facilitated by the fact that there is no difficulty in imagining a
world confined to only two dimensions--which we may take for
simplicity to be plane--though equally well it might be a spherical
or other surface. We may picture the inhabitants of flatland as
moving either on the surface of a plane or between two parallel and
adjacent planes. They could move in any direction along the plane,
but they could not move perpendicularly to it, and would have no
consciousness that such a motion was possible. We may suppose them
to have no thickness, in which case they would be mere geometrical
abstractions; or we may think of them as having a small but uniform
thickness, in which case they would be realities."
. . . . .
"If an inhabitant of flatland was able to move in three dimensions,
he would be credited with supernatural powers by those who were
unable so to move; for he could appear or disappear at will; could
(so far as they could tell) create matter or destroy it, and would
be free from so many constraints to which the other inhabitants were
subject that his actions would be inexplicable to them."
. . . . .
"Our conscious life is in three dimensions, and naturally the idea
occurs whether there may not be a fourth dimension. No inhabitant of
flatland could realize what life in three dimensions would mean,
though, if he evolved an analytical geometry applicable to the world
in which he lived, he might be able to extend it so as to obtain
results true of that world in three dimensions which would be to him
unknown and inconceivable. Similarly we cannot realize what life in
four dimensions is like, though we can use analytical geometry to
obtain results true of that world, or even of worlds of higher
dimensions. Moreover, the analogy of our position to the inhabitants
of flatland enables us to form some idea of how inhabitants of
space of four dimensions would regard us."
. . . . .
"If a finite solid was passed slowly through flatland, the
inhabitants would be conscious only of that part of it which was in
their plane. Thus they would see the shape of the object gradually
change and ultimately vanish. In the same way, if a body of four
dimensions was passed through our space, we should be conscious of
it only as a solid body (namely, the section of the body by our
space) whose form and appearance gradually changed and perhaps
ultimately vanished. It has been suggested that the birth, growth,
life, and death of animals, may be explained thus as the passage of
finite four-dimensional bodies through our three-dimensional space."
Attempts have been made to construct drawings and models showing a
four-dimensional body. The success of such attempts has not been very
encouraging.
Investigators of this class look upon the actuality of a fourth
dimension as an unsolved question, but they hold that, provided we could
see our way clear to adopt it, it would open up wondrous possibilities
in the way of explaining abstruse and hitherto inexplicable physical
conditions and phenomena.
There is obviously no limit to such speculations, provided we assume the
existence of such conditions as are needed for our purpose. Too often,
however, those who indulge in such day-dreams begin by assuming the
impossible, and end by imagining the absurd.
We have so little positive knowledge in regard to the ultimate
constitution of matter and even in regard to the actual character of the
objects around us, which are revealed to us through our senses, that the
field in which our imagination may revel is boundless. Perhaps some day
the humanity of the present will merge itself into a new race, endowed
with new senses, whose revelations are to us, for the present, at least,
utterly inconceivable.
The possibility of such a development may be rendered more clear if we
imagine the existence of a race devoid of the sense of hearing, and
without the organs necessary to that sense. They certainly could form no
idea of sound, far less could they enjoy music or oratory, such as
afford us so much delight. And, if one or more of our race should visit
these people, how very strange to them would appear those curious
appendages, called ears, which project from the sides of our heads, and
how inexplicable to them would be the movements and expressions of
intelligence which we show when we talk or sing? It is certain that no
development of the physical or mathematical sciences could give them any
idea whatever of the sensations which sound, in its various
modifications, imparts to us, and neither can any progress in that
direction enable us to acquire any idea of the revelations which a new
sense might open up to us. Nevertheless, it seems to me that the
development of new senses and new sense organs is not only more likely
to be possible, but that it is actually more probable, than any
revelation in regard to a fourth dimension.
FOOTNOTES:
[5] "History of Civilization in England." American edition, Vol. II,
page 342.
[6] "The Natural History of Hell," by John Phillipson, page 37.
[7] "Flatland," by E. A. Abbott. London, 1884.
[8] "From Adam's Peak to Elephanta--" page 160.
HOW A SPACE MAY BE APPARENTLY ENLARGED BY CHANGING ITS SHAPE
The following is a curious illustration of the errors to which careless
observers may be subject:
Draw a square, like Fig. 19, and divide the sides into 8 parts each.
Join the points of division in opposite sides so as to divide the whole
square into 64 small squares. Then draw the lines shown in black and cut
up the drawing into four pieces. The lines indicating the cuts have been
made quite heavy so as to show up clearly, but on the actual card they
may be made quite light. Now, put the four pieces together, so as to
form the rectangle shown in Fig. 20. Unless the scale, to which the
drawing is made is quite large and the work very accurate, it will seem
that the rectangle contains 5 squares one way and 13 the other which,
when multiplied together, give 65 for the number of small squares, being
an apparent gain of one square by the simple process of cutting.
[Illustration: Fig. 19.]
[Illustration: Fig. 20.]
This paradox is very apt to puzzle those who are not familiar with
accurate drawings. Of course, every person of common sense knows that
the card or drawing is not made any larger by cutting it, but where does
the 65th small square come from?
On careful examination it will be seen that the line AB, Fig. 20, is not
quite straight and the three parts into which it is divided are thus
enabled to gain enough to make one of the small squares. On a small
scale this deviation from the straight line is not very obvious, but
make a larger drawing, and make it carefully, and it will readily be
seen how the trick is done.
CAN A MAN LIFT HIMSELF BY THE STRAPS OF HIS BOOTS?
I think it was the elder Stephenson, the famous engineer, who told a man
who claimed the honor of having invented a perpetual motion, that when
he could lift himself over a fence by taking hold of his waist-band, he
might hope to accomplish his object. And the query which serves as a
title for this article has long been propounded as one of the physical
impossibilities. And yet, perhaps, it might be possible to invent a
waist-band or a boot-strap by which this apparently impossible feat
might be accomplished!
Travelers in Mexico frequently bring home beans which jump about when
laid on a table. They are well-known as "jumping beans" and have often
been a puzzle to those who were not familiar with the facts in the case.
Each bean contains the larva of a species of beetle and this affords a
clue to the secret. But the question at once comes up: "How is the
insect able to move, not only itself, but its house as well, without
some purchase or direct contact with the table?"
The explanation is simple. The hollow bean is elastic and the insect has
strength enough to bend it slightly; when the insect suddenly relaxes
its effort and allows the bean to spring back to its former shape, the
reaction on the table moves the bean. A man placed in a perfectly rigid
box could never move himself by pressing on the sides, but if the box
were elastic and could be bent by the strength of the man inside, it
might be made to move.
A somewhat analogous result, but depending on different principles, is
attained in certain curious boat races which are held at some English
regattas and which is explained by Prof. W. W. Rouse Ball, in his
"Mathematical Recreations and Problems." He says that it
"affords a somewhat curious illustration of the fact that commonly a
boat is built so as to make the resistance to motion straight
forward less than that to motion in the opposite direction.
"The only thing supplied to the crew is a coil of rope, and they
have (without leaving the boat) to propel it from one point to
another as rapidly as possible. The motion is given by tying one end
of the rope to the afterthwart, and giving the other end a series of
violent jerks in a direction parallel to the keel.
"The effect of each jerk is to compress the boat. Left to itself the
boat tends to resume its original shape, but the resistance to the
motion through the water of the stern is much greater than that of
the bow, hence, on the whole, the motion is forwards. I am told that
in still water a pace of two or three miles an hour can be thus
attained."
HOW A SPIDER LIFTED A SNAKE
One of the most interesting books in natural history is a work on
"Insect Architecture," by Rennie. But if the architecture of insect
homes is wonderful, the engineering displayed by these creatures is
equally marvellous. Long before man had thought of the saw, the saw-fly
had used the same tool, made after the same fashion, and used in the
same way for the purpose of making slits in the branches of trees so
that she might have a secure place in which to deposit her eggs. The
carpenter bee, with only the tools which nature has given her, cuts a
round hole, the full diameter of her body, through thick boards, and so
makes a tunnel by which she can have a safe retreat, in which to rear
her young. The tumble-bug, without derrick or machinery, rolls over
large masses of dirt many times her own weight, and the sexton beetle
will, in a few hours, bury beneath the ground the carcass of a
comparatively large animal. All these feats require a degree of instinct
which in a reasoning creature would be called engineering skill, but
none of them are as wonderful as the feats performed by the spider. This
extraordinary little animal has the faculty of propelling her threads
directly against the wind, and by means of her slender cords she can
haul up and suspend bodies which are many times her own weight.
Some years ago a paragraph went the rounds of the papers in which it was
said that a spider had suspended an unfortunate mouse, raising it up
from the ground, and leaving it to perish miserably between heaven and
earth. Would-be philosophers made great fun of this statement, and
ridiculed it unmercifully. I know not how true it _was_, but I know that
it _might have been_ true.
Some years ago, in the village of Havana, in the State of New York, a
spider entangled a milk-snake in her threads, and actually raised it
some distance from the ground, and this, too, in spite of the struggles
of the reptile, which was alive.
By what process of engineering did the comparatively small and feeble
insect succeed in overcoming and lifting up by mechanical means, the
mouse or the snake? The solution is easy enough if we only give the
question a little thought.
The spider is furnished with one of the most efficient mechanical
implements known to engineers, viz., a strong elastic thread. That the
thread is strong is well known. Indeed, there are few substances that
will support a greater strain than the silk of the silkworm, or the
spider; careful experiment having shown that for equal sizes the
strength of these fibers exceeds that of common iron. But
notwithstanding its strength, the spider's thread alone would be useless
as a mechanical power if it were not for its elasticity. The spider has
no blocks or pulleys, and, therefore, it cannot cause the thread to
divide up and run in different directions, but the elasticity of the
thread more than makes up for this, and renders possible the lifting of
an animal much heavier than a mouse or a snake. This may require a
little explanation.
Let us suppose that a child can lift a six-pound weight one foot high
and do this twenty times a minute. Furnish him with 350 rubber bands,
each capable of pulling six pounds through one foot when stretched. Let
these bands be attached to a wooden platform on which stand a pair of
horses weighing 2,100 lbs., or rather more than a ton. If now the child
will go to work and stretch these rubber bands, singly, hooking each one
up, as it is stretched, in less than twenty minutes he will have raised
the pair of horses one foot!
We thus see that the elasticity of the rubber bands enables the child to
divide the weight of the horses into 350 pieces of six pounds each, and
at the rate of a little less than one every three seconds, he lifts all
these separate pieces one foot, so that the child easily lifts this
enormous weight.
Each spider's thread acts like one of the elastic rubber bands. Let us
suppose that the mouse or the snake weighed half an ounce and that each
thread is capable of supporting a grain and a half. The spider would
have to connect the mouse with the point from which it was to be
suspended with 150 threads, and if the little quadruped was once swung
off his feet, he would be powerless. By pulling successively on each
thread and shortening it a little, the mouse or snake might be raised to
any height within the capacity of the building or structure in which the
work was done. So that to those who have ridiculed the story we may
justly say: "There are more things in heaven and earth than are dreamed
of in _your_ philosophy."
What object the spider could have had in this work I am unable to see.
It may have been a dread of the harm which the mouse or snake might
work, or it may have been the hope that the decaying carcass would
attract flies which would furnish food for the engineer. I can vouch for
the truth of the snake story, however, and the object of this article is
to explain and render credible a very extraordinary feat of insect
engineering.
HOW THE SHADOW MAY BE MADE TO MOVE BACKWARD ON THE SUN-DIAL
In the twentieth chapter of II Kings, at the eleventh verse we read,
that "Isaiah the prophet cried unto the Lord, and he brought the shadow
ten degrees backward, by which it had gone down in the dial of Ahaz."
It is a curious fact, first pointed out by Nonez, the famous
cosmographer and mathematician of the sixteenth century, but not
generally known, that by tilting a sun-dial through the proper angle,
the shadow at certain periods of the year can be made, for a short time,
to move backwards on the dial. This was used by the French
encyclopaedists as a rationalistic explanation of the miracle which is
related at the opening of this article.
The reader who is curious in such matters will find directions for
constructing "a dial, for any latitude, on which the shadow shall
retrograde or move backwards," in Ozanam's "Recreations in Science and
Natural Philosophy," Riddle's edition, page 529. Professor Ball in his
"Mathematical Recreations," page 214, gives a very clear explanation of
the phenomenon. The subject is somewhat too technical for these pages.
HOW A WATCH MAY BE USED AS A COMPASS
Several years ago a correspondent of "Truth" (London) gave the following
simple directions for finding the points of the compass by means of the
ordinary pocket watch: "Point the hour hand to the sun, and south is
exactly half way between the hour hand and twelve on the watch, counting
forward up to noon, but backward after the sun has passed the meridian."
Professor Ball, in his "Mathematical Recreations and Problems," gives
more complete directions and explanations. He says:
"The position of the sun relative to the points of the compass
determines the solar time. Conversely, if we take the time given by
a watch as being the solar time (and it will differ from it only by
a few minutes at the most), and we observe the position of the sun,
we can find the points of the compass. To do this it is sufficient
to point the hour-hand to the sun and then the direction which
bisects the angle between the hour and the figure XII will point due
south. For instance, if it is four o'clock in the afternoon, it is
sufficient to point the hour-hand (which is then at the figure IIII)
to the sun, and the figure II on the watch will indicate the
direction of south. Again, if it is eight o'clock in the morning, we
must point the hour-hand (which is then at the figure VIII) to the
sun, and the figure X on the watch gives the south point of the
compass.
"Between the hours of six in the morning and six in the evening the
angle between the hour and XII, which must be bisected is less than
180 degrees, but at other times the angle to be bisected is greater
than 180 degrees; or perhaps it is simpler to say that at other
times the rule gives the north point and not the south point.
"The reason is as follows: At noon the sun is due south, and it
makes one complete circuit round the points of the compass in 24
hours. The hour-hand of a watch also makes one complete circuit in
12 hours. Hence, if the watch is held with its face in the plane of
the ecliptic, and the figure XII on the dial is pointed to the
south, both the hour-hand and the sun will be in that direction at
noon. Both move round in the same direction, but the angular
velocity of the hour-hand is twice as great as that of the sun.
Hence the rule. The greatest error due to the neglect of the
equation of time is less than 2 degrees. Of course, in practice,
most people would hold the face of the watch horizontal, and in our
latitude (that of London) no serious error would thus be introduced.
"In the southern hemisphere, or in any tropical country where at
noon the sun is due north, the rule will give the north point
instead of the south."
MICROGRAPHY OR MINUTE WRITING AND MICROPHOTOGRAPHY
Minute works of art have always excited the curiosity and commanded the
admiration of the average man. Consequently Cicero thought it worth
while to record that the entire Iliad of Homer had been written upon
parchment in characters so fine that the copy could be enclosed in a
nutshell. This has always been regarded as a marvelous feat.
There is in the French Cabinet of Medals a seal, said to have belonged
to Michael Angelo, the fabrication of which must date from a very remote
epoch, and upon which fifteen figures have been engraved in a circular
space of fourteen millimeters (.55 inch) in diameter. These figures
cannot be distinguished by the naked eye.
The Ten Commandments have been engraved in characters so fine that they
could be stamped upon one side of a nickle five-cent piece, and on
several occasions the Lord's Prayer has been engraved on one side of a
gold dollar, the diameter of which is six-tenths of an inch. I have also
seen it written with a pen within a circle which measured four-tenths of
an inch in diameter.
In the Harleian manuscript, 530, there is an account of a "rare piece of
work, brought to pass by Peter Bales, an Englishman, and a clerk of the
chancery." D'Israeli tells us that it was "The whole Bible in an English
walnut, no bigger than a hen's egg. The nut holdeth the book: there are
as many leaves in his little book as in the great Bible, and he hath
written as much in one of his little leaves as a great leaf of the
Bible."
By most people, such achievements are considered marvels of skill, and
the newspaper accounts of them which are published always attract
special attention. And it must be acknowledged that such work requires
good eyes, steady nerves, and very delicate control of the muscles. But
with ordinary writing materials there are certain mechanical limitations
which must prevent even the most skilful from going very far in this
direction. These limitations are imposed by the fiber or grain of the
paper and the construction of the ordinary pen, neither of which can be
carried beyond a certain very moderate degree of fineness. Of course,
the paper that is chosen will be selected on account of its hard,
even-grained surface, and the pen will be chosen on account of the
quality of its material and its shape, and the point is always carefully
dressed on a whetstone so as to have both halves of the nib equal in
strength and length, and the ends smooth and delicate. When due
preparation has been made, and when the eyes and nerves of the writer
are in good condition, the smallness of the distinctly readable letters
that may be produced is wonderful. And in this connection it is an
interesting fact that in many mechanical operations, writing included,
the hand is far more delicate than the eye. That which the unaided eye
can see to write, the unaided eye can see to read, but the hand, without
the assistance or guidance of the eye, can produce writing so minute
that the best eyes cannot see to read it, and yet, when viewed under a
microscope, it is found to compare favorably with the best writing of
ordinary size. And those who are conversant with the more delicate
operations of practical mechanics, know that this is no exceptional
case. The only aid given by the eye in the case of such minute writing
is the arrangement of the lines, otherwise the writing could be done as
well with the eyes shut as open.
Since the mechanical limitations which we have noted prevent us from
going very far with the instruments and materials mentioned, the next
step is to adopt a finer surface and a sharper point. These conditions
may be found in the fine glazed cards and the metal pencils or styles
used by card writers. In these cards the surface is nearly homogeneous,
that is to say, free from fibers, and the point of the metal pencil may
be made as sharp as a needle, but to utilize these conditions to the
fullest extent, it is necessary to aid the eye, and a magnifier is,
therefore, brought into use. Under a powerful glass the hand may be so
guided by the eye that the writing produced cannot be read by the
unaided vision.
The specimens of fine writing thus far described have been produced
directly by the hand under the guidance either of a magnifier or the
simple sense of motion. Just how far it would be possible to go by these
means has never been determined, so far as I know, but those who have
examined the specimens of selected diatoms and insect scales in which
objects that are utterly invisible to the naked eye are arranged with
great accuracy so as to form the most beautiful figures, can readily
believe that a combination of microscopical dexterity and skill in
penmanship might easily go far beyond anything that has yet been
accomplished in this direction, either in ancient or modern times.
But by means of a very simple mechanical arrangement, the motion of the
hand in every direction may be accurately reduced or enlarged to almost
any extent, and it thus becomes possible to form letters which are
inconceivably small. The instrument by which this is accomplished is
known as a pantagraph, and it has, within a few years, become quite
popular as a means of reducing or enlarging pictures of various kinds,
including crayon reproductions of photographs. Its construction and use
are, therefore, very generally understood. It was by means of a very
finely-made instrument embodying the principles of the pantagraph that
the extraordinarily fine work which we are about to describe was
accomplished.
It is obvious, however, that in order to produce very fine writing we
must use a very fine pen or point and the finer the point the sooner
does it wear out, so that in a very short time the lines which go to
form the letters become thick and blurred and the work is rendered
illegible. As a consequence of this, when the finest specimens of
writing are required, it is necessary to abandon the use of ordinary
points and surfaces and to resort to the use of the diamond for a pen,
and glass for a surface upon which to write. One of the earliest
attempts in this direction was that of M. Froment, of Paris, who
engraved on glass, within a circle, the one-thirtieth of an inch in
diameter, the Coat of Arms of England--lion, unicorn, and crown--with
the following inscription, partly in Roman letters, partly in script:
"_Honi soit qui mal y pense_, Her Most Gracious Majesty, Queen Victoria,
and His Royal Highness, Prince Albert, _Dieu et mon droit_. Written on
occasion of the Great Exhibition, by Froment, a Paris, 1851."