Popular Science Monthly/Volume 52/March 1898/In a World Half as Large
| IN A WORLD HALF AS LARGE. |
By the Late M. J. DELBŒUF.
LAPLACE says, in his Exposition du Système du Monde: "The law of attraction inversely as the square of the distance is that of emanations starting from a center. It appears to be the law of all forces the action of which is perceptible at sensible distances, as we recognize in electrical and magnetic forces. This law, therefore, responding exactly to all phenomena, should be regarded in its simplicity and its generality as rigorous. One of its remarkable properties is, that if the dimensions of all the bodies in the universe, their mutual distances, and their velocities should be increased or diminished proportionately, they would describe curves like those they now describe; so that the universe, thus successively reduced to the smallest imaginable space, would always present to observers the same appearance. These appearances are, consequently, independent of the dimensions of the universe, since, by virtue of the law of proportionality of force and velocity, they are independent of the movement it may have in space. The simplicity of the laws of Nature thus permits us to observe and recognize only these relations."
This masterly page contains propositions of different natures. Some of them are of an exclusively mechanical order; as those which teach that attractive forces, emanating from a center, act in inverse proportion to the squares of the distances; that this appears to be the law of all forces acting at sensible distances; that its simplicity and generality should cause it to be regarded as rigorous; and that the consequence flows from it that we may conceive an infinity of universes mechanically alike—that is, built upon all imaginable scales. These propositions, even if they had not the support of Newton, would acquire an incontestable authority from the single fact that Laplace advanced them.
Other of these propositions lie in the domain of psychology and metaphysics. Such are those that assert that these infinitely numerous universes, built on different scales, enlarged or diminished, would always present the same appearances to observers; and that, consequently, these appearances are independent of the dimensions of the universe, because the simplicity of the laws of Nature permits us to observe and recognize only relations. From all this we are authorized to infer as a final consequence which Laplace does not deduce explicitly, but which was assuredly in his thought, that the universe has fundamentally no fixed, immutable, absolute dimensions; that it is, in short, a purely geometrical universe, constructed in a homogeneous space, of which the proportions have the same properties, whatever their extent. I propose to demonstrate the fallacy of these consequences.
For this purpose I reduce this proposition to its simplest dimension, and speak, in our planetary system, only of the sun and our globe. If this system and all it contains were reduced to one half the present linear dimensions, if the velocity of the earth in its orbit were one half less, the densities of the sun and the earth remaining the same in homologous points, there would be, according to the theorem of Laplace, no other change than of dimensions, and an observer belonging to the system would not perceive any; only an observer placed outside of the system and having a standard of comparison being competent to notice it.
Or the problem may be presented in another way. We might keep the two systems, the original and the reduced one, inclosing them, in thought, one within the other, with the centers of their two suns coinciding. If the two planets were in corresponding parts of their orbits at the same time, an observer at the common center would see only the smaller one, because it would always conceal the larger.
To make the matter plainer, let us call the fictitious planet Mars. In fact, what we are going to say will nearly apply to the real Mars, whose radius is 0.517 that of the earth, and its density 0.95 that of the earth. We remark, also, that Mars receives only half as much heat as the earth. Our imaginary Mars shall be an exact image of the earth; with the same seas and continents, the same flora and fauna, the same peoples, the same cities, and the same monuments; and a person who might be transported in his sleep would be carried from one to the other, provided his own size was correspondingly diminished, without perceiving that he had changed his abode, so long as he confined his attention to the phenomena of space. If we suppose the year to consist of three hundred and sixty-five days of the same length as our days, which we may legitimately do, there would be no change in the relations of time. Generally, there will be no change in the senses of touch and sight, so far as they relate to surfaces.
Supposing our imaginary Martians to have invented a system of weights and measures resting on a like basis with the French metric system, their measures of length will be of half the value of ours; of surface, one fourth; of capacity, one eighth; and their weights, into the valuation of which other elements will enter, one sixteenth.
Hence, if we suppose the mean weight of an earthly man to be eighty kilogrammes, that of the Martian would be only five kilogrammes.
The difference in the relations of the measure of capacity and the weight deduced from it, according to the rules of the metric system, which is brought out in the above, arises from the fact that capacity is a thing of three dimensions, while weight has four. Mass, being for the same density proportional to the volume, has, in kind, only three dimensions; but weight has four, because it is mass multiplied by a new factor, gravity, which is not the same on our Mars as on the earth. Some extremely curious consequences have their origin in this fact.
Megamicros, as we shall call our man of the earth transported in his sleep to the new Mars, wakes up, opens his eyes, and finds himself in bed in his room. All the things in it are familiar to him—the furniture, clothes, books, and wares are just where he had left them overnight. He does not suspect the trick that has been played on him. He stretches himself, throws up his arms, leaps from his bed, goes to the washstand, lifts the pitcher, puts on his clothes—and is greatly surprised.
All these actions are of a common character, and consist in raising masses to a certain height. His water pitcher, for instance, holds two litres, new measure. On the earth these two litres, representing two kilogrammes, require a certain effort to be raised, say, to the height of thirty centimetres. But on Mars these two litres weigh only two Martian kilogrammes, or sixteen times less in earthly weight. Further, he does not have to lift them to a height of thirty centimetres, but of only fifteen centimetres, his size being diminished one half; so that the work to be performed is reduced to one thirty-second. On the other hand, his strength, which is proportioned to the volume or the mass of his muscles, is only reduced to one eighth. Consequently, the effort he is required to make is four times less. His water pitcher seems extremely light; so do his clothes. He probably remarked the same thing when he threw up his arms and jumped from his bed, but simply thought he was in unusually good spirits.
If he is in the habit of practicing in gymnastics, and if, on the earth, he raised weights of fifty kilogrammes above his head, he is no little astonished to see that he can now play with weights four times as heavy, or of two hundred kilogrammes.
He prepares to go out. He walks across the length and breadth of his room. There is nothing unusual. The room is smaller, indeed, but his steps are correspondingly shorter. He goes downstairs. He feels again a wonderful lightness and spring. He hardly has to touch the steps. When he goes upstairs his astonishment increases, if that is possible, for he can go up four steps at a time. His muscular energy, it is true, is reduced in proportion to his volume, or to one eighth, but his weight is reduced to a sixteenth, and he only has to lift it half as high. If, feeling so spry, he thinks he will play a little, he is again surprised to find that while he could formerly leap only to the height of his hips, he can now jump twice as high as his head. If the Eiffel Tower is near by, and he climbs it, he gets to the top four times as quickly as formerly. If he lives in Savoy and climbs Mont Blanc, he feels only one fourth the fatigue of the olden time, and will be very apt to think that somebody has given him a very full dose of some invigorating extract while he was asleep. He is no less astonished when he finds how little danger there is in falling. His child falls down a whole story without being hurt, and he drops fragile things, his water pitcher, for example, without their breaking.
In the same way, but inversely, the inhabitant of Mars, transplanted to the earth, would feel four times as heavy: his leaps would be only a quarter as high; the steps, going up and down stairs, would be four times too high, although they would look the same as ever. Any climb, not to speak of the Eiffel Tower or of Mont Blanc, would take away his breath; and he would be ready to think he had all at once become decrepit.
Fairy stories, and such humorists as Swift and Voltaire, have made us familiar with the idea that there may be cities of dwarfs and of giants copied exactly from ours; and we do not perceive, at first sight, why they should not have their Paris, with the Louvre, boulevards, and hotels, built after earthly models. We could easily fancy that if Gulliver, arriving at Lilliput or Brobdingnag, had become smaller or larger, according to the measure of his hosts, he would not have remarked the diminutiveness of Lilliput and the Lilliputians, or the great size of Brobdingnag, and the Brobdingnagians. This fancy is the more natural at first sight because we have invented the art of drawing and other arts relating to it, and the microscope and photography show us every day considerable enlargements and diminutions without alteration of shapes.
It is, however, not consistent with the most incontestable results of science. The cat is not an exact reduction of the tiger, or the Lilliputian of the Brobdingnagian, any more than a small crystal of alum is a reduction of a large crystal—although one regular octahedron may be the exact image of another octahedron. For if they were, there would no longer be a question of atoms, or molecules, or cells. From the geometrical point of view the cell, the molecule, and the atom are infinitely divisible universes, and therefore capable of containing all imaginable figures within their limits; while from the chemical and physiological point of view they are absolute quantities not capable of reduction in their kind.
Now let us see if a Martian house can be constructed wholly on the plan of an earthly house—that is, if it can be made to present the same proportions in all its parts. The cube of the construction and the cubic dimensions of the rooms, as well as the number of windows and their superficial proportions, might indeed be the same; but we will consider here the details of construction in view of the materials used. Let us reduce the problem to its simplest expression: a beam on two supports in a condition to bear the weight of a man. Let P be this weight; l, b, and h, the length of the beam between the points of support, its breadth, and thickness; and R, the resistance of the wood. According to a well-known formula, we have: which means that the weight that can be supported increases directly in proportion to the resistance of the wood, the breadth of the beam, and the square of its thickness, and inversely as the distance between the supports.
Since on Mars, according to the data of the problem, R suffers no change, while b, h, and l become , , and , we see that the weight which the geometrically reduced apparatus can support will be equal to . We have just seen that the weight of a Martian is . Consequently, the apparatus will be four times as solid as necessary. The Martians could use joists and planks proportionately only half as thick or one quarter as broad, or with supports four times as far apart, or any other combination that would reduce the second member of the equation to one sixteenth.
From what we have said of the lightness of the Martians, it would appear that their division walls and fences would have to be relatively four times, absolutely twice as high as with us.
Now, suppose Megamicros preparing to continue on Mars some work he had begun on the earth. He has a bench, planks, nails, and a hammer. His hammer is of seven eighths less volume and mass, and its weight is reduced to one sixteenth. Himself smaller in size, he can no longer lift the instrument to the same height, so that on a final analysis the living force of the hammer reduced in weight is only one thirty-second. The nail is only half as long, and of only one fourth section; so that, supposing the same rigidity, he meets only one eighth of the resistance in driving it into the board. Megamicros then finds that his hammer is four times too light, and can not understand what makes it so.
If the real Martians have passed, like us, through a stone age and come to an iron age, they have had to work with implements relatively four times larger than ours, and are now using hammers of corresponding dimensions.
To return to our imaginary Martians. It may be objected that the mass of the hammer being eight times less, and the muscular force being no more than eight times less, the velocity impressed upon the hammer will not be half, but equal. The objection is well founded; but then the phenomenon of the fall of the hammer will not be the same as it would be on the earth. An observer situated in the common center of the two suns would see the Martian's hammer drop twice as fast as the terrestrian's. Or the objection might take another form. The velocity of the hammer may remain proportional, but the muscular exhaustion will be four times less, so that the workman can quadruple the number of his blows. In this case the temporal phenomenon, if I may call it that, will be changed. At any rate, if he does not multiply his blows, the nail will not be driven in in the same way. To whichever side we turn, we fall into the same definitive conclusion.
We further remark that the Martians can lift loads four times as heavy as ours: first, because they do not have to carry them so high; and, secondly, because the weight is only half as much. Thus the unfortunate people who built the pyramids on Mars would require only a quarter of the time. Consequently, Megamicros will see all tasks that consist in raising weights performed four times as rapidly. If he builds a house, it will be under roof before he could have got it aboveground on the earth. Life thus passes more rapidly on Mars than on the earth; and yet we can not think of diminishing the length of the days, for then we would increase the number in the year to fourteen hundred and sixty; for we have supposed the new Martian year to be of the same length as that of the earth.
Megamicros, who has learned on the earth to reckon by terrestrial measures, will have a new set to deal with when he is transported to the new Mars. Some may say that he will experience no difficulty in this. They are mistaken.
To speak first of measures of length and surface: If Megamicros required six square metres of cloth to make himself a complete suit of clothes when he was on the earth, he will need no more on Mars, because the surface of his body which he has to cover is diminished in the same proportion as that of the square metre, or as four to one. But as the sun sends him no more heat there than our sun does to us, the goods he will select must be at least as warm as on the earth. No reduction, therefore, is possible in their thickness. Hence, if he himself makes the goods he requires, if his wife knits his stockings and his vests, they will be surprised at the amount of labor they will have to expend, and the quantity of materials they will have to use for the same purpose. With a skein of a hundred metres of yarn, working with a double thread, they will not make more than half of what they expect to. It will be as if their metre, already shortened one half, were reduced another half. Hence the acre they may devote to the cultivation of flax or hemp or cotton will fall far short of furnishing them as many shirts.
With respect to measures of capacity and weight: On the earth Megamicros quenched his thirst with two litres of wine. These two litres restored to him the quantity of liquid which he lost by transpiration and excretion. Without speaking of excretions, the Martian man will lose perceptibly more by evaporation alone than he did on the earth; for while his mass is reduced to one eighth, his surface is only reduced to one fourth. He will, therefore, lose twice as much by transpiration as he did before, and a litre of wine will not seem to contain more than half the same sum of satisfaction. For a like reason, a kilogramme of bread will not appease hunger in the same measure as on the earth. For food, besides furnishing energy to the muscles, serves, by repairing the loss of caloric, to maintain the animal heat. The cooling surface of the body is twice as great in proportion to the mass; the kilogramme of bread will, therefore, not procure the same sum of muscular energy. We know, as a fact, that small animals have to eat and drink relatively more than large animals.
Megamicros will feel a change of temperature on Mars more than when he was on the earth. He gets cool and is warmed again in less time, when all other things are equal. If a cloud passes over the sun, he will immediately feel a depression of the temperature of his skin. It is a very sensitive thermometer. Two thermometers, geometrically alike, do not act in the same manner. There is no synchronism in their movements. All such disagreements arise from the fact that surfaces do not diminish in the same proportion as volumes.
The problem becomes more and more complicated as we address ourselves to more delicate phenomena. Muscular energy is due to the burning by oxygen of the carbon contained in the blood. This combustion is effected on the surface of the lungs. The quantity of blood of a Martian is eight times less than that of a being of the earth. But while the thoracic cage is diminished in the proportion of eight to one, the pulmonary surface is so only in the proportion of four to one. The combustion is therefore more complete with the Martian than with the man of the earth. Consequently his muscular energy, the effects of which were already so striking in consequence of the reduction of weight, will be still more marked by virtue of this circumstance. On the other hand, combustion being more active, the kilogramme of bread, which we have already found not enough, becomes still more insufficient—a new peturbation. Another consequence: we have just said that the cube of the houses might be, on Mars, proportional to the cube of our houses. But if boards of health should proceed there according to the same principles as with us, they would have to order larger and higher rooms. It may be suggested that the atmosphere of Mars is less dense than ours, because it is not so thick, and gravity is less there. Here is, indeed, a great difficulty, which, fortunately, it is not necessary to resolve, because the law of Laplace supposes that the density is the same at homologous points. On the surface of Mars the air, then, has the same density and the same composition as on the surface of the earth. But will the inspirations be as long? The diaphragm of the Martians has not the same work to do as that of the earth people. It is true that it is only half as thick, but its surface is only a quarter, and the amplitude of its movement only one half. But we must not discuss the rhythm of the diaphragm or the beatings of the heart unless we shorten the days and increase the number in a year.
We see that whichever way we turn we can not for an instant entertain the illusion that an earth-man could be put on Mars without knowing it.
We shall fall into more and more inextricable difficulties if we go further into the detail of the respiratory and circulatory phenomena. The capillary vessels of the Martians are four times narrower in section. The heart, then, would have to use more force to make the blood circulate; yet the heart is much weaker, with thinner walls, smaller cavities, etc. Even if the Martians resemble men externally, their whole interior organization must be decidedly very different.
The persistency will be remarked with which the number four recurs in these calculations. This is because weight is reduced one half on Mars, and also, because of the reduction in their linear dimensions, the Martians have to exercise only half the effort to produce the same apparent effect. It might be concluded from this that if the universe was reduced geometrically in the proportion of three or of five to one, we should immediately find ourselves nine or twenty-five times lighter, stronger, and more active. By an inverse conclusion, if we could imagine the case for a few moments, we should have to be three or five times larger. Hence the paradoxical conclusion that the smaller the world is, the larger its people should be; so that, if the smallest asteroids are inhabited by men, the inhabitants would be more important in size than their planet, and might, in case of extremity, take it in their arms. On the other hand, the larger the planetary mass, the smaller the men should be, if their muscular sensations are comparable to ours. If we have brothers living on Jupiter, they must be about the size of ants.
If, indeed, we go down to the bottom of the matter, we shall be forced to admit that there is, in a being capable of observing and discerning, something permanent, fixed, and superior to spatial dimensions, and independent of their variations. This something is the feeling of the required effort, of the executed movement, and the fatigue that follows it. The feeling appertains essentially to every muscle that works under the impulsion of the will. This feeling is neither line, nor surface, nor volume, nor weight. It is the same for small and large animals, for the child and the man, the dwarf and the giant. He who lifts the heaviest weight he can manage fatigues himself always just so much, whether he be weak or strong, fresh and nimble or tired and dull, sturdy or feeble. The ant dragging a straw experiences integrally the same feeling as the porter carrying a sack, or the horse pulling a wagon, if the straw, the sack, and the wagon are proportioned to the strength of the ant, the man, and the horse.
Magamicros, transported to Mars, has retained in himself the inner unity of measure that permits him to estimate masses according to his strength. When on the earth he lifts loads of fifty kilogrammes greatest weight, the equivalent of this weight on Mars is not for him fifty Martian kilogrammes, but two hundred kilogrammes greatest weight. Hence his astonishment. I say, purposely, sometimes weight and sometimes mass. Fundamentally, he estimates weight; but weight represents to him a certain quantity of matter as related to his strength and his requirements. No matter if the kilogramme of Mars is only equivalent to one sixteenth of the kilogramme of the earth. If a Martian kilogramme of meat, bread, or beans represents on Mars the same relative amount of work or pleasure as the earthly kilogramme of meat, bread, or beans does on the earth, it is equivalent, even equal to it; but if Megamicros does not find in it the same quality of restoring strength or producing pleasure, he will think it is different.
Pleasure, pain, and fatigue are not measured by the metre, litre, or kilogramme. The pleasure felt by a grasshopper in eating a blade of grass is not less than that of a cow with the range of a whole pasture. But we know that the grasshopper has a more pressing need of food than the cow, because it is smaller. If, then, by an ideal reduction of proportions, the cow fails to get the same satisfaction out of the meadow, she will not judge its size by sight, but by her stomach.
Let us go still further. None of my readers has thought of raising the objection that while transporting Megamicros to Mars I have diminished him in other respects but not in intelligence, and that I should have given him only a half, a quarter, or an eighth of judgment. There would be no reason in this. I have received from Biskra a uromastix—a kind of herbivorous lizard, with its tail armed with points. Not having any Algerian plants to feed it, I put it in a field where there were all kinds of wild flowers. The animal found the flowers of the smartweed to its taste. Wishing to vary its diet, and particularly to find something it would eat in winter, I tried to feed it other things; but, though it was docile and ate smartweed, fumitory flowers, and wood violets from the hand, it showed a marked aversion or indifference to clover. One day, in my impatience, not having found any smartweed, I opened its mouth and forced in a clover blossom which it finally swallowed. The next day, to my astonishment, having some clover blossoms in my hand, the animal seized them and devoured them with evident greedy pleasure. It recognized the plant it had been forced to swallow and had found good, though it had despised it before. It had got rid of a prejudice. Would a rhinoceros have acted more rationally? Who would have thought of its large size giving it more intelligence?
A very important conclusion results from our discussion. Laplace's law is true mechanically, within the strict limits in which it is announced. But the psychical consequences Laplace draws from it are fallacies, and the simplest phenomena of elasticity make the fallacy evident. Yet if the law of universal attraction were all we had by which to account for every kind of manifestations, psychical as well as physical, or, in other words, if there were nothing in the universe but material atoms situated at perceptible distances apart, and attracting one another in proportion to their masses and inversely as the squares of their distances, Laplace's conclusion would be impregnable; an observer could not perceive any diminution or augmentation in the universe. But why? Because there would be no longer an observer. As I have demonstrated, the moment there is an observer, he will perceive a change; and if he perceives it, it is undoubtedly because the faculty of observation escapes—with others—the law of universal attraction; because it does not depend solely upon the mass of the atoms and their distance. It is the same with the ant and with the elephant.
A final conclusion is that if all these deductions are exact, real space is different from geometrical space, and the dimensions of the universe are absolute.—Translated for the Popular Science Monthly from Ciel et Terre.
