City of London School Magazine 8(December 1885), 217--221.
We have strong reasons for believing that the author of the above is not unknown to
most of our readers: that this is not the first or the most philosophical production of his
pen: and, what is more to the point, that the name of A Square will be found in the
Mathematical Tripos list for the year 186- by any one who will consult the Cambridge
Calendar for that purpose.
The work is dedicated ``To the Inhabitants of Space in
general ... by a Humble Native
of Flatland, in the Hope that, even as he was initiated into the Mysteries of Three
Dimensions, having been previously conversant with only Two, so the Citizens of that
Celestial Region may aspire yet higher and higher to the Secrets of Four, Five, or even Six
Dimensions, thereby contributing to the Enlargement of the Imagination, and the possible
Development of that most rare and excellent Gift of Modesty among the superior races of
Solid Humanity.'' We hope the author will excuse our placing before a certain section of
the ``Inhabitants of Space'' --- that important section known as ``Our Readers'' --- a short
account of his experiences, contributing thereby to their amusement, and, it is to be
hoped, to their mental edification as well.
Part I. --- This world (from the point of view of
A SQUARE) enlarges upon the nature of
Flatland. For the benefit of our younger readers, we will give a short explanation of
Flatland (or the World of Two Dimensions). Imagine a vast sheet of paper --- the thinner
the better --- and on it, or in it (which comes to the same thing if the paper is very thin) a
number of figures drawn --- straight lines, triangles, squares, pentagons, hexagons, and
others --- moving about freely in the surface of the paper, but without the power of rising
above or sinking below it. ``Then,'' says the writer, ``you will have a pretty correct notion
of my country and countrymen. Alas, a few years ago, I should have said ``my universe;''
but now my mind has been opened to higher views of things.''
The inhabitants of Flatland are distinguished by the number of their sides. Women are
straight lines; soldiers and common workmen are isosceles triangles with very short bases;
the middle class consists of equilateral triangles; professional men and gentlemen are
squares or pentagons; lastly come the several degrees of the nobility, beginning at
hexagons, and rising by increase in the number of sides to unlimited polygons, which, we
scarcely point out to students of Euclid, finally approximate to perfect circles, the order
As a general rule, a son has one more side than his father, but this does not apply to
the lower classes. Nevertheless even they are not denied all hope of improving their
condition. After a long period of military success or diligent labour, it is generally found
that the more intelligent among the soldier and artisan classes manifest a slight increase of
their base, and a shrinking of their equal sides, thus bringing them nearer to the equilateral
type. ``A wise ordinance of nature has decreed that, in proportion as the working classes
increase in intelligence, knowledge, and all virtue, in that same proportion their acute angle
(which makes them physically terrible) shall increase also, and approximate to the
harmless angle of the equilateral ... How admirable is this law
The inhabitants of Flatland have two methods of recognizing one another. Firstly, by
the ordinary three-dimensional process of sight, considerably modified by the exigencies of
two-dimensional physical laws. On the least reflection it will be patent to any one that an
inhabitant of Flatland can only see another as a line --- reduced to a point in the case of a woman coming directly towards him. Whence the law in well-regulated states that a
woman shall keep herself perpetually oscillating to disclose her whereabouts to the other
sex, and save them from sudden and invisible annihilation. The aids to estimation of
distance afforded by differential muscular effort and parallactic displacement appear to
require a brain of three dimensions, and presumably an intelligence of more than three.
Circles, squares, etc., are distinguished through fog, which is introduced much more
opportunely in this universe than in ours, by the rapidity with which the line seen shades
off at its extremities.
The second method is known as ``feeling,'' which consists in feeling the angles of the
person to be recognized, and estimating the amount of their acuteness or obtuseness, as
the case may be. This method is in favour with the lower classes, because the other
requires a somewhat prolonged education. The following incident will serve to illustrate
the principles described, as well as their danger:---
``I have heard that my excellent grandfather --- one of the least irregular of his unhappy
isosceles class, who indeed obtained, shortly before his decease, four out of seven votes
from the Sanitary and Social Board for passing him into the class of the equal-sided ---
often deplored, with a tear in his venerable eye, a miscarriage which had occurred to his
great-great-great-grandfather, a respectable working man with an angle or brain of 59°30'. According to his account, my unfortunate ancestor, being afflicted with rheumatism, and in the act of being felt by a polygon, by one sudden start accidentally transfixed the great man through the diagonal; and thereby, partly in consequence of his long imprisonment and degradation, and partly because of the moral shock which pervaded the whole of my ancestor's relations, threw back our family a degree and a half in their ascent towards
better things. The result was that in the next generation the family brain was registered at
only 58°, and not till the lapse of five generations was the lost ground recovered, the full 60° attained, and the ascent from the isosceles finally achieved. And all this series of calamities from one little accident in the process of feeling.''
Space, to say nothing of a feeling that the author will not thank us for disclosing too
much of its contents, forbids us to linger over Part I., with its account of the Universal
Colour Bill, to aid recognition by sight, and the Suppression of the Chromatic Sedition
that ensued. We must hurry on to the second, and, on the whole, more interesting part of
the book, --- Other Worlds.
On the night of the last day but one of the year --- 1999, according to Flatland
SQUARE --- having to follow his experiences for some little time we feel
justified in calling him ours --- had a dream, in which he saw Lineland, and, having accosted
the monarch thereof as ``woman,'' for such he naturally appeared to be, discovering his
mistake, endeavoured to enlarge the said monarch's ideas, with very little result, especially
since the monarch could not even see his informant unless the latter crossed the path of
the line which constituted the former's universe. The most striking feature of Lineland is
the impossibility of passing your neighbours; nothing but oscillatory motion within small
intervals is possible.
On the last night of 1999 our
SQUARE was sitting alone with his wife, musing on the
events of the past and the prospects of the coming millennium, but more particularly
pondering over some words that had casually issued from the mouth of his youngest
grandson, a most promising young hexagon, of mathematical attainments far above the
average, who, when told that 32 represented the number of square inches in a square
whose side was 3 inches long, had perversely suggested that 33 must represent ``a square
of 3 inches each was, moving somehow parallel to itself (but I don't see how) making a
something else ( but I don't see what) of 3 inches every way'' --- a logical deduction which somewhat ruffled his grandfather. While in this state of mind he is visited by a sphere,
who naturally appears to him as a circle, but possessed of the remarkable power of
increasing his size from a point to a large circle, and vice versâ, and the still more
remarkable power of vanishing at will. The stranger in vain seeks to impress him with the
fact that there is a direction ``up and down,'' not to be confounded with north and south,
and only enrages him by taking a tablet out of a locked closet [from above] and by giving
him a dig in his stomach, or area, which he had hitherto naturally supposed inaccessible.
At last the sphere loses all patience, and hurling him perforce out of his plane conducts
him above it, and lets him look down on it and see the inside of everything, men and
women not excepted. Then, following up his advantage by argument, the sphere makes
his convert and replaces him in Flatland, coming to him again, however, and this time
showing him Pointland, whose monarch can only ring the changes on one all-embracing
sentiment, ``It fills all space,'' the It alluded to being of course himself. Our
SQUARE is then left by the sphere with the watchword, ``Upward, not Northward,'' to preach the new
``Gospel of the Third Dimension'' amongst his fellow-countrymen, a proceeding which brings down upon him the strong arm of the law, to whose tender mercies he is consigned, and where we must leave him, the prey of a diseased imagination in the eyes of his countrymen, but in reality, as we know, a pitiable example of that pitiable class of individuals whom the world persistently misunderstands and rejects, and of whose genius it is not worthy.
We must again crave pardon of the author for extracting the pith from his romance, and
would recommend any, especially mathematicians, who might appreciate a profitable
diversion from the all-absorbing festivities of the Christmas season, to invest in the book.
Its style in particular will amply repay them by its lucid simplicity and suppressed
The moral of the story, if indeed it have one, must lie in the existence of space of more
than three dimensions. But not even the concentrated wisdom of a president of the
British Association has been able as yet to do more than suggest in the most provoking
way that he himself has some conception of a fourth dimension. Such an assertion is of
course incontrovertible; but it does not help us much. The only process by which it
might be possible to attain to any conception is that of analogy. Whether we regard four
dimensions as the way from the inside to the outside of a closed surface without passing
through it, or as space the section of which by three-dimension space is a limited solid,
i.e. space whose shadow is of three dimensions; in both of these, and in similar arguments, we are confronted by a total lack of imagination to grasp the ideas and ``body them forth,'' an expression that has in itself a decided smack of three dimensions.
If, however, at any time the author is fortunate enough, in his capacity of a solid, to
receive a revelation from the universe next above us in the continuous scale, we entreat
him to lose no time in transferring to paper and transmitting to posterity his adventures in
that region also. We at least shall be grateful to him.