A Romance of Many Dimensions, by Edwin A. Abbot

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If however the Reader will take the trouble to refer to the passage
in which Recognition by Feeling is stated to be universal,
he will find this qualification -- "among the lower classes".
It is only among the higher classes and in our temperate climates
that Sight Recognition is practised.

That this power exists in any regions and for any classes
is the result of Fog; which prevails during the greater part
of the year in all parts save the torrid zones. That which is
with you in Spaceland an unmixed evil, blotting out the landscape,
depressing the spirits, and enfeebling the health, is by us recognized
as a blessing scarcely inferior to air itself, and as the Nurse
of arts and Parent of sciences. But let me explain my meaning,
without further eulogies on this beneficent Element.

If Fog were non-existent, all lines would appear equally
and indistinguishably clear; and this is actually the case
in those unhappy countries in which the atmosphere is perfectly dry
and transparent. But wherever there is a rich supply of Fog
objects that are at a distance, say of three feet, are appreciably
dimmer than those at a distance of two feet eleven inches;
and the result is that by careful and constant experimental
observation of comparative dimness and clearness, we are enabled to
infer with great exactness the configuration of the object observed.

An instance will do more than a volume of generalities to make
my meaning clear.

Suppose I see two individuals approaching whose rank I wish
to ascertain. They are, we will suppose, a Merchant and a Physician,
or in other words, an Equilateral Triangle and a Pentagon:
how am I to distinguish them?

<>

<>

C (1)
|\ - _ D
| \ ||- _
| \ || - _
| <--- >|| -----------+(> Eye-glance
___C' (2) | / A|| _ -
___--- \ - _D' | / ||_ -
__--- \ || - _ |/ _ - E
| \ || - _ B
| \ || - _
| Eye-glance \ || - _
| <----------- A'>|| ------------------------+(>
| / || _ -
| / || _ -
|__ / || _ -
---___ / || _ -
---___/ _ -E'
B'

It will be obvious, to every child in Spaceland who has touched
the threshold of Geometrical Studies, that, if I can bring my eye so
that its glance may bisect an angle (A) of the approaching stranger,
my view will lie as it were evenly between his two sides that are
next to me (viz. CA and AB), so that I shall contemplate
the two impartially, and both will appear of the same size.

Now in the case of (1) the Merchant, what shall I see? I shall see
a straight line DAE, in which the middle point (A) will be very bright
because it is nearest to me; but on either side the line will
shade away RAPIDLY INTO DIMNESS, because the sides AC and AB
RECEDE RAPIDLY INTO THE FOG and what appear to me as
the Merchant's extremities, viz. D and E, will be VERY DIM INDEED.

On the other hand in the case of (2) the Physician, though I shall
here also see a line (D'A'E') with a bright centre (A'),
yet it will shade away LESS RAPIDLY into dimness, because the sides
(A'C', A'B') RECEDE LESS RAPIDLY INTO THE FOG: and what appear
to me the Physician's extremities, viz. D' and E', will not be
NOT SO DIM as the extremities of the Merchant.

The Reader will probably understand from these two instances how --
after a very long training supplemented by constant experience --
it is possible for the well-educated classes among us to discriminate
with fair accuracy between the middle and lowest orders,
by the sense of sight. If my Spaceland Patrons have grasped
this general conception, so far as to conceive the possibility of it
and not to reject my account as altogether incredible --
I shall have attained all I can reasonably expect. Were I to attempt
further details I should only perplex. Yet for the sake of the young
and inexperienced, who may perchance infer -- from the two simple
instances I have given above, of the manner in which I should
recognize my Father and my Sons -- that Recognition by sight
is an easy affair, it may be needful to point out that in actual life
most of the problems of Sight Recognition are far more
subtle and complex.

If for example, when my Father, the Triangle, approaches me,
he happens to present his side to me instead of his angle, then,
until I have asked him to rotate, or until I have edged my eye
round him, I am for the moment doubtful whether he may not be
a Straight Line, or, in other words, a Woman. Again, when I am
in the company of one of my two hexagonal Grandsons, contemplating one
of his sides (AB) full front, it will be evident from
the accompanying diagram that I shall see one whole line (AB)
in comparative brightness (shading off hardly at all at the ends)
and two smaller lines (CA and BD) dim throughout and shading away
into greater dimness towards the extremities C and D.

<>

<>

/\ - _ C
/ \ || _
/ \ || - _
/ \|| - _
| A || - _
| || -+(> (Eye)
| B || _ -
\ /|| _ -
\ / || _ -
\ / || -
\/ _ - D

But I must not give way to the temptation of enlarging on
these topics. The meanest mathematician in Spaceland will readily
believe me when I assert that the problems of life, which present
themselves to the well-educated -- when they are themselves in motion,
rotating, advancing or retreating, and at the same time attempting to
discriminate by the sense of sight between a number of Polygons
of high rank moving in different directions, as for example in
a ball-room or conversazione -- must be of a nature to task
the angularity of the most intellectual, and amply justify
the rich endowments of the Learned Professors of Geometry,
both Static and Kinetic, in the illustrious University of Wentbridge,
where the Science and Art of Sight Recognition are regularly taught
to large classes of the ELITE of the States.

It is only a few of the scions of our noblest and wealthiest houses,
who are able to give the time and money necessary for the thorough
prosecution of this noble and valuable Art. Even to me,
a Mathematician of no mean standing, and the Grandfather of two
most hopeful and perfectly regular Hexagons, to find myself
in the midst of a crowd of rotating Polygons of the higher classes,
is occasionally very perplexing. And of course to a common Tradesman,
or Serf, such a sight is almost as unintelligible as it would be
to you, my Reader, were you suddenly transported into our country.

In such a crowd you could see on all sides of you nothing but a Line,
apparently straight, but of which the parts would vary
irregularly and perpetually in brightness or dimness. Even if you
had completed your third year in the Pentagonal and Hexagonal classes
in the University, and were perfect in the theory of the subject,
you would still find that there was need of many years of experience,
before you could move in a fashionable crowd without jostling against
your betters, whom it is against etiquette to ask to "feel", and who,
by their superior culture and breeding, know all about your movements,
while you know very little or nothing about theirs. In a word,
to comport oneself with perfect propriety in Polygonal society,
one ought to be a Polygon oneself. Such at least is
the painful teaching of my experience.

It is astonishing how much the Art -- or I may almost call it instinct
-- of Sight Recognition is developed by the habitual practice of it
and by the avoidance of the custom of "Feeling". Just as, with you,
the deaf and dumb, if once allowed to gesticulate and to use
the hand-alphabet, will never acquire the more difficult
but far more valuable art of lipspeech and lip-reading, so it is
with us as regards "Seeing" and "Feeling". None who in early life
resort to "Feeling" will ever learn "Seeing" in perfection.

For this reason, among our Higher Classes, "Feeling" is discouraged
or absolutely forbidden. From the cradle their children,
instead of going to the Public Elementary schools (where the art
of Feeling is taught), are sent to higher Seminaries
of an exclusive character; and at our illustrious University,
to "feel" is regarded as a most serious fault, involving Rustication
for the first offence, and Expulsion for the second.

But among the lower classes the art of Sight Recognition is regarded
as an unattainable luxury. A common Tradesman cannot afford
to let his son spend a third of his life in abstract studies.
The children of the poor are therefore allowed to "feel"
from their earliest years, and they gain thereby a precocity
and an early vivacity which contrast at first most favourably with
the inert, undeveloped, and listless behaviour of the half-instructed
youths of the Polygonal class; but when the latter have at last
completed their University course, and are prepared to put
their theory into practice, the change that comes over them
may almost be described as a new birth, and in every art, science,
and social pursuit they rapidly overtake and distance
their Triangular competitors.

Only a few of the Polygonal Class fail to pass the Final Test
or Leaving Examination at the University. The condition of
the unsuccessful minority is truly pitiable. Rejected from
the higher class, they are also despised by the lower.
They have neither the matured and systematically trained powers
of the Polygonal Bachelors and Masters of Arts, nor yet the native
precocity and mercurial versatility of the youthful Tradesman.
The professions, the public services, are closed against them;
and though in most States they are not actually debarred
from marriage, yet they have the greatest difficulty in forming
suitable alliances, as experience shews that the offspring of such
unfortunate and ill-endowed parents is generally itself unfortunate,
if not positively Irregular.

It is from these specimens of the refuse of our Nobility
that the great Tumults and Seditions of past ages have generally
derived their leaders; and so great is the mischief thence arising
that an increasing minority of our more progressive Statesmen
are of opinion that true mercy would dictate their entire suppression,
by enacting that all who fail to pass the Final Examination
of the University should be either imprisoned for life,
or extinguished by a painless death.

But I find myself digressing into the subject of Irregularities,
a matter of such vital interest that it demands a separate section.

Section 7. Concerning Irregular Figures

Throughout the previous pages I have been assuming --
what perhaps should have been laid down at the beginning as a distinct
and fundamental proposition -- that every human being in Flatland
is a Regular Figure, that is to say of regular construction.
By this I mean that a Woman must not only be a line,
but a straight line; that an Artisan or Soldier must have
two of his sides equal; that Tradesmen must have three sides equal;
Lawyers (of which class I am a humble member), four sides equal,
and generally, that in every Polygon, all the sides must be equal.

The size of the sides would of course depend upon the age of
the individual. A Female at birth would be about an inch long,
while a tall adult Woman might extend to a foot. As to the Males
of every class, it may be roughly said that the length of
an adult's sides, when added together, is two feet or a little more.
But the size of our sides is not under consideration.
I am speaking of the EQUALITY of sides, and it does not need
much reflection to see that the whole of the social life in Flatland
rests upon the fundamental fact that Nature wills all Figures
to have their sides equal.

If our sides were unequal our angles might be unequal.
Instead of its being sufficient to feel, or estimate by sight,
a single angle in order to determine the form of an individual,
it would be necessary to ascertain each angle by the experiment
of Feeling. But life would be too short for such a tedious grouping.
The whole science and art of Sight Recognition would at once perish;
Feeling, so far as it is an art, would not long survive;
intercourse would become perilous or impossible; there would be
an end to all confidence, all forethought; no one would be safe
in making the most simple social arrangements; in a word,
civilization would relapse into barbarism.

Am I going too fast to carry my Readers with me to these
obvious conclusions? Surely a moment's reflection, and a single
instance from common life, must convince every one that our whole
social system is based upon Regularity, or Equality of Angles.
You meet, for example, two or three Tradesmen in the street,
whom you recognize at once to be Tradesmen by a glance at their angles
and rapidly bedimmed sides, and you ask them to step into your house
to lunch. This you do at present with perfect confidence,
because everyone knows to an inch or two the area occupied
by an adult Triangle: but imagine that your Tradesman drags
behind his regular and respectable vertex, a parallelogram
of twelve or thirteen inches in diagonal: -- what are you to do
with such a monster sticking fast in your house door?

But I am insulting the intelligence of my Readers by accumulating
details which must be patent to everyone who enjoys the advantages of
a Residence in Spaceland. Obviously the measurements of
a single angle would no longer be sufficient under such
portentous circumstances; one's whole life would be taken up
in feeling or surveying the perimeter of one's acquaintances.
Already the difficulties of avoiding a collision in a crowd are enough
to tax the sagacity of even a well-educated Square; but if no one
could calculate the Regularity of a single figure in the company,
all would be chaos and confusion, and the slightest panic
would cause serious injuries, or -- if there happened to be
any Women or Soldiers present -- perhaps considerable loss of life.

Expediency therefore concurs with Nature in stamping the seal
of its approval upon Regularity of conformation: nor has the Law
been backward in seconding their efforts. "Irregularity of Figure"
means with us the same as, or more than, a combination of
moral obliquity and criminality with you, and is treated accordingly.
There are not wanting, it is true, some promulgators of paradoxes
who maintain that there is no necessary connection between
geometrical and moral Irregularity. "The Irregular", they say,
"is from his birth scouted by his own parents, derided by
his brothers and sisters, neglected by the domestics,
scorned and suspected by society, and excluded from all posts
of responsibility, trust, and useful activity. His every movement
is jealously watched by the police till he comes of age
and presents himself for inspection; then he is either destroyed,
if he is found to exceed the fixed margin of deviation,
or else immured in a Government Office as a clerk of
the seventh class; prevented from marriage; forced to drudge
at an uninteresting occupation for a miserable stipend;
obliged to live and board at the office, and to take even his vacation
under close supervision; what wonder that human nature,
even in the best and purest, is embittered and perverted
by such surroundings!"

All this very plausible reasoning does not convince me, as it has not
convinced the wisest of our Statesmen, that our ancestors erred
in laying it down as an axiom of policy that the toleration
of Irregularity is incompatible with the safety of the State.
Doubtless, the life of an Irregular is hard; but the interests of
the Greater Number require that it shall be hard. If a man with
a triangular front and a polygonal back were allowed to exist
and to propagate a still more Irregular posterity, what would become
of the arts of life? Are the houses and doors and churches
in Flatland to be altered in order to accommodate such monsters?
Are our ticket-collectors to be required to measure every man's
perimeter before they allow him to enter a theatre or to take
his place in a lecture room? Is an Irregular to be exempted
from the militia? And if not, how is he to be prevented from
carrying desolation into the ranks of his comrades? Again,
what irresistible temptations to fraudulent impostures must
needs beset such a creature! How easy for him to enter a shop
with his polygonal front foremost, and to order goods
to any extent from a confiding tradesman! Let the advocates of
a falsely called Philanthropy plead as they may for the abrogation
of the Irregular Penal Laws, I for my part have never known
an Irregular who was not also what Nature evidently intended him to be
-- a hypocrite, a misanthropist, and, up to the limits of his power,
a perpetrator of all manner of mischief.

Not that I should be disposed to recommend (at present)
the extreme measures adopted by some States, where an infant
whose angle deviates by half a degree from the correct angularity
is summarily destroyed at birth. Some of our highest and ablest men,
men of real genius, have during their earliest days laboured under
deviations as great as, or even greater than, forty-five minutes:
and the loss of their precious lives would have been an irreparable
injury to the State. The art of healing also has achieved
some of its most glorious triumphs in the compressions, extensions,
trepannings, colligations, and other surgical or diaetetic operations
by which Irregularity has been partly or wholly cured.
Advocating therefore a VIA MEDIA, I would lay down no fixed
or absolute line of demarcation; but at the period when the frame
is just beginning to set, and when the Medical Board has reported that
recovery is improbable, I would suggest that the Irregular offspring
be painlessly and mercifully consumed.

Section 8. Of the Ancient Practice of Painting

If my Readers have followed me with any attention up to this point,
they will not be surprised to hear that life is somewhat dull
in Flatland. I do not, of course, mean that there are not battles,
conspiracies, tumults, factions, and all those other phenomena which
are supposed to make History interesting; nor would I deny
that the strange mixture of the problems of life and the problems
of Mathematics, continually inducing conjecture and giving
the opportunity of immediate verification, imparts to our existence
a zest which you in Spaceland can hardly comprehend. I speak now
from the aesthetic and artistic point of view when I say that life
with us is dull; aesthetically and artistically, very dull indeed.

How can it be otherwise, when all one's prospect, all one's
landscapes, historical pieces, portraits, flowers, still life,
are nothing but a single line, with no varieties except degrees of
brightness and obscurity?

It was not always thus. Colour, if Tradition speaks the truth,
once for the space of half a dozen centuries or more,
threw a transient splendour over the lives of our ancestors
in the remotest ages. Some private individual -- a Pentagon
whose name is variously reported -- having casually discovered
the constituents of the simpler colours and a rudimentary method
of painting, is said to have begun decorating first his house,
then his slaves, then his Father, his Sons, and Grandsons,
lastly himself. The convenience as well as the beauty of the results
commended themselves to all. Wherever Chromatistes, --
for by that name the most trustworthy authorities concur
in calling him, -- turned his variegated frame, there he at once
excited attention, and attracted respect. No one now needed
to "feel" him; no one mistook his front for his back;
all his movements were readily ascertained by his neighbours
without the slightest strain on their powers of calculation;
no one jostled him, or failed to make way for him; his voice was saved
the labour of that exhausting utterance by which we colourless Squares
and Pentagons are often forced to proclaim our individuality
when we move amid a crowd of ignorant Isosceles.

The fashion spread like wildfire. Before a week was over,
every Square and Triangle in the district had copied the example
of Chromatistes, and only a few of the more conservative Pentagons
still held out. A month or two found even the Dodecagons
infected with the innovation. A year had not elapsed before
the habit had spread to all but the very highest of the Nobility.
Needless to say, the custom soon made its way from the district of
Chromatistes to surrounding regions; and within two generations no one
in all Flatland was colourless except the Women and the Priests.

Here Nature herself appeared to erect a barrier, and to plead
against extending the innovation to these two classes.
Many-sidedness was almost essential as a pretext for the Innovators.
"Distinction of sides is intended by Nature to imply distinction
of colours" -- such was the sophism which in those days
flew from mouth to mouth, converting whole towns at a time
to the new culture. But manifestly to our Priests and Women
this adage did not apply. The latter had only one side,
and therefore -- plurally and pedantically speaking -- NO SIDES.
The former -- if at least they would assert their claim to be
really and truly Circles, and not mere high-class Polygons
with an infinitely large number of infinitesimally small sides --
were in the habit of boasting (what Women confessed and deplored)
that they also had no sides, being blessed with a perimeter of
one line, or, in other words, a Circumference. Hence it came to pass
that these two Classes could see no force in the so-called axiom about
"Distinction of Sides implying Distinction of Colour"; and when
all others had succumbed to the fascinations of corporal decoration,
the Priests and the Women alone still remained pure from
the pollution of paint.

Immoral, licentious, anarchical, unscientific -- call them
by what names you will -- yet, from an aesthetic point of view,
those ancient days of the Colour Revolt were the glorious childhood of
Art in Flatland -- a childhood, alas, that never ripened into manhood,
nor even reached the blossom of youth. To live was then in itself
a delight, because living implied seeing. Even at a small party,
the company was a pleasure to behold; the richly varied hues
of the assembly in a church or theatre are said to have more than once
proved too distracting for our greatest teachers and actors;
but most ravishing of all is said to have been the unspeakable
magnificence of a military review.

The sight of a line of battle of twenty thousand Isosceles suddenly
facing about, and exchanging the sombre black of their bases for
the orange and purple of the two sides including their acute angle;
the militia of the Equilateral Triangles tricoloured in red, white,
and blue; the mauve, ultra-marine, gamboge, and burnt umber
of the Square artillerymen rapidly rotating near their vermilion guns;
the dashing and flashing of the five-coloured and six-coloured
Pentagons and Hexagons careering across the field in their offices
of surgeons, geometricians and aides-de-camp -- all these may well
have been sufficient to render credible the famous story
how an illustrious Circle, overcome by the artistic beauty
of the forces under his command, threw aside his marshal's baton
and his royal crown, exclaiming that he henceforth exchanged them
for the artist's pencil. How great and glorious the sensuous
development of these days must have been is in part
indicated by the very language and vocabulary of the period.
The commonest utterances of the commonest citizens in the time
of the Colour Revolt seem to have been suffused with a richer tinge
of word or thought; and to that era we are even now indebted for
our finest poetry and for whatever rhythm still remains
in the more scientific utterance of these modern days.

Section 9. Of the Universal Colour Bill

But meanwhile the intellectual Arts were fast decaying.

The Art of Sight Recognition, being no longer needed,
was no longer practised; and the studies of Geometry, Statics,
Kinetics, and other kindred subjects, came soon to be
considered superfluous, and fell into disrespect and neglect even at
our University. The inferior Art of Feeling speedily experienced
the same fate at our Elementary Schools. Then the Isosceles classes,
asserting that the Specimens were no longer used nor needed,
and refusing to pay the customary tribute from the Criminal classes
to the service of Education, waxed daily more numerous
and more insolent on the strength of their immunity from
the old burden which had formerly exercised the twofold
wholesome effect of at once taming their brutal nature and thinning
their excessive numbers.

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