas de
telle quantité dans la nature, il s'ensuit que la quantité infinie
n'est proprement que dans notre esprit, et n'existe dans notre
esprit que par une espèce d'abstraction, dans laquelle nous
écartons l'idée de bornes. L'idée que nous avons de l'infini
est donc absolument négative, et provient de l'idée du fini, et le
mot même négatif d'infini le prouve. (Encycl. Meth. INFINI.)
He then draws a clear distinction between infinite and indefi-
nite.
2 "Si enim quantitas tam fuerit parva, ut omni quantitate
assignabili sit minor, ea certe non poterit non esse nulla; nam-
que nisi esset=0, quantitas assignari possit ipsi equalis, quod
est contra hypothesin." (Calc. Diff. cap. iii.) Equally con-
secutive in its logic is his well-known proof that less than
nothing has an absolute existence.
2
10
MR DE MORGAN, ON INFINITY;
All our
I now come to a distinction which runs through the whole subject. A concept, a thought
in the mind about an object of thought, is to us a constituent of knowledge when we know it
as an attribute of other concepts, or other concepts as attributes of it.
So long as the concept
stands alone it cannot be a subject or a predicate: affirm or deny another concept of it, and
we have a proposition, which we may combine with others. But some concepts, over and
above their connexion with others, belong to objects of thought which have images, which we
can place, as the phrase is, before the mind's eye. We make a kind of image where there is
none, in such a manner that the word imagine becomes a synonyme of conceive.
senses have their images; we can image a cry of fire, and the inmates of the house rousing up
in alarm. But, though we can conceive their alarm, we cannot image it; by the depravation
of a word we can imagine it. The distinction is the heading of a wide chapter in psycho-
logy, and one which is little read. When we think of that which can be imaged, we may
have the concept before us either with or without mental representation: for instance, we do
not want a picture of the human animal before the mind's eye, when we discuss the dis-
advantages of arbitrary power. We then take the concept as that of a being with certain
faculties and tendency to excesses of self-love: and our pros and cons are the same as they
would be if his figure were that of an ox or of an elephant. But good authorities tell us that
we cannot think of a man except as an individual, who must be tall or short, dark or fair,
clothed or naked, &c. I am prepared to believe that any one, who declares it to be so with
himself, must summon up figure and features when he is told to imagine a person invested with
irresponsible power; he being, in fact, a kind of Laird of Ellangowan, who could not separate.
the revenue from the exciseman. But I claim for myself, and, so far as my observation
goes, for educated persons in general, the power and the habit of conceiving by abstraction of
such attributes of the concept as relate to the matter in hand. Thus, for myself, I am quite
clear that I conceive colour without extension: though I quite agree¹ that no image of colour
can be without it. But I take it for certain that when attention is directed to the question,
the image must come, if it can come. Call on me, in the way of defiance, to think of man
without an image, and I cannot do it. Nevertheless, if the current of thought be undisturbed
by foreign introduction of the human figure, I speak of a despot, meaning a man as a despot,
with no more perception of the presence of two legs than of the absence of feathers.
It is not very easy to find in philosophical writings either admission or refusal of this
distinction between imaged and unimaged concepts: there is occasionally what can only be
construed as implied admission or implied refusal; and very often there is such confusion of
phraseology as would lead to the supposition that the distinction is not seen. By Hamilton 2
1 Hamilton affirms, conversely, that we cannot conceive
space without colour: I think he ought to have said that we
cannot image it.
2 "Your minds are not infinite, and cannot, therefore, posi-
tively conceive infinite space. Infinite space is only conceived
negatively, only by conceiving it inconceivable; in other
words, it cannot be conceived at all. But if we do our utmost
to realise this notion of infinite extension by a positive act of
imagination, how do we proceed? Why, we think out from a
centre, and endeavour to carry the circumference of the sphere
to infinity. But by no one effort of imagination can we accom-
plish this; and as we cannot do it at once by one infinite act,
it would require an eternity of successive finite efforts, an end-
less series of imaginings beyond imaginings, to equalise the
thought with its object. The very attempt is contradictory."
An excellent description of the failure of image: but the
thought has an 'object' all this time: that is, the infinite is a
concept, and can be conceived.'
(
AND ON THE SIGN OF EQUALITY.
11
the concept and the image are not distinguished. In my quotation we see an 'object' of
thought, a concept, which the mind cannot realise by any succession of imaginings: here is
the concept without image, by reason of necessary failure in the attempt to form an image.
But it is declared to be inconceivable,' no concept at all. Hobbes' seems clearly to make no
distinction. Mr John Mill², who is of one mind with Hobbes on several important points, does
not agree with him here. Both the quotations use the words conceive and imagine in one
sense, without any words implying concept without image. When Mr John Mill says we
cannot conceive length without breadth, he concedes that, by abstraction, we can attend to
length without breadth: we must therefore have the concept: but we are declared to have
it without image. I digress to say that, on referring to my own experience, I find that I
have always had the image of "length without breadth.” I remember when I first opened
Euclid, at thirteen years of age. I am sure I had no bias to admit anything which should
make mathematics "exist as a science:" for I should have been better pleased if it had not
existed at all, science or no science. I thought I had studies enough; and Walkingame, who
I understood was a cousin of Euclid, had given me no prejudice in favour of the family.
But in the first glance at the book, when I came to a line is length without breadth,"
I felt that I had gained expression for an idea which I distinctly possessed by image, but
could not have put into words. And so, in a small way, I found that geometry did exist as
a science.
Among the images connected with space, is that of quantity. I can, as to length, for ex-
ample, put before the mind's eye an inch, or a hundred yards, with more, or less. I cannot go
down to the millionth of an inch, nor up to a million of miles; all image has become indefinite
confusion long before I reach anything so small or so great. Nevertheless, I have a definite
concept of both extremes, on which I can reason: I know quantitative attributes, those of
definition and others, both of a millionth of an inch and of a million of miles. I need not
:
prove that we can reason on a million of miles. Now with respect to infinite space or time,
as a quantity, it is said that conception of it is impossible. This is not true, and palpably
false for those who can combine arguments against infinity must have the concept; to say
nothing of the notion of finites being only conceived as the negation of infinites. Though
we deal with finites before we deal with infinites, it is not as finites: the moment we think of
them as finites, we are distinguishing them from infinites.
Those who affirm that infinity is inconceivable, ought to mean that it is not adequately
1 "Quicquid imaginamur, Finitum est. Nulla ergo est
idea neque conceptus quae oriri potest a voce hac, Infinitum...
Nulla inesse homini potest imaginatio rei quæ non sit percipi-
enda sensibus. Nemo itaque concipere aliquid potest, nisi ut
in loco.”—(Leviathan, ch. 3).
2 "A line, as defined by geometers, is wholly inconceivable.
We can reason about a line as if it had no breadth; because
we have a power, which is the foundation of all the control we
can exercise over the operations of our minds; the power, when
the perception is present to our senses, or a conception to our
intellect, of attending to a part only of that perception or con-
ception, instead of the whole. But we cannot conceive a line
without breadth; we can form no mental picture of such a line:
all the lines which we have in our minds are lines possessing
breadth. If any one doubts this, we may refer him to his own
experience. I much question if any one who fancies that he
can conceive what is called a mathematical line, thinks so from
the evidence of his consciousness: I suspect it is rather because
he supposes that, unless such a conception were possible,
mathematics could not exist as a science."-(Logic, Book 11.
ch. 5).
2—2
12
MR DE MORGAN, ON INFINITY;
<
imageable as a quantity in relation to finite quantities. This is true: but reasoning requires
only such conception as gives propositions. Let A, B, C be incomprehensible; still I know
that A is B and B is C,' necessarily infer A is C;' so that knowledge of the premises
-though no other knowledge of the concepts exist is knowledge of the conclusion. The
mathematician, accustomed to reason upon imaged concepts, may, if he like, declare that he
will not reason upon any others. Let there be two sects': and let them be distinguished
by the acceptance and rejection of unimaged concepts. The rejectors, if consistent, will
decline all magnitudes which cannot be placed before the mind's eye: they will accept a
circle of 50 feet radius as actually coinciding with I know not what inscribed polygon; but
will demand a larger number of sides when the radius is 100 feet.
There is a certain inactivity of conception about mathematical writers, fostered by their
teachers, and transmitted to their readers. Their diminution without limit is no more than
can be shewn in a diagram of perceptible parts: their increase without limit never gets them
outside the paper. Accordingly, the infinitesimal of copperplate is less than that of woodcut:
the infinite of quarto is greater than that of octavo. When the line a diminishes without
limit, it ought to become to the unit what a grain of sand is to a rock, and then ought to
become the grain of sand to the first grain taken as a rock; and so on. When a few
thousands of such successions have been supposed, we have but a feeble idea of diminution
without limit, but much more nearly adequate than that usually given. But how are we
to conceive all this? I answer, in the manner proposed. We can express, on a reasonable
hypothesis as to the number of grains in a rock, the number of times the line of the
ten thousandth supposition is contained in a unit: I need not write an Arenarius to prove
this. How are we to image it? I answer, not at all; for we cannot do it.
No more, then, of our having no conception of infinity; its quantity will never be
adequately imaged, but it always will be, as it always has been, the concept out of which
finitude is carved, with attributes affirmed and denied. If there be no infinite, neither is
there finite: if there had been but one colour, there would have been no colour at all.
Some say that infinity is indefinite, and therefore cannot be a definite notion. Descartes
(Princ. Part 1. § 26) says "Ita nullis unquam fatigabimur disputationibus de infinito: Nam
He
¹ These sects exist, though not under definite names. The
word infinite is never rejected, but many treat it as a sound to
which it is competent to them to fit a meaning at convenience.
D'Alembert, in one of his moods, is a leader of this sect.
explains how "on peut attacher des notions nettes, simples, et
précises, aux expressions dans lesquelles entrent le terme ou
l'idée d'infini." But what right has any one to attach notions,
except by absolute treatment of the concept, to a word which
expresses a well-defined concept? I say well defined: for a
privative notion is as well defined as its correlative. If I have
a perfectly definite knowledge of each one notion, as to whe-
ther it be or be not wholly in the notion man, I have equally
definite knowledge with respect to not-man. That is, I am in
that case as clear about what is or is not not-man, as I am about
what is or is not man. D'Alembert, to avoid the difficulties of
the positive notion of infinity, describes it negatively, and
affirms that he gives a neat, simple, and precise notion of it by
making it an abstract idea of an intellectual limit which no
finite quantity can ever attain. This is a neat and precise
notion of a way of treating infinity which has a large, but not
perfect, measure of success: it does not give a notion of infin-
ity, but of a mode of evading it. Prester John might have
been treated in the same way: he was not the King of France,
not the Czar, not the Grand Turk, &c. This would have been
a sufficient description for politics, because Prester John was
not in communication with any one of whom we knew. But
infinity actually exists and acts, as Newton said of gravitation:
of which he felt justified in determining the properties, though
he did not know what it was. Infinity is a pertinacious med-
dler, who will not be turned out: we must find out what he
wants, and give it him.
AND ON THE SIGN OF EQUALITY.
13
sane cum simus finiti, absurdum esset nos aliquid de ipso determinare, atque sic illud
quasi finire ac comprehendere conari...de iis nulli videntur debere cogitare, nisi qui mentem
suam infinitam esse arbitrantur." This, in English, is
Who drives fat oxen should himself be fat.
We are apt, by an easy transfer, to associate with the idea the quality of its concept: thus
our idea of the horrible is a horrible idea. But we do not say that the idea of blue is a blue
idea, nor the idea of tallow a greasy idea. Nor should we say that the concept of infinity
is an infinite concept: still less that, if it be an infinite concept, it requires an infinite¹ mind;
the concept blue does not require a blue mind. Least of all may we call infinity an in-
definite concept: it is one of the most definite we have; we know perfectly how to distinguish
in thought the infinite from the finite: and the definiteness of a concept is measured by the
clearness of the distinction between it and its privative. Our perception that it is indefinite
is a definite perception. In like manner we cannot have a good understanding of the old
relations between England and Scotland without a definite idea of the indefinite Debateable
Land; it is the boundary which is indefinite, the concept of the character of the boundary
is definite. To have a good notion of the spectrum, we must have before our minds in a
definite form, the indefiniteness of the passage from green into blue.
Whether the concept
of the spectrum be itself variegated, is a question for those who shall continue to discuss the
resemblance of ideas and prototypes: may all success attend them!
When asked for my own way of illustrating the word infinity, I appeal to space, in
connexion with which the concept necessarily comes before the mind. The parallels which
X
α
0 | 0 |
d
101
f
Y
A
B
C
D
help to determine the strip A-they determine
it in thought, though they do not complete a
boundary—are to be continued ad infinitum.
But how can this be done? Certainly not
at so much an hour: if required to draw the
figure in a finite time, I must postulate an
infinite velocity. But I have a conception of
the thing done: I distinguish with clearness
If allowed to set out with knowledge of the infinity
of A, I know that I cannot fill up A by the repetition a, b, c, d, &c.; and I know that
I cannot fill up the angular space X by A, B, C, D, &c. I have therefore a perception, of
which no objection can deprive me, of A as infinite compared with a; of a as infinitesimal
compared with A; of X as infinite compared with A; of A as infinitesimal compared with X.
And I believe other persons have the same: but if they deny it, I will concede to them self-
knowledge on the point, because I know I have it myself.

what I mean from everything else.
1 This notion, which runs through many writers, from Des-
cartes to Hamilton, that the mind must be big enough to hold
all it can conceive, leads to curious conclusions. If it cannot
contain infinity without being infinite, it cannot contain a mil-
lion of cubic miles without having at least that bulk: now it
does contain a million of cubic miles, as all admit; or if not,
no one will deny that it contains 50 cubic feet. Consequently,
the mind does not inhabit the body, as commonly supposed.
For myself, I am content to feel that a mind may contain space
as a ledger contains sovereigns.
14
ן
MR DE MORGAN, ON INFINITY;
But the angular space X may be made to coincide with the angular space Y by super-
position; and yet they differ by A: shall they be called equal? Certainly not, and for this
reason, They are not equal. Leibnitz, in my opinion, made a most unfortunate mistake when
he laid it down-or at least proceeded as if it had been laid down-that A+ a = A if a be
an infinitesimal of A. Choose another word, say equivalent, and it will be seen that most of
the axioms of equality will bear the transfer. Things which are equivalents of the same
are equivalents of one another; if equivalents be added to equivalents, the sums are equi-
valents; and so on. All this is true when by equivalents we mean magnitudes which differ
by an infinitesimal of either, if by anything: equal is a species of the genus equivalent. Do
I then mean to say that there are spaces which can be made to coincide and are not equal?
I answer that I do, and I ask, Why not? I am referred to Euclid's axiom, and the common
reason on which it is founded. I rejoin that Euclid's axiom speaks only of finite spaces,
Euclid's word figure meaning nothing else; and also that the common reason will see in my
figure the truth of what I say.
•
<
But if most of the axioms suggested by the species 'equal' be true of the genus equi-
valent,' is there anything to hinder an enlargement of the specific word into coextension with
the generic? I think not, in routine operation: but in matter of thought it is necessary to
keep the distinction before the mind. The symbol . might designate equivalence, the point
(.) denoting the infinitesimal by which the two sides of (=) differ. If from X a line be
drawn within the angular space X, it makes with the outer line of A an angle which by
multiplication will exceed X: and the same of the angular spaces.
Hence the smaller an-
gular space is infinitely greater than that of A, and the first cannot be contained within the
second that is, the line drawn within X must cut the inner side of A. This is Bertrand's
proof of Euclid's axiom, and I hold it valid, though I admit the expediency of not allowing
beginners to rely on reasoning of the kind. To define parallels by help of the notion of
infinity, and to refuse admission to that notion in deduction from the definition, is logically
absurd; as absurd as would be the demand that the equality of the radii should be excluded
in reasoning on the circle: and it has produced a discussion many centuries long.
The distinction of concepts with and without images ought to be a turning point of the
discussion, I do not say that another person ought to arrive at one conclusion or the other
about the consequences of this distinction: I do say that two disputants who either do not
see it, or neglect to employ it, will soon be at cross-purposes.
A second discussion arises upon the introduction of number. This word and multitude
are often confounded: but their meanings are quite distinct. Multitude may be used in
an objective sense: pluralities might exist, according to our ideas, without mind to think of
them but number' is a mental affection: it is the distinction made by thought between
¹ We are apt to pronounce that the admirable pre-established
harmony which exists between the subjective and objective is
a necessary property of mind. It may, or may not, be so.
Can we not grant to omnipotence the power to fashion a mind
of which the primary counting is by twos, 0, 2, 4, 6, &c.; a
mind which always finds its first indicative notion in this and
that, and only with effort separates this from that. I cannot
invent the fundamental forms of language for this mind, and
so am obliged to make it contradict its own nature by using
our terms. The attempt to think of such things helps towards
the habit of distinguishing the subjective and objective.
AND ON THE SIGN OF EQUALITY.
15
one plurality and another, the register of pluralities. Common language, which in general
confounds the two ideas, knows how to distinguish there is no number without multitude,
but multitude without number is in intelligible use. Infinite multitude is a necessary
adjunct of our idea of infinite space, which has an absolutely unlimited plurality of
cubic feet, for example. But whether we can justify the phrase, an infinite number
of cubic feet' demands a separate consideration.
In thinking of space, we feel sensible that we carve the finite out of the infinite, which
was there, an existing notion, not of our own composition, but a portion of our mental nature.
But number of necessity sets out from 1, and is piled up by units; and we learn the possi-
bility of enumeration by gradual steps, aided by the invention of language: mere multitude
or plurality is a concept prior to language. We do not, therefore, set out with an idea of
infinite enumeration, composed of an unlimited succession of simple units; we build up this
notion, and we discover, by aid of our conception of infinite multitude, the unlimited extent
of possible enumeration. I have tried the following on many persons. All admit that the
universe of space holds an infinite multitude of quarts, and an infinite multitude of pints;
but few are ready with their answer to the question whether the universe of space has twice
as many quarts as pints. Some answer affirmatively; some negatively, because the pints
and quarts cannot be numbered, i. e. counted. Innumerable, is the answer; what is a number
which cannot be numbered? Though perfectly satisfied, for my own share, that there are
twice as many pints as quarts in the universe, I admit the force of the objection: but for the
considerations to which I presently proceed, it would be unanswerable.
Locke will have it (Book 11. Ch. xvII.) that we get our idea of infinity from experience
of the interminable character of space, time, and number: and this in a way which makes
reference to number only. We continue adding space to space, time to time, or unit to unit,
without coming nearer to conception of an end than we were when we began; and so we get
the notion of infinity. This is preposterous. If the notion of infinity be derived from ex-
perience of succession without end, either attained or indicated as existing, it is an induction
which further experience may deny: we only know it as we know that iron gravitates. I
tell the partisan of Locke that I have counted farther than he, and have found the natural
terminus of numeration, preceding the arrival at a million of millions, a so-called number
which is a mere delusion equipped with language. Should he ask me how this can be, I
tell him to count on, that he may be taught by that experience on which he relies. How can
I teach à priori, on a question of experience? I do not say how or why: I only declare, on
experience, for one settlement of a question on which it is admitted that experience is the
teacher. Nor am I met by the argument that 1012+ 1 is as conceivable as 1012; this I grant:
what I deny is the existence of 1012. Should he utterly disbelieve me, as probably he will,
I mean no disrespect; I only restore the sense of the
word, as Burke did. He wanted to read a letter written by
Hastings, and then to prove its animus: the Lords decided he
should prove first, and then read. This he styled preposterous,
and was called to order: but he silenced the Chancellor by
informing him that he had said no more than that their Lord-
ships had placed the cart before the horse.
16
MR DE MORGAN, ON INFINITY;
it is upon a consciousness of the interminability of enumeration, not derived from experience:
if not prepared to admit this, I leave him to count out the question, confessing to those who
know better that I did not quite adhere to truth.
That any number, however great, may be augmented, is, in arithmetic, either the child
or the parent of the concept of infinite multitude. Locke says it is the parent, and that it is
ascertained by trial. I have known persons, both children and adults, who thought that they
had arrived, by trial, at the conclusion that there is no end of number: but what they really
had been proving by induction was this, that there is no end of our power of inventing¹
names of sufficient brevity for numbers. This capability is by no means fundamental know-
ledge: the Arenarius of Archimedes is the standing proof that it was not self-evident. The
very difficulty lay in the necessity of granting infinite multitude, and this in a manner derived
from the infinity of space: there was the bulk of the solar system of the day, and grains of
sand to fill it. Then arose the question of representing by words or signs the number of
grains in the multitude.
Many writers, for instance Maclaurin throughout his work on Fluxions, takes Locke to
be the representative of the mathematical mind in this matter. They imagine that the ma-
thematicians strive to prove the existence of infinity by their not being able to get hold of it
piece-meal. If there be any one who is quite sure he has not the concept, I do not recom-
mend him to look for it. I write for those who, like myself, are quite sure both that they
have the concept, and that they cannot get rid of it. It is not our fault that the finite and
the infinite are necessary parts of our minds: and, these concepts being necessary, we are
obliged to acknowledge them. In the necessity of a concept we read the expediency of its
use. All the speculations about infinity, from Leibnitz to D'Alembert (partly inclusive² and
partly exclusive) are infested with, and spoiled by, this determination to call it a manufac-
tured article. Who can wonder that Dugald Stewart should have founded geometry upon
definitions, having a mind fashioned upon such teaching as that the infinity of space and
time was a consequence of and conclusion from, an inductive negation of terminability.
We have seen that even Leibnitz is asserted to have laid down that all infinites are
equal: if not Leibnitz, certainly many others. The schoolmen had the start of him, with
Infinito nihil infinitius. The source of this confusion is a view of infinity from which ma-
thematicians have escaped in practice, but not always in theory: namely, Locke's genesis of
infinity from number. They talk of 1, 2, 3, 4, &c., ad infinitum, they make number go up
to infinity; and if n be the infinite number, they ought to acknowledge that n − 1 is finite,
an obligation³ which has been fixed upon them long ago.
1 Some may think that invention of brief names is an essen-
tial of numeration. Practically it is so, but we could discrimi-
nate multitude from multitude by one, one and one, one and
one and one, &c. if time would allow, and retentive power
were great enough.
2 D'Alembert objects to Fontenelle d'avoir voulu réaliser
l'infini and Fontenelle does not sufficiently maintain the à
priori reality of the notion of infinity. If the notion be not of
f
necessity an accompaniment of our notions of space and time
the less said about it the better: if it be, there is no occasion to
make it real; this is done for us, and all attempt to unrealise
it is futile. We can as soon extract a bit of space, leaving the
place it occupied empty of space, as destroy the concept of
infinity in its connexion with space.
3 In the Turin Memoirs (1760-61) is a paper by Father
Gerdil, Sur l'infini absolu, written against what he takes to be
AND ON THE SIGN OF EQUALITY.
17
The question of infinite number, a thing the existence of which I will not admit, any
more than that of a watch or of a telescope, until it is manufactured, will be best introduced
after certain other considerations. Right or wrong in so doing, it is a fact that number,
abstracted from the concrete, has made incursions into its native land, and has returned
laden with other abstractions which it never could have fashioned by help of its own resources;
and with tremendous effect. The two great instances are as follows.
A
The unit of enumeration is indivisible: we can reckon 4 or 5, but nothing between.
clock may strike 7 or 8, but arithmetic cannot teach the bell to indicate the passage of any
portion of an hour. We may indeed invent another sound, be it a smaller bell, or a dog
trained to
Bow for the quarters, and wow for the hour,
as in the parody on Coleridge. And thus we may strike, or bark, three-quarters past seven.
But perhaps such an invention would never have been accomplished, had not minds become
accustomed very gradually to think of magnitude as divisible into smaller and equal magni-
tudes capable of enumeration.
Our an-
The history of fractional arithmetic, words which were never put together in old time, is
curious. It was long before numbers and fractions-arithmetic and logistics-coalesced
into one system. In our day, we have the arithmetic of integers and fractions.
cestors would have spoken of numbers and integers. They first reckoned in numbers: when
they came to divide, they dropped the unit, which they could not divide-nor we either—and
invented the integer, the divisible whole, of which they formed the fractions. We say that
is twice the third part of 1: Tonstall and Finæus said it is twice the third part of a divisible
integer: but they would not denote that integer by 1. In their books, the word integer
does not appear until fractions are wanted, and then not as 1, but as the divisible whole which
contains the fractional parts.
We have drilled ourselves-in spite of unitas non habet partes, unitas importat indi-
visibilitatem, and the like—into a divisible 1. We explain it by reference to magnitude or
to collections of secondary units. But, come how it may, we have added to the abstract notion
of unity, the lowest component of numeration, a still lower component'. The time was when
the division of the unit was utterly inconceivable; so that the point of geometry was declared
the analogue of the unit of arithmetic. Stevinus first declared that the point was the 0 of
the prevailing sentiment among infinitarians. To overturn them
he gives seven proofs, in 36 pages, that every number has a
finite ratio to the number which precedes. He proves it from
arithmetic, from elementary geometry, from logarithms, from
the hyperbola, from increasing progressions, from decreasing
progressions, and from methods of approximation. I think he
establishes his point; and I therefore acknowledge his priority
in right to the theorem that there is no number at which fini-
tude ends, and after which infinitude begins. Nor was the
theorem uncalled for: there was Leibnitz, with "On conçoit un
dernier terme, un nombre infini;" and there was Fontenelle,
with the notion that n must be infinite before n, and a squad
of finite numbers with infinite squares. (Geom. de l'Inf.
p. 63).
It is a very remarkable incident of human thought that
we arrive at completion of system, with reference to multipli-
cation and division, earlier than we do as much with addition
and subtraction. Less than one is for arithmetic: less than
nothing for algebra. There is an analogous point in the com-
plete algebra. Multiplication and division ask nothing but
multiplication of lengths, and addition of angles: the effect of
addition and subtraction requires complicated formula and
trigonometrical symbols.
3
18
MR DE MORGAN, ON INFINITY;
arithmetic but he did not hope to make his contemporaries see this; so he contented himself
with a prayer that their unfortunate eyes might be opened by Almighty power.
There is another maxim which we have equally got over; quantitas non habet contra-
rium. Undoubtedly there are magnitudes which, besides quantity, possess the relation of
contrariety, as distance up and down, time before and after, and so on. But this is a relation
perfectly distinct from quantity; and number, considered as representing the quantum upon a
convention as to what shall be called 1, cannot be made, per se, adequate to the distinction of
contraries. Nevertheless we have brought out of the concrete the distinction we call that of
positive and negative, and have fastened it on to abstract numeration, not entirely without
opposition', but with a general consent which has never been accorded to any theory of infinity.
Here is an unimaged concept of a very startling character at first. Concrete reality has
taught us that though numbers cannot go below 0, having no other side to fall back upon,
we can reason upon an invented representation of another side which has no image until we
return with our abstract numerals, and apply them again to their concretes.
If there be any
one who cannot place himself in the position of a beginner, he may think that 1 is naturally
divisible, and also that - 7 and +7 may he conceived in their relation of opposites without any
conception of magnitudes enumerated. I will not liken him to those who think that 10 of
its own nature is ten: because I believe that the power of abstraction which has produced
algebra, though the consequence of a legitimate use of thought, has its abstractions open to
fair though answerable objection. I cannot here pursue this subject: it is enough that number
is, I have no doubt, fairly proved to have taken on attributes, as number, which cannot be
imaged except when the number inheres in a concrete magnitude.
I look upon infinite number as a third derivation from the concrete, a symbol of nume-
ration allowed to represent infinite multitude. We have a kind of image of infinite
multitude, or at least we have no power of refusing the concept: if we must accept infinite
space, we must also accept an infinite multitude of parts. But we cannot say we set out with
any kind of image, or even concept, of unending enumeration: by the very notion of enume-
ration we stop somewhere. In our ideas of external things there is infinite multitude; but
we cannot subjoin, and we have counted it. And number, except as counted, is not number:
it is vague multitude under the name of number. Multitude may be counted: number has
been counted.
C
I should be quite content to abandon the phrase infinite number, and to say, for instance,
letbe the symbol of infinite multitude.' But, however important in psychological dis-
cussion, the distinction is of a kind which will certainly be passed over in practice. We shall,
1 There has not been much of argued opposition to the doc-
trines of algebra. But, in our own country, among those who
study mathematics for the purposes of application, there has
been a subdued feeling, checked by authority, I suppose, which
ranks the distinctive parts of algebra among needless mysteries
or conventional absurdities of high convenience. Much excuse
for this feeling may be found in old algebraical works, which
lay down less than nothing as to be accepted in an early defi-
"
nition. All those "practical men to whom mental specula-
tion is a scoff and a hissing, have this feeling. Thirty-five
years ago, I overheard, in a scientific society, some sotto voce
conversation about an individual who was described by one as
attending to a "parcel of stuff". "Indeed!" said the other,
with an expression of pity. "Yes! to be sure" rejoined the
first, "Lorgna on Series, and negative quantities."
1
AND ON THE SIGN OF EQUALITY.
1
19
therefore, have to put up with the misnomer: but we must not draw difficulties out of it
wherewith to oppose the notion of infinity.
If we admit the infinite number, or symbol of infinite multitude, we must also admit its
reciprocal, the infinitesimal fraction. I shall presently have to distinguish -1 from 0. The
following difficulty of thought seems to drive us to the infinitesimal fraction if we do not
admit the infinite number, while we are otherwise driven to the same fraction if we do.
All the points of a given line may be distinguished into those which divide it into com-
mensurable segments (call them As) and those which divide it into incommensurable segments
(call them Bs). Any portion of the length, no matter how small, contains an infinite
multitude of both and between each of one kind is an infinite multitude of the other
kind. Proceed from one A to another through all the intermediates; we can thus approach
to the other without limit, but we cannot reach it; for between any two As lie an unlimited
number both of As and of Bs. If we take the usual licence of saying 'Proceed in this way
without end,' we do not, as a limit, attain the second A, for we cannot eliminate the inter-
vening Bs: neither do we stop short of it by any finite distance. Can we say that the
interval diminishes without limit and at last vanishes, when the Bs can never disappear from
between two As, however near? The four words in Italics are not mine, but are commonly
used and according to ordinary notions we cannot refuse an infinitely small distance as the
ultimate separator of As from each other. There are difficulties against this admission as
well as in favour of it; nor can either be met incontrovertibly. But the instance is a good
exercise for those who are adepts in the art of seeing one difficulty and being blind to others.
tion.
The preceding case requires, not a continuous transit through all the points of a line, but
progression through all the points of one class, with recognition of the intervening points of
another class. If we admit the notion of infinite number, we confound the two classes, and
thus give what is, I suppose, the real solution of all the difficulty that arises from the distinc-
With a denominator 100000, &c., ad infinitum, we still have a true distinction between
commensurable and incommensurable divisions, but with less of importance. This supposition
was made by Halley, and others of his time, and, though staved off by those who allow an
interminable decimal fraction, and interpret it place by place, must or ought to be in thought
to all who speak of ½ as ·3333... ad infinitum. With this infinite denominator the commensur-
able fractions are merely those in which the numerators are or finally become periodic; and
they take their places among the rest without any difference of great specific note with refer-
ence to our subject.
3
Closely connected with this part of the subject is a difficulty which relates, when properly
viewed, to the connexion of infinite multitude, objectively possible, and infinite number, not a
subjective reality of thought except by an extension which demands hesitation.
A ball rolls along the ground; it describes a finite space in a finite time. Let space,
time, the ball, and the floor, be true external realities. Does the ball really take an infinite
number of positions in a finite time, being only a moment, that is, no time at all, in each?
We grant you a subjective infinity of positions; the mind cannot refuse the concept: but a
position of the ball is a real external phenomenon; and two different positions of one ball are
3-2
20
MR DE MORGAN, ON INFINITY;
things as different as two different' balls, which they might have been. Are we really to
believe a genesis of an infinite number of things in a finite time? Say a particle moves along
a straight line. That particle either does or does not take an infinite number of positions in
a finite time: which? If not infinite, numerable; how many? If infinite, how conceivable?
And as to the points, or moments of time, how can a particle be said to occupy a position for
no time at all, which a moment is, as much as a point of space is no length at all.
There are some who, pronouncing for the subjective reality of the infinite number, cannot
tolerate the notion of an infinite number of positions being assumed by an actual object in
objective space.
Some of these avoid it by assuming that what we take for the continuous
existence of the material universe is nothing but a succession of moments of existence-the
smallest finite time would reproduce the difficulty-divided by intervals of non-existence.
Moving particles, which are annihilated in given positions, commence re-existence a little
further on. Nobody can undertake to disprove this hypothesis; and if any, feeling the very
presence of the subjective concepts, should be staggered by objective deductions, they may
find comfort in the following hypothesis.
For clearness, I state it as if I believed it: but, as may be supposed, I neither affirm nor
deny it.
Space, the recipient of matter, is full of-we must not say particles, but-potentials.
These potentials, which must be supposed excessively near, are fixed, and become possessed
of the properties of matter when under certain influences which we may call material. A
material particle, posited at a certain point, means a potential for the moment under material
The following extract from my old notes was written in
1853:-"Two balls of the same size and material, perfect fac-
similes of each other, are distinguishable only by difference of
position in space; nor this except when seen together. If one
be seen after the other, we say we do not know whether we
have seen the same ball twice, or each of the two different
balls once. What makes this great distinction between space
and time? Why do we not say that in either case we have
seen two different balls, in one case distinguishable in space,
in the other distinguishable in time? If resemblances in space
be cumulated into a class, but not resemblances in time, do we
not apply an attribute drawn from consciousness of personal
identity to any existence which is of one form through a por-
tion of time? We have the subjective consciousness of iden-
tity lasting through change of time: we have not the con-
sciousness of personal identity simultaneously present in dif-
ferent places. We may conceive this last without contradic-
tion: we may imagine all the parts of a certain space simul-
taneously possessed of a common consciousness-an ego ex-
tended through space by aid of a faculty which must bear the
same relation to space (whatever that may be) which memory
does to time: and such that we could no more call this collec-
tion of consciousnesses so many different persons than we could
say the same of the consciousnesses of one person at different
times. Such a person would no more object to the idea of the
same ball in two different places at one time, than do we to the
idea of the same ball at different times in one place.”
This was written on the first of April, and I could not but
sinile when I remembered how many would think I had been
doing due honour to the day. But it so happened that I was
interrupted, a few moments after the above remarks had been
put on paper, by the arrival from a bookseller of Mr Hey-
wood's Analysis of Kant's Critick..., 1844. In turning over the
pages I found an account of Kant's discussion of the paralo-
gisms of the reason taken from the first edition, in which the
treatment differs from that of the later editions. And here 1
saw (p. 109) the very same idea of multipresent conscious-
ness communicated-but not in time-through space. Kant
uses it to prove that the "I think" might be identical, the
thinking subject being variable: to me it had appeared that
the multipresent consciousness would constitute a personality
of the same relation to space which such personality as we
know to be in ourselves bears to time. I believe there is no
such argument in the later editions: it may be that Kant after-
wards saw the manner in which his position might be turned.
However this may be, some such consideration is indispen-
sable to those who would make that distinction between pluri-
presence and extension in space which it seems to me is as
necessary to the philosophical contemplation-if I really attach
any notion to the phrase-of "the unconditioned," as to the
theological concept of the personal Creator. And ten years of
observation, with this notion in my head, have led me to a
strong suspicion that omnipresence and infinite extension are
very commonly confounded, while those who distinguish want
language to make the distinction clear.
AND ON THE SIGN OF EQUALITY.
21
influence. These influences must be capable of communication from one potential to another,
not by traversing the intermediate space, but per saltum, subject to the condition that a
recipient particle must retain the influence a certain small time before it can part with it.
This is a strong supposition, but, infinite subdivision is to be avoided at any cost. Proceeding
in this way we may contrive that the transmission of an influence shall give motion of a par-
ticle, as we call it, at the rate of only a million of million of positions to the inch, or a larger
number if wanted, but something very different from an infinite number. This is enough for
my purpose: those who please may speculate on the law of distribution of the potentials, and
by variation of it may make the translation of the solar system work any gradual change in
our order of things, be it development of gorillas into men, or subsidence of men into gorillas.
Whichever way we may be going, we have at least deprived motion, objectively considered, of
that necessity for the admission of infinity which our minds must recognise in it, considered
as matter of thought.
For myself, I never took the preceding difficulty as anything but a warning' not to let
any difficulty lead to the denial of a law of thought, nor to the refusal of anything which is
possible in thought as possible in objective reality. I have no doubt that in the motion of
matter an infinity of positions is assumed in a finite time, an innumerable multitude; that is
to say, innumerable in any finite time at any finite rate. Multitude is objective; but number
is not; it is the function of mind. I can reject infinite number, and in one sense I do
reject it but infinite multitude may still have external reality. The whole of the difficulty
which I now leave behind lies in the assumption that what we cannot count cannot have any
existence.
&
In some of the preceding exemplifications, we have gone beyond the simple infinitesimal,
and have arrived at that extreme case in which we see no representative except 0.
It may be
apparent, with slight reflexion, that there is a difficulty which may be expressed by saying
that we are hesitating at (de)', while Euclid did not hesitate at (dx). I say a difficulty,
and one which occurs several times; namely, that there is a sort of clearness in the higher
notions which does not exist in the lower. I speak of the consideration by force of which
indivisibles are prior to infinitesimals in the history of the subject. The equivalence of a
point² and an infinitesimal line is the first view which suggested itself. I do not speak only
1 I should recommend those who are much perplexed
by the objective to study Berkeley, and become idealists in
mathematics, which is entirely comparison of ideas, and does
not require an external world. Kant's space and time and
Berkeley's matter are sufficient outfit for a mathematician.
2 I cannot find that Leibnitz speaks expressly on this point
either in the disclosure to Newton or in the paper on maxima
and minima of 1684. But in his answer to Nieuwentiit, he
speaks as follows (Dutens, Vol. 111. p. 328). "Cæterum
æqualia esse puto, non tantum quorum differentia est omnino
nulla, sed et quorum differentia est incomparabiliter parva;
et licet ea Nihil omnino dici non debeat, non tamen est quan-
titas comparabilis cum ipsis, quorum est differentia. Quem-
admodum si lineæ punctum alterius lineæ addas, vel superficiei
lineam, quantitatem non auges. Idem est, si lineam quidem
lineæ addas, sed incomparabiliter minorem... Et quæ tali [finita]
quantitate non differunt, æqualia esse statuo, quod enim Ar-
chimedes sumsit, aliique post ipsum omnes... Et si quis talem
æqualitatis definitionem rejicit, de nomine disputat." Here
Leibnitz makes a claim to be the representative of Archi-
medes, as Kepler had done before him. Archimedes infers,
between assignable quantities, either equality or assignable
difference: and he is right. Leibnitz makes him give equality
to all magnitudes which have not a finite difference: but Leib-
nitz has introduced quantities immeasurably small, and there-
fore unassignable in terms of a finite unit. And he says that
any one is only disputing about a word who gives to "equality"
any other definition except "not having a finite difference.
There is more than verbal controversy here, when infinitesi-
mals are allowed to enter in thought. But it will be observed
"
22
MR DE MORGAN, ON INFINITY;
of Cavalieri, in whose mind the connexion had a peculiar strength. The first who broached
the subject was Kepler, and its first appearance in his writings is in the Stereometria (1615),
Theorem 2:"Circuli area ad aream quadratam diametri comparata rationem habet eam quam
11 ad 14 fere. Archimedes utitur demonstratione indirecta, quæ ad impossibile ducit: de
qua multi multa: Mihi sensus hic esse videtur. Circuli BG circumferentia. partes habet
totidem, quod puncta, puta infinitas, quorum quælibet consideratur ut basis alicujus trianguli
equicruri..." It is well worth notice that Kepler, in his bold innovation, considers himself
as only giving the substance of Archimedes. I believe he was right, as fully as I believe
that Archimedes would have got wrong if he had adopted Kepler's form.
6
Kepler, Cavalieri, Wallis, and Newton, distinctly recognize the moment; Leibnitz, and
(I think) Barrow, make it of no account: though Leibnitz, as we have seen, uses it in illus-
tration. Newton, who overtly employed infinitesimals in his early fluxions, in his private
explanations to Locke, and in his communication upon quadratures to Wallis, introduces
moments into the first edition of the Principia (Book 11. Lemma 2) in a manner which can
be only infinitesimal: in the second edition there is a change of phrase, which seems to have
reference to the express renunciation of infinitesimals which had taken place a few years
before. But in both they are principia jamjam nascentia finitarum magnitudinum. As late
as 1714 Newton' writes to Keill, Moments are infinitely little parts.' It cannot therefore
be shewn that Newton ever completely abandoned the notion of infinitesimals: but it is pro-
bable that he halted between this notion, and that of which Leibnitz makes use when he adds
a point to a line in illustration of incomparably small augmentation. What I have to do is
to prevent any reader of mine from carrying this ambiguity into his interpretation of my view
of infinitesimals. The O must be the ultimate infinitesimal, the point, if we be speaking of
length; the correlative of, the ultimate infinite, the whole length which the parallel never
does meet, as distinguished from the infinites at which lines meet under an infinitely small
angle.
Take a square, and let all the lines be drawn which are parallel to one pair of sides and
bounded by the other. We cannot think of these lines as successively drawn, but we can take
them as we take the individuals when we think of a species. This I say because there are
those who will tell me that I cannot conceive all these lines: nor can I singulatim; but I
can do with them what I do with the human species in the word man. Now these lines are,
in logical aggregation, the square; arithmetical enumeration being impossible. Give me the
whole square, you give me each line; give me every line, you give the whole square. Here
again may arise an objection. It may be said that if all the lines be given except one, the
area of the square is fully given: this is true as to area, in quantity; but a square is not
given, if there be points in it which are not given. Both parties are right enough: the op-
ponent as to quantity of area, the respondent as to power over the space within the boundary.
It is idle to deny that all the lines make up the square: if not, what more is wanted? It is
that Leibnitz, by way of preparation, has compared his infini-
tesimal line to, or even identified it with, a point.
¹ See Phil. Mag. Nov. 1852, "On the Early History of
Infinitesimals in England."
AND ON THE SIGN OF EQUALITY.
23
:
worse than idle to say that each line is represented in area by any symbol except 0. Where
is the escape from the assertion that an infinite multitude of parts of space-logically, parts
—each of which has no area, compose an area? If any one should feel difficulty about the
word part, let him call each line an internal of the square: the difficulty remains as before.
It is not of conventional origin; there is no power to refuse any one of its premises. There
is no area in any one line: the infinite multitude of lines, coexisting in thought, give the
square. Some have denied the concept of a line: they may be right for themselves. Others,
who may be equally right, have denied the joint concept of all the lines in simultaneous
existence. To myself, in whose mind a line is length without breadth, and the joint existence.
of an infinite multitude is well secured by the multitude of parts which compose my concept
of space, I cannot refuse either notion because the two, jointly, give a conclusion of difficult
form. I accept the conclusion as one of the obscurities of the subject, a concept without an
image.
I now
come to another branch of the subject, the consideration of the relations of
infinites. The notion of many mathematical writers seems to be that infinity is a kind
of terminus which we approach without limit, but never reach. They also say, Let a
quantity increase without limit, and at last become infinite: and those who do not say it,
allow a symbol to arrive at the state and privileges of infinite quantity by augmentation
without limit. Now there is no such thing as quam proximè approach to infinity: any
finite quantity, however large, is but an infinitesimal of the infinite, which can only be
attained by help of the infinite itself. The increasing quantity must take an infinite
velocity, if it be to arrive at infinite in a finite time: and it must therefore be considered
as one of any number of infinites, having, if we so please, finite ratios to one another,
depending upon the ratios of the infinite velocities we think of, and on the times we suppose
them to have been in existence.
Infinity must be considered, not as a terminus of the finite, but as a different status
of quantity, of which the finite is an infinitesimal. I now suppose that my reader is prepared
to accept the absolute infinite, and to examine its attributes. If prepared to admit an infinite
at all, as an object of mathematical reasoning, he either admits it absolutely, in the manner
of which he has an image in the diagram; or else by arrival at infinity of an increasing
magnitude, in which case he admits an infinite velocity. The idea of increase without
limit, with finite rapidity, and an attainment of infinite at last, is the stumbling-block in
the way of definite conception of the new status into which the symbol is introduced. It
should be clear that absolute infinite is not a notion to which we gain a gradually acquired
right.
The diagram shows two infinites, of which one is infinitely greater than the other: not
by any contrivance of our own, but as a necessity of thought. That the angular space
contains the strip between the parallels an unlimited multitude of timès, is the consequence
1 The attempt to get into or near to infinity by help of
finite velocity is analogous to the attempt to turn an infinitesi-
mal into a finite by help of an infinitesimal velocity. Every
notion connected with the relation of finite to infinite must be
tested by comparison of infinitesimal with finite, and of grades
of either extreme with other grades.
MR DE MORGAN, ON INFINITY;
24
of concepts over which we have no more power than over the postulate that two straight
lines cannot inclose a space. There is no hope in this subject for any one who approaches
it as a judge, and not as a learner. All that is seen in the diagram is well pictured to the
mind, which must be prepared to go on in the path pointed out, even beyond the point of
carrying images with the concepts. If we once admit two infinites, one infinitely greater than
the other, the beginning is made, and we have that concept which is forced on me, and I
believe on all who can arrive at the unimaged concept, by a species of rule of three. If A
and B can be conceived, and also C, then D can be conceived so that A, B, C, D are
proportionals.
If be conceived as infinite compared with 1, then for every value of n, integer or
fractional, commensurable with 1 or incommensurable, we have an order of infinites repre-
sented by ax", a being finite. And any two values of n being taken, n and n + v, x² (log x)”
represents, for all different values of n', orders of infinites which are infinites of x and
infinitesimals of "+".
nt v And so on without end, the successive factors of discrimination
being log x, log log a, log log log æ, &c. This for a beginning: by exponentials founded upon
the different orders thus obtained, we may produce a new succession of still higher orders;
and so on.
And we can superadd other methods, which we may again make the bases of
others. The number of orders of infinity is to be conceived as infinitely-as co ly-
exceeding the unlimited multitude of values which a letter may take. To all this there is,
of course, no image: but the concept is as definite as that of any long operation in algebra
in which the whole can only be distinctly apprehended by junction of parts too many to be
grasped at once. Does any one who restricts himself to conception of approach by limits
complain of the above being inconceivable as well as unimageable? Let him then take
the following, to which his own admissions bind him. Select a few millions of the preceding
formulæ, ranged in order of algebraical degree. Take a so great that each term shall contain
the preceding as many times as the earth would contain a grain of sand, or more. It is,
as he knows, a question of time and trouble only to determine the value of x which will
suffice. There are problems, if wants were wide enough to need them, mind strong enough
to grasp them, and life long enough to state them, which would require the determination
of this value of x. The impossibility of an image, and the complexity of the concept, are
things of the same kind whether we suppose a infinite, or only very great, to all who have
mastered the single passage from the lower infinite to the higher which is seen in the diagram.
There is a point which I find more clearly conceived by the logicians than by the mathe-
maticians. It is that all division is reducible to simple dichotomy, and its repetitions. When
the logical writer has once shewn division, difference, he does not trouble himself with the
difficulty of accumulation which arises out of a million of repetitions of the process. I mean
that he does not make this a difficulty of theory: he is the practical acceptor of a concept
in which image is lost by multiplicity of details. The mathematician, who, up to a certain
stage of progress, lives in imaged concepts, is apt to fancy that reason ought to refuse to
proceed without a mental picture.
We may imagine the orders of infinites laid down as follows. Take a marginal line, as
AND ON THE SIGN OF EQUALITY.
25
On one
Each
in ruled writing paper, and in one direction let all possible perpendiculars be drawn. Let
finite lengths, set off from the perpendicular on any one line, represent infinites having finite
ratios, of the order denoted by an ordinate taken on the marginal line, drawn from a point
through which passes the perpendicular on which finite quantities are laid down.
side and the other of the line of finitude are the loci of infinites and infinitesimals.
line contains all that come before in its infinitesimal abscissæ, and all that come after in its
infinite abscissæ: the whole system being intended to present a certain fictitious image of
what takes place on any one line. The purely relative character of finite and infinite is
illustrated: finitude is but a word applied to abscissæ on one of the lines taken at pleasure,
for distinction between infinites on one side and infinitesimals on the other. Nothing
short of this system will satisfy the wants of thought, if there be any wants at all. Το
accept any one infinite whatever, and to reject the mare infinitorum from which it comes, is to
act the timid bather who tries the water with his foot and shrinks back from the cold, instead
of plunging boldly in.
I may surprise some, perhaps, when I say that the above was the plan adopted by Newton
in his early writing on fluxions. He has but one infinitesimal; nothing of the second or
third order. By help of x + xo, y + yo, &c., o being infinitely small, he gets his first fluxions,
which, but for division by 0,-a disguised 0, I fear-would have been first differentials.
This division by o enables him to ascend to the line of finitude from a lower one. Having
thus, as it were, gained the finite platform, he makes another use of o to determine the second
fluxion, which is saved from being a second differential by means of two consecutive divi-
sions by o.
But what is left behind in a mathematical process is not annihilated: if o be
first made to yield x, from which to yields, there is an implicit use of ao from which
r is really obtained. There is no real relinquishment of o²; but there is gained some safety
of operation, and some facilitation of thought. The same, I feel sure, may be said of the
mode of representation adopted above.
X
Those who have seized the spirit of the relation between the different forms of algebra,
the ascent from arithmetic to single and thence to double algebra, and to such triple algebra
as has been given, the divergence to the calculus of operations, the algebra divested of some
of its laws which has been made an extension of the calculus of operations, and the method
of quaternions, seek for illustration of difficulties by allowing the formal science to remain
untouched, and looking for other matter of meaning to the symbols, under which all the
relations of form shall be preserved. Algebra is a deduction of complex relations from simple
relations, in which the meaning of the complex is derived from that of the simple, but the
meaning of the simple requires material introduction, and cannot be deduced from the form
itself. Set a person to interpret the hieroglyphic A + B = B+ A, telling him no more than
that is a sign of sameness, and he may come out successful by attributing to represen-
tation of any convertible relation whatever; as that A cannot walk side by side with B unless
B walk side by side with A. The fundamental forms of arithmetic and algebra are so few,
that no doubt many systems of meaning are capable of giving life to them. Now there is
one system of interpretation, whether ever before proposed I cannot tell, which puts common
4
26
MR DE MORGAN, ON INFINITY;
-
arithmetic upon a very different footing, without altering any equation which was ever ex-
pressed in numbers. Let every unit have a local value; thus in 4, or 1 + 1 + 1 + 1, each
of the units has a value of position, and the separate components of 4 are 0, 0 + 1, 0 + 0 + 1 and
0 + 0 + 0 + 1. In every symbol let position previously attained be that from which the new
operation commences; thus 5 – 4 = 1, not 0 + 0 + 0 + 0 + 1, the first unit removed being
the last that was put on. In multiplication, let each o required by any unit multiplier be
transferred to the product; thus 1 × 4 is 4, but 2 × 4 is 1 × 4 and (0+1) × 4 or 4 and
(0 + 4) or 4 + 4, or 8. Resolving into fractions, 1 would have two unequal halves, and
but altered in meaning. If the nth unit mean a" in common symbols,
a being <1, the first n units represent (1 − a") : (1 − a); and if we make the extension
+ 1 = 1 would be true,
2
to fractions, we express that unity is the sum of its two halves, in common symbols, by
1
1
α
1.
1 + √ a
1 + √ a
This system being completed with due care, every equation of arithmetic may be inter-
preted in it. We might imagine it laid down on a ruler in which, as in a scale of logarithms
or of chords, equally differenced numbers are not equally spaced on the ruler.
Our infinite
1 + 1 + 1 + 1 + ... is (1 − a)-¹, which we may denote by . No multitude of graduations.
will bring us to this point: but if, having got the limit by reasoning, we make it the unit of
a new scale, we have (1 − a)-2, for "; and so on. The second half of n would be
2
a'n (1 - a¹n): (1 − a),
and the second half of would be infinitely small. The common system is the limit of
this, when a = 1: and the whole forms a good illustration of the scales of infinites, the
ascent from one to another being through unlimited multitude, without unlimited magnitude.
The difficulties which arise from the attempt to image that which cannot be imaged,
though it can be conceived as possessing some attributes and not possessing others; from the
introduction of number and its infinites, as if they could have that à priori conception which
we feel as to space and time; from the tendency to make infinity a value, an unattainable
terminus of the finite-these difficulties, I say, may be conquered with comparative ease.
But something remains behind which it is far more difficult to bring to a satisfactory settle-
ment. The satisfaction I speak of is not the removal of difficulty, but the removal of ap-
parent contradiction.
I speak of the connexion of infinity and nothing. If any one should
finally pronounce that my views end with greater difficulties than they began with, I am
quite willing to say that this may be the case. I do not propound any system for others:
I only affirm that I have, certainly, and from observation I believe others have, mental infi-
nites of both kinds in their minds, as subjectivé realities.
On these realities reasoning may
be employed that this reasoning should remove difficulty is not to be asserted à priori;
still less that it should remove difficulties of one class without introducing others. It is not
the law of human knowledge that we come to a final clearance of all that is obscure: we
AND ON THE SIGN OF EQUALITY.
27
cannot name a single subject in which this has taken place. If we could be initiated into the
true conciliation of that free-will which our consciousness tells us we have with that necessity
the logic of which would be victorious, if logic could avail anything against consciousness, I
venture to surmise that some yet greater puzzle would arise out of our new knowledge.
If we suppose a to pass through all stages of diminution, and to become 0, having pre-
viously been infinitely small of every kind, we must suppose x-¹ to pass through every stage
of infinitely great, up to a terminal order of infinity, which is as unapproachable by passage
from order to order of infinites, as any infinite by passage from value to value of x. This is
the of algebra, usually called infinite. When no distinction of orders is made, when we
have but finite and infinite, the difficulties of this subject hardly appear. When we find it
necessary to compare infinites, and their reciprocal infinitesimals, we are obliged to admit that
O and 0-¹ are not adequately extensive representations of infinitesimal and infinite quantity.
And inasmuch as 0 really means the total absence of quantity, we need not be much surprised
if we find that this symbol will not obey all the laws of quantity, even in an extreme and
forced sense. But at the same time, so wedded are we by algebraical usage to 0 converted
into 0-¹ by law derived from the concept of division, carried to its extreme, that it may be
desirable to institute a close inquiry into the whole point. On the one hand, I postulate the
true meaning of 0, a total absence of quantity; on the other, I am fully satisfied that we are
never to consider algebra as satisfactorily established until every symbolic change has its in-
terpretation. But we are not to be sure that every interpretation in use is consistent with
itself and others.
(
Under the strictest meaning, we certainly apprehend 0 + a, a +0, a 0; and 0 a pre-
sents no one of the difficulties of this subject. Neither do we find anything to stop at in
0 + 0, 0 + 0 + 0, &c., nor in n × 0. At 0 × n we pause for a moment, and see the interpreta-
tion in 'n to be proposed for operation, but, as it happens, no part whatsoever of it to be
taken.' Even 0 x 0 is a true translation of there is none, and if there were, you would not
have any of it.' In all these cases, the 0 signifies a term from which something might be
obtained, but, so it happens, nothing. This is the reason why we have to write, perhaps,
2.700, instead of its equivalent 27, meaning that we have examined three decimal places, with
a result between 2-699 and 2.701.
But when we place 0 in a denominator, we must, by the very meaning of the symbol, ask
the question, How many total absences of quantity will jointly give the numerator. If the
numerator be also 0, the true answer is, Any number you please, be the same more or less.
But if the numerator be other than 0, the true answer is, No number, however great, not
even an infinite number. But we are taught in algebra to say that an infinite number is the
answer. This arises from a use of the fallacy which Leibnitz afterwards employed, and
allowably, in one sense; for he was applying, not correcting, the algebra in use. Lagrange
wrote his Théorie des Fonctions on the same principle: he took algebra as he found it, and
shewed that it was, if true, a basis for the differential calculus. But both Leibnitz and
Lagrange required an algebra founded on a better view of infinity: and both have been
criticised as if they had intended a special maintenance of the algebra they used. The
4-2
28
MR DE MORGAN, ON INFINITY;
≈ =
solution by rule of a + 1 = + 2 gives a 0-¹; and the truth of what I say is manifest. It
is said that this equation is satisfied when a is infinite: this is not true. All that is true is
that a + 1 and a + 2, when x is infinite, differ by a relative infinitesimal of the same order.
As a mounts through the grades of infinity, x + 1 and x + 2 differ by one grade after another
of infinitely small difference. This is the true meaning of = 0. We must, without any
x
.attempt at an image, treat it as superior to any infinite, just as 0 is inferior to any infini-
tesimal. We must treat this final symbol, 0-', not as a quantity, in any sense, but as indi-
cating that relation of an equation to possibility of solution which I have described. Is
2.0-¹ double of 0-¹? Is the number of quite empty purses out of which 40s. may be ob-
tained double of that which will yield 20s.? This is the question asked; we know it from
the definition of the symbols: and I think the question must be pronounced to yield no con-
cept on which to found an answer. If we could only, by multiplication of purses, get hold
of an infinitely small fraction of a farthing, we might manage all the rest. But, seeing that
we are sure of never arriving at any concept connected with number, any attribute of number,
which will shew how to produce this first infinitesimal, we must content ourselves with de-
claring that 2.0-1 and 0-1 tell us exactly the same about the equations which gave them, and
nothing at all about themselves. Observe that the difficulty is not the presence of the idea
of infinity, but the total want of all numerical concept, with attribute, attaching to the symbols.
But we are not without something to teach us that the above results of strict reasoning
on the meaning of symbols have their concrete analogies. The point is truly the 0 of length :
and points, though they may in logical conjunction be declared the materials of a length,
are not aggregately any length, be they ever so many. Are there twice as many points in two
feet as in one foot? We answer that we have as much right to say that two unequal
lengths contain the same number of points as that they contain different numbers. Let the
two lengths be bases of two triangles, with a common angle and sides in the same two lines.
By drawing lines from the common vertex through the bases we shew that each point o the
larger base has its own point belonging to it in the smaller. And yet it is easy, if the larger
base be double of the smaller, to assign two distinct points of the larger to each one of the
smaller. From this mode of viewing the subject we may conclude, if not that 0-¹ is k.0−¹,
at least that any pretension to the first is as good a pretension to the second: which is pre-
cisely what we may say of k.0 and 0, but more intelligibly and with more easily demonstrable
equality of pretension.
1
But is it not just as easy to conceive that 2.0-¹ is double of 0-¹, as that there are twice
as many pints as quarts in the whole universe of space? I do not know how it may be to
any one who cannot divest himself of the notion that 0-1 is the infinite which represents, for
instance, the relation of A to a. To me there is this difference. If we begin to measure
out the whole universe in quarts, we can make some way: if we only complete the interior
of St Paul's it is something done; and more if we add the Abbey. But in putting toge-
ther noughts to arrive at 1, we never make any way at all; the second thousand processes
gives no more than the first. And yet this difference, though apparently clear, does not
dispose of the question.
AND ON THE SIGN OF EQUALITY.
29
There are several points which appear to me to be divested of some obscurity by the
above distinctions.
First, the change of sign in passing through infinity. How is it that x-1 goes through
+∞ when goes through +0, while -2, greater than 1, does not become nega-
tive? The reason is that + 0-1 is the true representation of the change of a¹, which goes
through all the grades of infinity before it changes sign at 0-¹.
1
Secondly, it is said that parallel lines meet at an infinite distance. I was long puzzled
to know on which side the meeting took place, or whether on both; for both have equal
claim. Algebra tells us that 0-1 is the symbol which represents the attempt to assign the
distance; and the true interpretation is that "parallels never meet, though ever so far pro-
duced." That 0-¹ is as indifferent to sign as it is to value in comparison with k.0-1 is well
illustrated in the figure formed by parallels: and we are spared the admission that because
+0 = − 0, therefore + × = − ∞, which is a symbolical inference from common interpretation.
That + 0-1 and 0-1 are undistinguishable, is but the extension of the theorem that 0-¹ and
k.0-1 are undistinguishable for all values of k. But the right of confusion is not the
agreement of the concepts as to quantity; it is the equal failure of the attempt to attach
any such concept to either.
I can conceive the following question put to me. Are you sure that your infinites of
both kinds really mean anything but degrees of smallness and greatness which, like the
mirage, are subject to removal as you approach them? Take the smallest fraction that ever
was assigned by man: and diminish it until it is to what it was as a grain of sand compared
with the bulk of the solar system. Call this a, a fraction which can be reasoned on, though
unimageably small: and let a be the infinitesimal until need of less arises, taking a³, a³, &c.,
as wanted for higher orders. Let a¹, a', &c., be the corresponding orders of infinites.
Can you be certain that in your own mind you are not really dealing with such conceptions
as the above? I reply that I go further; that a may be the fraction which a is of A in the
diagram. It is certainly some fraction, because its removal makes the whole smaller, and
because it is clear that removal of multiples of a is removal of more than a. But it is not
numerically assignable: what fraction then is it? This question objects to the unassignable
that it is not assignable: I should feel more objection to the unassignable if it were also
assignable. And in like manner a great many objections to infinity resolve themselves into a
charge of not being finite; a charge which infinity will never repel. There is the infinitely
small fraction; and the true point of controversy is the invention of expression for it: the
true stumbling-block is the assumption of expression as a matter of course.
The theory of limits is the usual substitute for that of infinitesimals. It is supposed
to be strictly logical, and I am satisfied that it is so, and no wonder, for I find strict logic in
the doctrine of infinitesimals, valid argument upon concepts in possession. But limits are
taken as strictly logical by many who deny that infinitesimals furnish any basis of reasoning.
This last discrimination I proceed to examine.
There is an old phrase about a quantity less than any assignable quantity.' Here
assignable means given by a determinate fraction of an assigned unit. Taken literally
30
MR DE MORGAN, ON INFINITY;
the phrase is infinitesimal: what is wanted for dc, in all the preceding remarks, except
permission to invent an arithmetical symbol to represent something which is less than any
determinate fraction? But the rigorous users of the phrase alter the adjective of potenti-
ality into the participle of past action: they say 'less than any assigned quantity" meaning
less than any quantity that shall have been assigned, no matter how small. The ratio of arc
to chord may be made 1+a, where a is put before our eyes, no matter how small, but defi-
nite. There is just this difference between the person who has to meet 'assignables,' and the
one who has to meet 'what has been assigned:' the first has to prove against a fraction which
he sees; the second has to prove against a fraction in a sealed packet, which is not to be
opened until his proof is done. Neither must diminish the arc until it is absolutely 0; for
the evanescent arc and chord are not magnitudes with a ratio: neither can stop at any deter-
minate distance from 0, for the quantity which has been assigned might have been smaller
than his smallest fraction, and his proof must satisfy the receiver that any case could have
been met. Anything assignable might have been assigned; the proof is to hold good against
all that might have been assigned; therefore actual assignment and possible assignability
stand on none but a verbal difference. Permit the prover of the proposition to say that,
without knowing what is coming, he will-as he can prove the proposition for a less frac-
tion, namely that arc chord may be less than 1 + ẞ, where ẞ is less than the coming frac-
tion a: you grant him the infinitesimal. For do is nothing but the symbol of less than the
determinate fraction which you are going to assign, be that what it may or what it can be:
this do is nothing but such as the ratio of a to A in the diagram.
We may talk as much as we please about the limiting ratio; but we must not forget that
there is no state of things in which the limiting ratio is the ratio. No doubt the limiting
ratio is ready to be the ratio, if the magnitudes can be found of which it is to be the ratio.
But they cannot be found at the moment when they should be caught and equated, there is
no longer either arc or chord. Those who would, to avoid infinitesimals, put up with a ratio
of equality between a point of arc and a point of chord, must be prepared to find a point of
arc which contains a point of sagitta an infinite number of times. A glorious way out of
difficulty: the ratio of a to A in the diagram is imageable compared with this.
The final theorem is that the ratio of an infinitesimal arc and chord is infinitesimally near
to a ratio of equality. All the mathematical world is agreed not to allow the last feather to
break the camel's back, but to invent an ultimate equality, and to allow the limiting ratio
a sort of attainable existence. Nothing is clearer than that quantities which, being always
equal, approach limits, approach equal limits; and this though the limits be unattainable,
except quam proximè. But they must be quantities, neither zeros at last, nor ratios of zeros.
The fiunt ultimo æquales of Newton's first lemma may be tolerated, as only slightly objection-
able, when the sums of inscribed rectangles in two curves are in question: but the ultimate
equality of the arc and chord is a very different thing. Leibnitz and Newton had each what
I must call his sophism on the subject; and this though both were logicians among logicians,
as well as mathematicians among mathematicians. Leibnitz would have 'infinitely near to
equal' to be convertible with equal:' this is not only actually untrue, but not even capable
6
AND ON THE SIGN OF EQUALITY.
31
of adaptation to symbolic reasoning under extension of language.
it does not follow that a
b
= =
For if a b and c d,
= C c – d. Newton allowed himself to say that he did not mean
the arc and chord to be equal before they vanish, nor after they have vanished, but when
This strange pair, O the chord and 0 the arc, are fresh at the moment in which
they vanish.
they are laid, and stale in all time following.
disgust at infinitesimals which grew out of the
But the authority of Newton, and the national
quarrel, have fixed the notion of vanishing in
a ratio among the fundamental positions of the subject.
The notion of vanishing in a ratio, applied to two diminishing quantities, requires that
addition to the following phrase which I put in italics, 'Let the chord and arc diminish with-
out limit, and finally become nothing. Some reject the last four words, as not to their pur-
pose and they are not necessary to any logical view of limits. Now I say that the following
proposition is either true or false: before the arc and chord vanish, they become as A to
Aa in the diagram. Is this proposition true? Then there exist true propositions about
infinitesimals. Is it false? then the arc and chord do not vanish in a ratio of equality. The
only escape from this is to ask, What is the ratio of A to Aa? This means, by what
symbol of ratio is it to be denoted? I answer that I will not introduce number until the
concretes are settled to which it is to be applied, at least until positive and negative number
are explained before application. When our concrete ideas are settled, and not till then, will
it be time to make our abstract numeration coextensive with our wants of application.
Some have used the phrases 'indefinitely small' and 'indefinitely great,' intending thereby
to avoid the infinite, without admission of the properties of the finite. A very simple
dilemma disposes of this language. An indefinitely small fraction-not indefinite and small,
as it might be, for instance, if it lay between 0001 and 00001, but of indefinite smallness-
is either capable of expression in arithmetical form, or not capable. In the first case it is not
indefinite; in the second case it is infinitesimal. If the phrase mean that the fraction is
small, but of what degree of smallness cannot be settled until the exigencies of the case shall
have been examined, there is no difference between the indefinitely small fraction and a
fraction less than one assigned, but not known; on which I refer to previous remarks.
I now proceed to consider a possible fallacy with which I advisedly commenced, as I
hinted, namely, the proof that a is more of A than a point is of a line. The reasoning there
given is not to be despised, looking at the way in which infinity is often treated. The
abstraction of a point at the end does not, in any way we can conceive, take away more of the
line when done a second time than when done a first time. What is the line PZ which ends,
not at Z, but one point short of it? Neither reason nor imagination knows of any answer
except-PZ, as before. But reason, which doubts about dr, or at least confesses want of
power to decide at once, ought not to be very clear, at the first glance, about (da)". We shall
see that we must decline to pronounce, until taught, what a point is in relation to a line'.
It is said that there are three dimensions of space: it
ought to be said that there is an infinite number, reducible to
three by a process first learnt in the composition of pressures
and velocities, but as yet inadequately introduced into pure
mathematics. And direction is a better word than dimension.
Of dimensions more properly-but perhaps not quite properly—
32
MR DE MORGAN, ON INFINITY;
2
1
The
I shall assume the notion of infinity, and of its reciprocal infinitesimal: that a line can
be conceived infinite, and therefore having points at an infinite distance. Image apart, which
we cannot have, it seems to me clear that a line of infinite length, without points at an infinite
distance, is a contradiction. And I find I cannot rid myself of the concept of a line of
infinite length, nor of certain attributes which it must have. But there is one conclusion of
many writers which I throw away entirely, as absolutely false: it is that parallel lines¹ meet
at an infinite distance; they never meet, though ever so far produced. For myself, I date
the commencement of views which, however difficult, agree with each other, from the time
when I got rid of this noxious theorem, and of the confusion between 0-¹ and
intersection of parallels is an offence against a warning which never misleads: whatever may
be the conclusions which interpretation of forms may finally demand, no à priori reasoning
can conclude about infinity, unless increase without limit shew approach. What is true
without change of the 'however great' must not be declared false of the infinite: now parallel
lines preserve their distance at any departure, however great; and therefore at an infinite
departure. This being premised, let PR be the altitude of a in the
former diagram, and, Z having arrived at its position in this diagram, let
QT be parallel to ZR. Then PR is the same part of PT which PZ is
of PQ. When T recedes to an infinite distance, the angle ZRP is
infinitely small, and PZ is an infinitesimal of PQ. When QT then
makes an infinitely small revolution, and becomes parallel to PR, we
cannot announce any conclusion, because we cannot follow the reasoning
by which the proportion is established. Consequently, we cannot either
affirm o deny, from this proposition, that the final value-so to call it-
of PZ, the point P, is not that part of PQ which a is of A. Reasoning by limits, carried
through all the grades of infinity, would lead to the conclusion that a: 4 and 0: PQ are
convertible symbols.

T
R
P
Z
so called there are four, point, line, surface and solid. The
relation of each to the one above it is the same for point, line,
and surface. In the right solid represented by (x+a) (y + ß)
(x+y), when a, ß, y, vanish, there is the solid zyx
&c. become surfaces; xßy, &c. become lines; aßy becomes a
point.
α,
We are so reconciled to this supposition by usage, that
we cease to recoil from the double meeting of parallel lines,
one on each side. Let us then take a case which has novelty
against it. A cylinder is a cone with a vertex at an infinite
distance. The geodesic line of the cylinder, the screw, there-
fore travels as follows. After ascending through an infinite
number of coils, it turns round, and descends, the descending
coils belonging to parts of the developed straight line at an
infinite distance, beyond the perpendicular let fall from the
infinitely distant vertex upon the developed line. On the de-
scending side, the screw takes an infinite number of coils, which
at last become unequal, gradually widen, and end in the screw
becoming one generating line of the cylinder; and the return
coils become another generating line. And as the infinitely
distant ortex may be on either side of the cylinder, all this
happens on both sides: that is, the screw which begins to return
also goes on in a generating line. Half of this is true for a
cone of infinitely distant vertex; all is false for a cylinder.
2 I understand that Professor Ohm contended for the dis-
tinction between the two: but I have not seen any of the writ-
ings in which he developed this part of his system. It is put
to me whether I grant that theorems on lines which meet in a
point are, mutatis mutandis, true of parallels. Such theorems
are of two kinds, of position and of quantity. I grant those of
the first kind; that which is true wherever the lines meet, is
true when they meet at the distance (dx)-", however great n
may be; and is finally true at the limit-however improper
the phrase may be-of the grades of infinity, when the lines do
not meet at all. I grant the theorems of the second kind,
which no opponent of the view in the text has any right to do.
If the intersection of two lines be one of two conjugate har-
monic dividers, the other, I say, lies in the middle of the
divided line when the intersecting lines become parallel. But
those who affirm that the parallels meet at an infinite dis-
tance have no right to say more than that it is infinitely near
the middle.
AND ON THE SIGN OF EQUALITY.
33
This was to me for a long time a difficulty which seemed to defy explanation. I had
ceased to be frightened at the unimageable character of any concept; and cared nothing for
the incomprehensibility of subject and predicate, provided only I had knowledge that they
were subject and predicate. But absolute inconsistency was inadmissible: granting that the
coexistence of two conclusions might be obscure, it could not be tolerated that their contra-
diction should be manifest. And contradiction is here, to all appearance, as plain as can be.
In the first diagram, a + a certainly is more of A than a is of A, or 2a A is double of
aA; while 0+0 is not more of PQ than 0 is of PQ: and yet a÷A and o÷ PQ are
convertible symbols. All those to whom infinitesimals present themselves as zeros are,
ought to be, in precisely the same difficulty, only more easily got at.
or
Let us for
The solution of the puzzle seems to lie in the removal of the numerical notion.
a while waive the question what is to be the symbol of a point, whether 0 or anything else.
We easily seize the notion that a line may be extended, an elastic thread for example. A
small line may be lengthened into a great one; an infinitely small line may be lengthened into
a finite line; an infinitesimal of any order into an infinite of any order. A point may be
conceived as extended into a straight line. Geometrical extension of a line is inconceivable
except upon the supposition that each point of the unextended is represented by an infinite
multitude of points upon the extended: we have not the resource of mechanical thought, in
which we leave the number of material particles unaltered, and increase the distances between
them. When we come to speculate upon the extension of a finite line into an infinite, upon
the condition that each point shall be represented by a length a, we are encountered by the
difficulty of every point which is distinguishable from others being represented by an a
which is separated from the others by infinite intervals. And we are compelled, if we
persist in introducing the enumerative system, to resort to the notion of 0 + 0 + 0
representing the length ma.
+
ma
in number
We have seen that in considerations connected with number necessity derived from
the concrete is the mother of invention derived from the concrete. To raise the difficulty
upon enumeration is to take ground upon which battle need not be accepted, until it is
offered upon all the points at once. Let the distinction of positive and negative be fought
upon pure number, let a tenable account of the distinction be thus found, and I predict that
as good an account of 0+0+... shall be given by fair parallel. Return to the concrete,
and the opposition of directions gives meaning to + and Return also to the concrete,
and 0 + 0 + ... takes a signification, though full of arguable points. There is no duration
in a moment of time; there is no length in a point of line; there is no area in the distance
between points. But, as hereinbefore shown in the third case, time is, in a certain
enumerative sense, composed of moments, length is composed of points, area is composed of
parallel lines. In all cases or none, the difficulties of the numeral system are explained by con-
crete application: that is, abstractions of one kind, used jointly with abstractions of another
kind, are legitimately defended as to their conjunction by reference to the concrete in which
both were found together; and may not be defensible without it. This necessity for trans-
5
34
MR DE MORGAN, ON INFINITY;
ferring notions to the abstract numeration, where there is no meaning, from the original con-
crete, in which there is some meaning, is the great source of difficulty. But the greatest difficulty,
with
It will not be easy
many, will be the separation, when demanded, in the familiar case.
—I found it uncommonly difficult,—to look at + 1 and − 1, and go back to the time at which
these symbols were first used independently of the things represented.
The following argument is difficult to admit, and difficult to reject. A length is com-
posed of points; a point has no length: therefore a length is composed of points without
length. What is a length but all the points in it taken together? Is any of it wanting
when no point is wanting? Is it not all given when every point is given? The mathe-
matician feels bound to find a denial for the arithmetical part of the conclusion, which is that
0+0+0+... is a finite quantity: and so he is driven to the declaration that a line is not
composed of points, which he supports by the assertion that the composition cannot be
effected by points taken one by one. He will say, and very justly, a line is composed of all
the points you shall assign, and all the lengths between them. But this can be said, again,
of all these lengths, and again, of all subdivisions of these lengths; and so on for ever. But
still it remains undeniable that a line is composed of all the points you can assign, without
any lengths between them. Between these difficulties the subject will remain suspended till
the end of time. But though a point may be the zero of length, there is a something about
it; the point is a position-mark: now the 0 has not any such something about it, so that
0+0+ is much more distinct from a finite quantity, as to any notion of a finite being
composed of zeros, than a collection of points from a line. In like manner, you can no more
subtract 12 feet from 8 feet than 12 counters from 8 counters: but 8
12 = - 4 has an
explanation upon the length which is wholly incomprehensible on the mere counters or on the
pure counting.
...
-
We arrived at the negative quantity from actual problems which presented it, and
demanded explanation. Had we as urgent a necessity for all kinds of series as for all kinds
of equations, the finite 0+0+... would have made the same demand. Convergent series.
have been found in which approach to divergency¹ of form is accompanied by diminution of
every term, in such manner that the final form, when all the terms vanish, is a finite case of
0x∞.
It is not my object to extrude, or even to make light of, the difficulties of this subject.
They are of the kind which make their appearance in every subject, at first or at last; that
they appear almost at first is, as already hinted, a presumption that the early teaching of
consciousness on the subject of space and time puts us farther forward than is done in other
subjects by long reflexion. I take it as a reasonable supposition that we shall never conquer
these difficulties, and never evade them, and that no systematic treatment of them can be
pronounced successful unless it lead to more difficulties still. Do we really expect that we
shall ever arrive at the ontological ultimatum of any of our notions? We are down upon
the granite: shall we ever get through it? If we do, shall we come upon some pre-primitive
1 The time will arrive when, in extreme cases of this kind, it will be necessary to define the order of the infinite which
expresses the number of terms of an infinite series.
AND ON THE SIGN OF EQUALITY.
35
And if the
rock, or shall we find, as some have supposed, that we are poking into nothing?
last, by what kind of psychological action is the whole shell kept together? What we may
hope for, in time, is either agreement on the precise character of the difficulties, and on the
considerations which incline us for and against this or that way of interpreting symbolic
language; or, reasonable assurance that the want of agreement is a consequence of difference
of structure between one mind and another. We shall get no agreement of the first kind
until we have a system which, final or not, is complete and self-consistent: that is, which
does not take upon itself to exclude anything on à priori views of imageable and unimageable,
conceivable or inconceivable; and does not admit in one part what it denies in another.
There is no other alternative. We have, in practice, a system which gives true results: and,
to use the words of a writer to whose Analytical Calculations elective affinity led me when
I was an elementary student, "Since it leads to truth, it must have a logic." Woodhouse
said this of imaginary quantities, which are now certainly as fully explained as the distinction
of positive and negative.
I may digress to remark that the system of explanation of imaginary quantities, the
complete algebra, though not actively opposed, meets with a passive resistance, and occasionally
a Parthian arrow, which may be traced to a want of perception of the true footing on which
stands the algebra of positive and negative distinction. Those who have forgotten the
concrete basis of this distinction, and in whose minds it has attained a purely arithmetical
character as abstract as number itself, recoil from the geometrical basis on which a + b√-1
is made to take any direction whatever. They want some signification of symbols which shall
give that à priori view of a + b-1 which they think they have of a in common algebra.
Let them set out this view, and I will answer for it that as good a view shall be given of
the complete algebra: hypothesis for hypothesis, evasion for evasion, suppression for suppres-
sion, abstraction for abstraction, indirect allusion to the concrete for indirect allusion. The
pure number is as much entitled to its 1 as to its 1: show me a way of giving the
second without any reference to supposition of concrete magnitude, and I undertake to bring
in the first on the same terms.
<<
To return to the subject. I have spoken of a point extended into a straight line, which
I purposely presented without example: but geometry will give us instances. The pole and
polar line, any given conic section being the standard, form an extreme case of two polar
reciprocals. The single pole has all the points of the polar line for corresponding points;
and all the lines drawn through it are tangents to the evanescent ellipse for which there is only
the pole to show. Here a point is replaced by, extended into, an infinite line. Now let one
polar reciprocal be that extreme of an hyperbola which consists in two intersecting straight
lines, of which we call the intersection the centre. The other reciprocal now corresponds to
a couple of points, the poles of the intersecting lines, with as much of their straight line as falls
between them, or else all that does not; say the first. Here the two intersecting lines are
compressed each into a point, except their intersection, which is elongated into the line between
the points. Compare the hyperbola which is infinitely near the intersecting lines with the
infinitely flattened ellipse which is its polar reciprocal, and the extreme case will be manifest.
5-2
36
MR DE MORGAN, ON INFINITY;
I end with the production of a very obvious objection, and reply. You have used, it
may be said, a notion of an infinitesimal drawn from the relation of a to A in the diagram, and
you end with the admission that, when the sides of the strip are parallel, a is to A what 0 is
to PQ: consequently, your infinitesimal of finite is 0. The objection is valid: but it is to
be remembered that the suggestions of a fallacious representation may be good suggestions.
Those who make the objection either do or do not distinguish parallels from lines making an
infinitely small angle at an infinite distance. If they do not, the objection falls: if they do,
then they must be asked to substitute for A the infinitely elongated triangle of lines infinitely
near to parallel, and to draw suggestion from thence. With those who do not concede an
infinite line at all, I have no case whatever: with those who do, I can force the concepts of
points at an infinite distance, necessary adjuncts of the concept of infinite length.
SECTION 2. On the sign of equality.
THE late Dr Peacock, whose thoughts on first principles were the results of an erudition
which I suppose was not equalled in his day, and of a judgment which was held second to
that of no other person in the affairs of life, may be regarded as a most remarkable specimen
of mathematical and general learning combined in a man of cautious and reflective tempera-
ment. If I speak very decidedly about the consequences of the neglect of pure logic by
mathematicians, as I have done elsewhere about the neglect of mathematical thought by
logicians, I shall not be supposed to have any disrespectful intention, when it is seen that
my strongest case in point is a friend whom I so highly valued, and to whose thoughts I have
been so much indebted.
:
=
I know of no mathematical writing which more strongly illustrates my opinions on this
point than the first edition of Dr Peacock's Algebra. It is founded on the basis of the
permanence of equivalent forms. This is a very near approach to the assertion that algebra
is, like logic, a formal science: nothing was wanted but an introduction and incorporation
of that distinction between form and matter which now rules in the definition of pure logic.
My mode of statement would be that algebra ought to be a formal science: I do not maintain
that it is. It will become a formal science when all its forms, without exception, shall be
true of every material instance, equally without exception. I may be asked whether I mean
that, because abb gives a = 1, it is to follow that 20 a gives 2 = 1.
I shall discuss
this point algebra is certainly not a formal science-independently of all question what it
ought to be until the symbols are so understood that 2x = x
x gives 2 = 1.
There is nothing
in a purely formal identity which admits of particular and exceptional cases: that "something
which is both A and B is something which is both A and C necessitates something B is
also C," is an assertion which cannot be denied by finding out some particular A for which it
cannot be true. Falsehood may be thrown into the propositions; but not into the connexion
stated.
We may, if we please, say that there is a nonexisting white which is a nonexisting
black; if so, there is a white which is black. We may have a quarrel with the first pro-
position, and grave doubts upon there being a white or a black which does not exist. But
we cannot quarrel with the infererce as inference: all our difficulty is with the matter of the
=
AND ON THE SIGN OF EQUALITY.
37
premise. How can white be black? This does not touch the point: it was not asserted that
white can be black, but only that the first assertion necessitates the second. How can we
predicate of non-existences? This question is equally irrelevant, for the same reason. I
have chosen this instance because it seems to offer a sort of resemblance to the case already
put. Here is a form of algebra ab = b necessitates a = 1': but the form is not universal;
it fails when 6 = 0. We can have 2x = x; we cannot have 2 = 1.
6
Again, I may be asked whether I mean that the failing case of Taylor's theorem,
$ (x + h) = ∞ +∞.h + ... is finally to become a truth of formal algebra. It is so already,
in some cases, and will be so in all which are fairly examined. In (x < 1)
(1 + 1 + 1 + ...) − (1 + 2 + 3 + ...) x + (1 + 3 + 6 + ...) x² − ( 1 + 4 + 10 + ...) ∞³ +
let every term of every series be included.
We have thus a true instance of co
But we have here precisely the same terms, neither more nor fewer, that are in
1
2
(1 − x + x² - ...) + (1 − 2x + 3x³ — ...) +
or (1 + x)−¹ + (1 + x)−² + ….. a convergent series, of which the sum is x-¹.
8+∞
The conclusion which I shall endeavour to establish is that in the algebra of quantity,
when the different orders are considered, both of infinites and infinitesimals, which lie between
0-¹ and 0, as already explained, every order has its own sign of equality. In fact there are
as many signs of equality as there are orders of magnitude: and in algebraical transfor-
mations, every change which alters the order of one or both sides of the equation, may or
may not necessitate, and may or may not permit, an alteration in the order of the sign of
equality, in a manner pointed out by fixed rules, which must be accounted part of the formal
laws of algebra, explained as to quantity, and to be explained in any system, if such there be,
in which numerical symbols are all signs of operation.
The cultivators of algebra divide into two classes.
The first look to all symbolic results
as things to be explained; if not now, at a future time. They show their principle rather in
practice than in profession: when they state their creed, they appear as no other than be-
lievers in the future advent of permanent algebra. But, being rather intimidated, I suspect,
by the superior numbers and confidence of the other party, they reserve all maintenance of
their own guiding convictions, and content themselves with declining to acquiesce in the
opposite conclusions.
The second class,-whom I should propose to call algebra teachers, the others being
algebra learners-dictate final acceptance or rejection of symbolic difficulties upon what they
take to be legitimate à priori conceptions of algebra. I am satisfied that their reasons are
very often à posteriori, suggested by the forms which are to be rejected, and moulded into
preliminary maxims upon the very consequences which are to be drawn from them. This is
not only quite fair, but absolutely necessary: the new ideas which are started by new cases.
are as much entitled to suggest new principles of objection to one as new extensions of
meaning to another. And it would be quite unreasonable to propose to any one to lay down
beforehand all the postulates which are to govern his limitations; unless the demand were
38
MR DE MORGAN, ON INFINITY;
accompanied by the counter proposal that the advocate of the other side should be equally
universal and final with his permanent and all-inclusive laws. Probably, indeed, if the
algebra teacher could exhibit his whole array of objections, the algebra learner would at once
see how to gather together his whole system of laws. What is not quite so fair, and certainly
not the least necessary, is the assumption of the à priori form, the implied assertion that the
mode of refutation is what a trader would describe as already in stock. On the other hand,
the algebra learner may equally sin against candour by starting from the extension which is
to cover the result, in such a manner as to prevent the difficulty from appearing, or to make
the difficulty, with its solution, only an example of the method. The only algebra teacher
who has any right to assert a full and fair previous code of interpretation, is the pure¹ arith-
metician, who admits no use of symbols which outsteps the limits of absolute number, and
forbids all extension of language.
For myself, I can say that the following extensions are derived from the instance above
given. If, as I felt compelled to prognosticate, there be an algebra of which the forms are
absolutely unlimited, then the interpretation of those forms must give meaning to the deduc-
tion of 2 = 1 from 2x = x. This is absurd under the usual extent of meaning of the sign =:
is there then any extension which, preserving the usual meaning when is placed between two
finite quantities, will give the necessary latitude of interpretation? I found that not only is
the requisite extension more nearly in the field than might be supposed, but that some con-
fusions and irregularities in the use of the existing sign are due to a want of perception of
the question, and of decision one way or the other.
I use the uncloaked language of infinites and infinitesimals.
I have stated the grounds
on which I assert the subjective reality of both extremes of quantity. In the mean time,
everything which I say, or which can be said, in this broad and easy language, can be trans-
lated into the language of limits by those who will take the trouble, and who may fairly be
expected to take it for themselves.
I now proceed to point out that we have really different definitions of the sign in actual
=
1 Perhaps this sect is extinct. During the last century,
its chief writers were Robert Simson, Francis Maseres, and
William Frend. So far as these opponents set out their ob-
jections, it is seen that there is much force in them against the
mode of elementary writing then in vogue. Having been
casually brought into contact with Mr Frend in early life, at a
time when I was engaged in examination of first principles,
and having had many discussions with him, and having been
led thereby to an attentive examination of Maseres, Simson,
and others, I long ago came to the conclusion that in those
minds which are irresistibly led to a sweeping condemnation
of the minds of all but the smallest few, on matters of which
the evidence is that of consciousness, the bias is a craving for
simplicity, ending in the discovery of means of rejection for all
that cannot be immediately reduced to the earliest axioms.
The distinction of greater and less than nothing owes its
establishment in this country to the authority of Newton.
Among the definitions given in the Arithmetica Universalis
we find, "quantitates vel affirmativæ sunt seu majores nihilo,
vel negativæ seu nihilo minores. Sic in rebus humanis pos-
sessiones dici possunt bona affirmativa, debita vero bona nega-
tiva." We may remember, though it could hardly be expected
of Newton's immediate successors in algebraical writing, that
the work was only a collection of notes for lectures, "currente
calamo pro officii urgentis ratione compositus," published with
a very unwilling consent wrung from the author. Had Newton
been his own editor, there would have been a Scholium.
Descartes lays down the negative as less than nothing
without explanation: his commentator Schooten, and his very
prolix commentator Rabuel, follow him without an additional
word. To Descartes must be assigned that curious contradic-
tion which produces a"-Σa."-1+...=0 by multiplying toge-
ther x-a=0, x-b=0, &c. He is followed by a great many
writers, down to Waring inclusive. Harriot is quite free
from it.
AND ON THE SIGN OF EQUALITY.
39
use and I request particular attention to this point. Whether what I shall ultimately pro-
pose be judged sound or unsound, I can easily make it appear that I am not corrupting any
perfection of system which actually exists, but only endeavouring to introduce system where
none is now to be found. In the differential calculus, we are told by the most careful
writers that pv fx, when in the form 0:0 or ∞∞, is equal to 'a: '; and this
without exception or reservation. I take Cauchy's latest work, after looking through its
predecessors to satisfy myself that there is no accidental relaxation of rigour, and I find
(Moigno, 1. 42) as follows:-
: co:
ar
a la
"On a pour ∞ = ∞
a ∞, 1º.
= 8
L'exponentielle l'emporte sur le variable x, ou
X
1
Lx
1
= 0. Le logarithme croît moins rapidement que
a
xla
croît beaucoup plus rapidement, 2º.
le nombre."
Here L and I are both symbols of the logarithm, L having the base a. We see then that
infinites, as such, are held equal. But ax-¹ and a log a are infinites of different orders;
and aa-¹- a log a = 0 would be rejected at once. Accordingly A = B, as used, does not
always give A – B = 0.
x =
But infinites are not held equal in the lower algebra. Thus we do not admit x² = x to
be satisfied by x = ∞ Again, 0, which is held to satisfy x = x, seems to fail in a very
x²
important point: as x approaches to 0, the equation, instead of approaching to truth, abso-
lutely recedes from it. Is da² = dx held nearly true? Certainly not. According to the
usages of algebra, we infer A B from A - B = 0, but we must not infer 4 - B=0 from
A
A = B. Again, there are two hyperbolas y = (1 + x)¹ and y = (1 + 2x)-¹, of which the
X
1
=
1
axis of x is a common asymptote. Have they a common point when a = ∞ the algebraic
geometer answers in the affirmative; and he knows that in all the oblique projections of these
curves the point at an infinite distance becomes a determinate point, at which the projections.
are touched by the projection of the common asymptote. Does the pure algebraist admit
x = ∞ as satisfying (1 + x)-¹ = (1 + 2x)−¹?
(1 + 2x)-¹? At a look he says yes, for 00: but his pupil,
by common rule, finds only = 0, and his teacher does not instruct him to the contrary.
x
The same with y=x-1 and y = (2x)~¹. Here x-¹ (2x)-1 is satisfied by a = ∞ in geo-
metry, and the rules give (2x) = 0, or x = ∞, by one mode of application, and 12 by
another. But = 2x gives only = 0; though by rules it is a consequence of a-¹
x a
Every person has his own way of meeting these difficulties, when they appear: but it must
not be forgotten that the processes may be unseen in the heart of a complex operation.
Algebra, be it what it may, is not what it pretends to be until it ceases to have cases of
failure.
1
=
1
=
(2x)-¹.
Before I explain the extension I propose, I must say something about the result 2 = 1,
which would at first sight appear either a joke or an absurdity. Stated by a teacher to a
pupil of the faith common in learners, it would be a mystery equal to, but not exceeding,
that of the quantity less than nothing which writers on algebra sometimes announced in the
very first pages of their books. The extensions of the meanings of words which algebra
40
MR DE MORGAN, ON INFINITY;
categorically demands are in almost every instance found in common thought, mixed up with
the notions of usual things: the phrase less than nothing was not the invention of mathe-
maticians. The following paragraph might have passed without remark, if it had occurred
with a suitable context.
If the sum of two quantities be undistinguishable from the double of one of them, it
must be because the quantities themselves are undistinguishably near to each other. But if
one of them be undistinguishable from its own double, it is either because the quantity and
its double are both so small as to be immeasurable, or because they are both so great that
measurement of either by ordinary means is useless. That is, our measuring standard either
contains so many units that 2 and 1 are equally insignificant, or is a fraction so small that
2 and 1 are both merged in the idea of practically useless enormity. And we recognize all
these cases in common usage. To a carpenter the 500th of an inch is convertible with the
1000th; both are equally uncared for, so that they are, to him, equals in the only point of
view from which he need look at them. But still they are not the less 2 and 1. In thinking
of the solar system, the astronomer is content to put all the fixed stars on the celestial
sphere, that is, to make them all equidistant. Should it be that the pole-star is twice as far
from us as Sirius, he would, even if he knew it, never think of the two as unequally distant,
when contemplating the sun and the earth. But are we then to say that 2 and 1 are equal?
Not as quantities, but as to importance, looking at the quantities which it is worth while to
distinguish.
Again, the following would excite no particular remark. When we use a certain stan-
dard, and by aid of it pronounce upon an equation, we must, if we should apply a magni-
fying or a diminishing glass to our quantities, apply the same glass to our standard also. In
what manner we speak of very small fractions by reference to an inch, so only can we speak
of inches by reference to a large multiple of an inch. Carry these practical notions of
common life all the way to infinites and infinitesimals, and we have the system which I now
propose.
=
What we strictly call equality is but the most frequent case of what we have come by
slow steps, to designate by the usual sign (=). If we take all the cases in which the sign is
used, we arrive at the conclusion that A B means-actually means according to custom-
that A and B are both infinite, both finite and equal, or both nothing. But this with certain
inconsistencies. Thus∞∞ is a form of satisfaction of an equation in some applications
to geometry, and some parts of the differential calculus, but not in the general theory of
equations. The notion attached to the sign = is then that of signifying that A and B, in
A = B, cannot be distinguished, whether by greatness, by want of difference, or by smallness.
When to this we add the very frequent occurrence of the Leibnitian equality A+ a = A + a,
when a and a are infinitesimal parts of A, we see what makes up altogether a very large
amount of accordance with the definition I shall propose.
In algebra
We may distinguish the differential calculus from pure algebra, as follows.
the only distinctions are 0-", finite quantity, and o", n being positive: in the differential
∞ In algebra 0-
calculus, we have all grades of co" and ", infinites and infinitesimals.
70
AND ON THE SIGN OF EQUALITY.
41
t
has no distinction arising out of variation of n; neither has o". Interpolated grades, of the
form " (log ∞)", &c. need not be considered here, as being of rare occurrence. I shall
consider first the forms of the differential calculus, after which those of algebra will be
easily disposed of.
O
- N
>
=
n
n
Let the units of our several orders of magnitude be ... ∞, ∞, 1, ∞, ∞, &c., with
interpolates of fractional dimension when necessary. Let ∞ be supposed of one assigned
meaning, so that a. ∞, b. ∞, &c. are other infinites of the same order. Thus we should say,
being infinitely near to a right angle, sec : tan 0 a. cob. c, where a = b+a, a being
infinitely small. Let every grade of infinite and infinitesimal," or ∞", have its own sign
of equality =" or =
connected with others in the manner to be shown.
When two quan-
tities of the order n differ by quantities of a lower order, let it be said that the symbol
may be normally applied to their relation of magnitude. Thus, A and B being of the
order n, we have, C-k being their difference, A =" B
B. As the language here employed need
not be used, the principles laid down being easily applied to ordinary symbols in all the
cases wanted, we need not invent any other symbol for A, - B₁ than C₁-k, and the normal
equation A, B, leads to the normal equation A, B, Cn-k
An
– =
When both sides of the
equation are of a higher order than the sign of equality, or of a lower order, let the equation
between them be said to be supernormally satisfied, or subnormally. Thus A, B, is super-
normally satisfied; 4 =* B, subnormally. Thus A, =" B, leads to the subnormal equation
A 2
An - B₂="0.
n
n
ル
​4
n
n-k
n
=
1
2
-1
Thus da da + da² and also mdx dx + dr, both subnormally: but dx =−¹ dx + dx²
=ºdx =ºdx
normally, which is not true of mdx. And A + B_₁ =° A normally, as used by Leibnitz :
from which follows B₁ = 0, subnormally.
-1
0
1
The equation Anta="Bns is always true, supernormally, when a and ẞ have positive
value; and also An-a" B-s, subnormally.
B₂-8, subnormally. It is as if the sign of equality—or undistin-
guishability— carried with it a reference to an order of magnitude understood, and pronounced
the two sides undistinguishable, as being themselves either immeasurably great when the
measure is as given, or immeasurably small, or having an immeasurable difference.
n
n
Let A =" B and C₁ =" D₁, two equations of the ordinary type. We have then, from
these two normal equations,
n
ກ
n
n
n
A₂ ± C₁ =" B₂ ± Dn
And
in which the equation may be subnormal: for A, B, is of the nth order or a lower. And
A C₁ =" BD₂ is supernormal, unless n = 0: but A, C, BD, is normally satisfied.
=²™
generally Am=" B, C," D, give AmCn="+" BD, all being normal.
B„Dŋ, all being normal.
absolutely equal, we have A„ - B₁ = 0, for every value of m, subnormally.
176
=
m
-m
n
m-n
m
When A and B, are
n
When A, B and C-n="D_n, we have Am. C_₂ ="-" B„D_„, all being normal. Any
Am
further development is needless: I shall proceed to some examples.
When an equation is proposed, it may either be that the index of equality is given, or
that it may be what we please. If the first, the equation may be impossible, even when held
possible in ordinary algebra. Again, it may be that normal solution is demanded; or that
6
42
MR DE MORGAN, ON INFINITY;
0
0
either of the three kinds is conceded. Generally speaking, normal and subnormal solutions
are considered legitimate: but supernormal solution is sometimes admitted, sometimes excluded.
Let us have 2ª = x. If we require 2x = x, we may satisfy this equation supernormally, by
x = α∞ m: and this we actually do when we solve (1 + x) −¹ =° (1 + 2x)-¹ subnormally in
application to geometry. We may say it being what we mean that the second equation
is subnormally satisfied by the supernormal solution of the first. Again, we have subnormal
solution of 2x = x, when ≈ =
=0
x a(dx)". But we cannot have a normal solution of 2x
any finite value of k.
k
x, for
When we multiply or divide both sides of an equation, we must raise or lower the index
of equality by that of the multiplier or divisor, in order to make the step complete.
m
x
Thus 2x=x, being of the orderm, is subnormally satisfied; and 2" 1 follows,
which is true, the solution being still subnormal: for 2 and 1 are undistinguishable with
reference to ∞™. The normal solution may appear true when both sides are of a different
order from that of the equality. Thus + x + 1 =¹ x² + 2x is normally true, as well as
x²
supernormally, when a co¹.
¹. Thus y = px and y = a being two curves, there is contact
of the nth order when y(x+dx)=-" p(x+dx) is normally satisfied for all values of dx
contained under a∞ 1. But I think it would be best to restrict the use of the word normal
to cases in which the order of the sides is that of the sign of equality. And, calling" the
metre of an equation containing ", we might divide equations whose sides have the same
order into supermetrical, metrical, and submetrical. Thus
1
(0 ∞ ) x² + x + 1 = ¹ ∞² + 2x
0
1
- :
1
is supermetrical, reducible to normal by changing into = x + a =¹ x + b, is metrically
normal, or simply normal; da = da + da² is submetrically normal.
=' dæ + dæ² is submetrically normal. I treat metre as a Latin
word, to avoid the resemblance of hyper and hypo.
∞
ε° = €°
The supernormal solution of an equation is very sparingly admitted. All the theories
of algebra are founded upon the convertibility of AB and A – B = 0: and all these
theories may be preserved untouched. For some purposes, as seen, we pronounce
and ∞2 = ∞, but not e € 0, nor 2 - ∞ € 0. But these supernormal equations
occur only in the results of the differential calculus.
∞
∞
∞
But is it not a formal law of algebra that A = B and A – B = 0 are convertible ? First,
I do not find it so: for Cauchy's ∞ ∞ would not be translated into ∞ ∞ = 0, either
e°
e°
by him or by any one else. Secondly, I do not make it so, because I can show that the
metrical relation of A, B, and A- B, is material, and not formal. Given the orders of A
and C, that of AmC" is given, in all cases: but, given the orders of A and B, that of A ± B
is not thereby given, in all cases. If A and B differ in order, the order of A± B is the
greater of the two orders: but if A and B be of the same order, it is a question of leading
coefficients, having nothing to do with the orders, whether A± B be of the common order of
A and B, or of a lower order. And if of a lower order, it depends on the inferior terms of
A and B, what the order of AB shall be. Consequently, A = B, C = D do not of necessity
infer A + B = C + D. All we can say is that if both equations be normal, or both sub-
AND ON THE SIGN OF EQUALITY.
43
1
normal, A + B = C + D is satisfied, either normally or subnormally: while, if one be normal
and the other subnormal, A + B = C + D is normal. All this applies to AB, B = B,
applies_to_A
and ABB- B.
But should not the supernormal be entirely excluded? Is it not an inadvertence?
Ought we not to say that p↓x and p'x: y'x, when px and y both vanish, are both
infinite, both finite and equal, or both vanish? I think it would be difficult to preserve
algebra entirely free from the supernormal: but it is in our power to try. The case is
parallel with that of infinites and infinitesimals: we shall find it hard to exclude one without
the other. But, decide in what way we may, a clear view of what we exclude and of what we
retain is essential. Whence arises the disposition of mathematical writers not to be very
definite about the boundary of an excluded region? Is it a feeling that well-defined exclusion
is inclusion? I believe this maxim is pretty near the truth. In elementary geometry, for
example, the extreme cases are very properly considered as undemonstrated, and demonstrated
separately. Thus the square on the tangent is not allowed to enter as the rectangle under
the external segments of a vanishing chord. So long as the learner is not allowed to see what
he excludes, all is well make him sensible of what he is not to admit, and he cannot help
admitting it; that is, he sees it, and knows it. In pure algebra we recognize only the
extremes O and 0-1, which are the infinitesimal and the infinite of the subject. The explana-
tions are unaltered. Thus 2° is normally impossible: but = 0 solves it subnormally
and = 0¹ supernormally. And 2 = 1 is admissible under symbols which, as they will not
be used, might be 21 and 2 =-1. The transition to the extreme case of pure algebra is
only made with clearness by ascending and descending through all the grades of infinitely
great and infinitely small.
x
1
I shall not stop to point out how many common processes are no more than translated
into downright language in the preceding system. Nor shall I multiply instances of my own
selection; I prefer to await those which may be brought forward in the way of objection.
The absolute identification, A = B, of common algebra may be fully retained: in this
system it is A =" B, normally satisfied for all values of n: that is, normally for one value,
supermetrically or submetrically for all others.
n
0
I have elsewhere¹ expressed my belief that the values given to forms which calculation
refuses to determine, from and its congeners to sin ∞, &c. are the alternatives of indeter-
minateness: that is, for instance, that (a² — a²): (a − a), when not what we please, must
(a−
be 2a. This alternative may receive some explanation as follows. When an equation is
given in all its generality, we have not merely the solutions, normal and abnormal, but also
the possibility of treating the abnormal cases normally. Which of the two the problem may
be satisfied by is of the matter of the problem. If we have Ax = B, A and B being func-
0
¹ In a paper (Qu. Jour. Math. 1. 204) in which I attempted—and I believe with success-to amend the demonstration
of the properties of &c. I may also refer to that paper for some consideration of the principle of mean value, as connected
with sinc, &c.
02
6-2
44
MR DE MORGAN, ON INFINITY;
k k
tions of which vanish when = 0, the equation Ar = B, k being infinitely small, is sub-
normally satisfied by all finite values of x. But if we require that A B shall be normally
satisfied when k is infinitely small, we ask for the solution of
k
1
-1
Ах
Ax=-¹ B, or (0 + A₂dk)x =−¹(0 + B₂dk),
or, dividing by dk, Ar=° Br. If we propose the following-Required a number of yards
such that a yards less at a shillings a yard shall cost the same as b yards less at b shillings a
yard-we find the equation a (x − a) = b (x — b). When a and b are infinitely near, being
finite, this equation is subnormally satisfied by every value of x. If now two straight lines cut
off from the axes of x and y the lengths 1, a, 1, b, and if two perpendiculars be drawn to
those lines from their points of intersection with the axis of x, the abscissa of intersection of
those perpendiculars is determined by a ( a) b (x − b). If a and b be infinitely near,
•
any finite value of a is the abscissa of two points, one on each line, which are infinitely near.
But when we want what is called the ultimate intersection, we demand the normal solution of
a (x
G
a)
=
−1 b (x − b): that is, that a (x − a) and b(x - b) being finite, shall differ by a
quantity below. This gives only a = 2a.
Were it necessary to distinguish the solutions, I should propose to denote the subnormal
and supernormal solution by thickening the lower and higher bars of the sign of equality; but
as the distinction does not arise in operation, this will not be necessary.
It must be remembered that when the two sides of the equation are of the same order,
and the solution normal, it remains normal after multiplication or division by infinites, and
corresponding change of index. Thus x + α =¹ x, x being infinite, is normal if a be finite:
and so is 1 + ax¯¹ =º 1. The first equation is looked upon as defining the relation of the
finite to the infinite: and it is given in the system of Leibnitz as expressing that a is infi-
nitely small compared with a. What is most wanted in that system, as first given, is pre-
cisely the extension of the sign which is here made, or something equivalent. A word
signifying an infinite degree of nearness, and a sign to express it, would have avoided the
memorable stumbling-block of a quantity which, not being 0, is too small to destroy equality
by its addition to one side of an equation.
The scale of infinites is as long as the scale of number itself: nor can
we stop at the
first, which may be made to lead to a second, and so on. The extreme infinite is which
this?
α
1
0
cannot be otherwise represented. The equation a∞ may give a =
+
, which is one
form of solution: but all the lower infinites will do as well. It may be asked how we prove
To which I answer that it is in the meaning of the equation: x + a="x is normally
solved by any infinite of the nth order, for the symbols are meant to tell us that a + a and x
must differ by a lower quantity than x. All these things I find set out, as wanted,
in algebraical reasoning, which is very often an attempt to snatch a fearful explanation, as
if the writer were afraid of a reproving voice in the wind. And then the notion of infinity
is laid for the time, and the spells by which it is evoked are buried until wanted, like the
AND ON THE SIGN OF EQUALITY.
45
mighty book which was to rest interred with the wizard " save at his chief of Branksome's
need."
I have had this system before me more than eleven years, during which time I have used
it with increasing confidence. For others the present question is, not whether it must be
received, but whether it does give an account, sound or unsound, of all those ambiguities
which the use of infinites of both kinds has actually introduced into the treatment of the sign
of equality. Are there yet any difficulties which this system has no pretension to include?
If this question should be answered in the negative, the rest will follow.
On all the points of this paper one remark may be made, That they are truths in the
mind of the writer is not the point on which I insist. They form a body of considerations
which are constantly occurring, each in many different forms, and for the most part singly
and in isolation. The inquirer who meets with one, separate from the others, will quell the
difficulty by conclusions of new thought or old habit which he would be warned to revise
if their bearing on other points were suggested or remembered.
I do not know of any
modern work in which kindred difficulties are confronted with each other; I have endeavour-
ed to shew them in natural junction to those who are inclined to think of them as realities
which are not to be disposed of by conventional assumptions or conventional exclusions.
UNIVERSITY COLLEGE, LONDON,
April 2, 1864.
A. DE MORGAN.
་་
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De Morban, A
MATHEMATICS
QA
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On infinity
1
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