it according to the laws of perfectly elastic bodies, or according to
those of bodies devoid of elasticity, or according to an intermediate
law? The corpuscles of Lesage can not act as perfectly elastic bodies;
otherwise the effect would be null, since the corpuscles intercepted by
the body _B_ would be replaced by others which would have rebounded from
_B_, and calculation proves that the compensation would be perfect. It
is necessary then that the impact make the corpuscles lose energy, and
this energy should appear under the form of heat. But how much heat
would thus be produced? Note that attraction passes through bodies; it
is necessary therefore to represent to ourselves the earth, for example,
not as a solid screen, but as formed of a very great number of very
small spherical molecules, which play individually the rôle of little
screens, but between which the corpuscles of Lesage may freely
circulate. So, not only the earth is not a solid screen, but it is not
even a cullender, since the voids occupy much more space than the
plenums. To realize this, recall that Laplace has demonstrated that
attraction, in traversing the earth, is weakened at most by one
ten-millionth part, and his proof is perfectly satisfactory: in fact, if
attraction were absorbed by the body it traverses, it would no longer be
proportional to the masses; it would be _relatively_ weaker for great
bodies than for small, since it would have a greater thickness to
traverse. The attraction of the sun for the earth would therefore be
_relatively_ weaker than that of the sun for the moon, and thence would
result, in the motion of the moon, a very sensible inequality. We should
therefore conclude, if we adopt the theory of Lesage, that the total
surface of the spherical molecules which compose the earth is at most
the ten-millionth part of the total surface of the earth.
Darwin has proved that the theory of Lesage only leads exactly to
Newton's law when we postulate particles entirely devoid of elasticity.
The attraction exerted by the earth on a mass 1 at a distance 1 will
then be proportional, at the same time, to the total surface _S_ of the
spherical molecules composing it, to the velocity _v_ of the corpuscles,
to the square root of the density [rho] of the medium formed by the
corpuscles. The heat produced will be proportional to _S_, to the
density [rho], and to the cube of the velocity _v_.
But it is necessary to take account of the resistance experienced by a
body moving in such a medium; it can not move, in fact, without going
against certain impacts, in fleeing, on the contrary, before those
coming in the opposite direction, so that the compensation realized in
the state of rest can no longer subsist. The calculated resistance is
proportional to _S_, to [rho] and to _v_; now, we know that the heavenly
bodies move as if they experienced no resistance, and the precision of
observations permits us to fix a limit to the resistance of the medium.
This resistance varying as _S_[rho]_v_, while the attraction varies as
_S_{sqrt([rho]_v_)}, we see that the ratio of the resistance to the
square of the attraction is inversely as the product _Sv_.
We have therefore a lower limit of the product _Sv_. We have already an
upper limit of _S_ (by the absorption of attraction by the body it
traverses); we have therefore a lower limit of the velocity _v_, which
must be at least 24·10^{17} times that of light.
From this we are able to deduce [rho] and the quantity of heat produced;
this quantity would suffice to raise the temperature 10^{26} degrees a
second; the earth would receive in a given time 10^{20} times more heat
than the sun emits in the same time; I am not speaking of the heat the
sun sends to the earth, but of that it radiates in all directions.
It is evident the earth could not long stand such a régime.
We should not be led to results less fantastic if, contrary to Darwin's
views, we endowed the corpuscles of Lesage with an elasticity imperfect
without being null. In truth, the vis viva of these corpuscles would not
be entirely converted into heat, but the attraction produced would
likewise be less, so that it would be only the part of this vis viva
converted into heat, which would contribute to produce the attraction
and that would come to the same thing; a judicious employment of the
theorem of the viriel would enable us to account for this.
The theory of Lesage may be transformed; suppress the corpuscles and
imagine the ether overrun in all senses by luminous waves coming from
all points of space. When a material object receives a luminous wave,
this wave exercises upon it a mechanical action due to the
Maxwell-Bartholi pressure, just as if it had received the impact of a
material projectile. The waves in question could therefore play the rôle
of the corpuscles of Lesage. This is what is supposed, for example, by
M. Tommasina.
The difficulties are not removed for all that; the velocity of
propagation can be only that of light, and we are thus led, for the
resistance of the medium, to an inadmissible figure. Besides, if the
light is all reflected, the effect is null, just as in the hypothesis of
the perfectly elastic corpuscles.
That there should be attraction, it is necessary that the light be
partially absorbed; but then there is production of heat. The
calculations do not differ essentially from those made in the ordinary
theory of Lesage, and the result retains the same fantastic character.
On the other hand, attraction is not absorbed by the body it traverses,
or hardly at all; it is not so with the light we know. Light which would
produce the Newtonian attraction would have to be considerably different
from ordinary light and be, for example, of very short wave length. This
does not count that, if our eyes were sensible of this light, the whole
heavens should appear to us much more brilliant than the sun, so that
the sun would seem to us to stand out in black, otherwise the sun would
repel us instead of attracting us. For all these reasons, light which
would permit of the explanation of attraction would be much more like
Röntgen rays than like ordinary light.
And besides, the X-rays would not suffice; however penetrating they may
seem to us, they could not pass through the whole earth; it would be
necessary therefore to imagine X'-rays much more penetrating than the
ordinary X-rays. Moreover a part of the energy of these X'-rays would
have to be destroyed, otherwise there would be no attraction. If you do
not wish it transformed into heat, which would lead to an enormous heat
production, you must suppose it radiated in every direction under the
form of secondary rays, which might be called X'' and which would have
to be much more penetrating still than the X'-rays, otherwise they would
in their turn derange the phenomena of attraction.
Such are the complicated hypotheses to which we are led when we try to
give life to the theory of Lesage.
But all we have said presupposes the ordinary laws of mechanics.
Will things go better if we admit the new dynamics? And first, can we
conserve the principles of relativity? Let us give at first to the
theory of Lesage its primitive form, and suppose space ploughed by
material corpuscles; if these corpuscles were perfectly elastic, the
laws of their impact would conform to this principle of relativity, but
we know that then their effect would be null. We must therefore suppose
these corpuscles are not elastic, and then it is difficult to imagine a
law of impact compatible with the principle of relativity. Besides, we
should still find a production of considerable heat, and yet a very
sensible resistance of the medium.
If we suppress these corpuscles and revert to the hypothesis of the
Maxwell-Bartholi pressure, the difficulties will not be less. This is
what Lorentz himself has attempted in his Memoir to the Amsterdam
Academy of Sciences of April 25, 1900.
Consider a system of electrons immersed in an ether permeated in every
sense by luminous waves; one of these electrons, struck by one of these
waves, begins to vibrate; its vibration will be synchronous with that of
light; but it may have a difference of phase, if the electron absorbs a
part of the incident energy. In fact, if it absorbs energy, this is
because the vibration of the ether _impels_ the electron; the electron
must therefore be slower than the ether. An electron in motion is
analogous to a convection current; therefore every magnetic field, in
particular that due to the luminous perturbation itself, must exert a
mechanical action upon this electron. This action is very slight;
moreover, it changes sign in the current of the period; nevertheless,
the mean action is not null if there is a difference of phase between
the vibrations of the electron and those of the ether. The mean action
is proportional to this difference, consequently to the energy absorbed
by the electron. I can not here enter into the detail of the
calculations; suffice it to say only that the final result is an
attraction of any two electrons, varying inversely as the square of the
distance and proportional to the energy absorbed by the two electrons.
Therefore there can not be attraction without absorption of light and,
consequently, without production of heat, and this it is which
determined Lorentz to abandon this theory, which, at bottom, does not
differ from that of Lesage-Maxwell-Bartholi. He would have been much
more dismayed still if he had pushed the calculation to the end. He
would have found that the temperature of the earth would have to
increase 10^{12} degrees a second.
IV
_Conclusions_
I have striven to give in few words an idea as complete as possible of
these new doctrines; I have sought to explain how they took birth;
otherwise the reader would have had ground to be frightened by their
boldness. The new theories are not yet demonstrated; far from it; only
they rest upon an aggregate of probabilities sufficiently weighty for us
not to have the right to treat them with disregard.
New experiments will doubtless teach us what we should finally think of
them. The knotty point of the question lies in Kaufmann's experiment and
those that may be undertaken to verify it.
In conclusion, permit me a word of warning. Suppose that, after some
years, these theories undergo new tests and triumph; then our secondary
education will incur a great danger; certain professors will doubtless
wish to make a place for the new theories.
Novelties are so attractive, and it is so hard not to seem highly
advanced! At least there will be the wish to open vistas to the pupils
and, before teaching them the ordinary mechanics, to let them know it
has had its day and was at best good enough for that old dolt Laplace.
And then they will not form the habit of the ordinary mechanics.
Is it well to let them know this is only approximative? Yes; but later,
when it has penetrated to their very marrow, when they shall have taken
the bent of thinking only through it, when there shall no longer be risk
of their unlearning it, then one may, without inconvenience, show them
its limits.
It is with the ordinary mechanics that they must live; this alone will
they ever have to apply. Whatever be the progress of automobilism, our
vehicles will never attain speeds where it is not true. The other is
only a luxury, and we should think of the luxury only when there is no
longer any risk of harming the necessary.
BOOK IV
ASTRONOMIC SCIENCE
CHAPTER I
THE MILKY WAY AND THE THEORY OF GASES
The considerations to be here developed have scarcely as yet drawn the
attention of astronomers; there is hardly anything to cite except an
ingenious idea of Lord Kelvin's, which has opened a new field of
research, but still waits to be followed out. Nor have I original
results to impart, and all I can do is to give an idea of the problems
presented, but which no one hitherto has undertaken to solve. Every one
knows how a large number of modern physicists represent the constitution
of gases; gases are formed of an innumerable multitude of molecules
which, at high speeds, cross and crisscross in every direction. These
molecules probably act at a distance one upon another, but this action
decreases very rapidly with distance, so that their trajectories remain
sensibly straight; they cease to be so only when two molecules happen to
pass very near to each other; in this case, their mutual attraction or
repulsion makes them deviate to right or left. This is what is sometimes
called an impact; but the word _impact_ is not to be understood in its
usual sense; it is not necessary that the two molecules come into
contact, it suffices that they approach sufficiently near each other for
their mutual attractions to become sensible. The laws of the deviation
they undergo are the same as for a veritable impact.
It seems at first that the disorderly impacts of this innumerable dust
can engender only an inextricable chaos before which analysis must
recoil. But the law of great numbers, that supreme law of chance, comes
to our aid; in presence of a semi-disorder, we must despair, but in
extreme disorder, this statistical law reestablishes a sort of mean
order where the mind can recover. It is the study of this mean order
which constitutes the kinetic theory of gases; it shows us that the
velocities of the molecules are equally distributed among all the
directions, that the rapidity of these velocities varies from one
molecule to another, but that even this variation is subject to a law
called Maxwell's law. This law tells us how many of the molecules move
with such and such a velocity. As soon as the gas departs from this law,
the mutual impacts of the molecules, in modifying the rapidity and
direction of their velocities, tend to bring it promptly back.
Physicists have striven, not without success, to explain in this way the
experimental properties of gases; for example Mariotte's law.
Consider now the milky way; there also we see an innumerable dust; only
the grains of this dust are not atoms, they are stars; these grains move
also with high velocities; they act at a distance one upon another, but
this action is so slight at great distance that their trajectories are
straight; and yet, from time to time, two of them may approach near
enough to be deviated from their path, like a comet which has passed too
near Jupiter. In a word, to the eyes of a giant for whom our suns would
be as for us our atoms, the milky way would seem only a bubble of gas.
Such was Lord Kelvin's leading idea. What may be drawn from this
comparison? In how far is it exact? This is what we are to investigate
together; but before reaching a definite conclusion, and without wishing
to prejudge it, we foresee that the kinetic theory of gases will be for
the astronomer a model he should not follow blindly, but from which he
may advantageously draw inspiration. Up to the present, celestial
mechanics has attacked only the solar system or certain systems of
double stars. Before the assemblage presented by the milky way, or the
agglomeration of stars, or the resolvable nebulae it recoils, because it
sees therein only chaos. But the milky way is not more complicated than
a gas; the statistical methods founded upon the calculus of
probabilities applicable to a gas are also applicable to it. Before all,
it is important to grasp the resemblance of the two cases, and their
difference.
Lord Kelvin has striven to determine in this manner the dimensions of
the milky way; for that we are reduced to counting the stars visible in
our telescopes; but we are not sure that behind the stars we see, there
are not others we do not see; so that what we should measure in this way
would not be the size of the milky way, it would be the range of our
instruments.
The new theory comes to offer us other resources. In fact, we know the
motions of the stars nearest us, and we can form an idea of the rapidity
and direction of their velocities. If the ideas above set forth are
exact, these velocities should follow Maxwell's law, and their mean
value will tell us, so to speak, that which corresponds to the
temperature of our fictitious gas. But this temperature depends itself
upon the dimensions of our gas bubble. In fact, how will a gaseous mass
let loose in the void act, if its elements attract one another according
to Newton's law? It will take a spherical form; moreover, because of
gravitation, the density will be greater at the center, the pressure
also will increase from the surface to the center because of the weight
of the outer parts drawn toward the center; finally, the temperature
will increase toward the center: the temperature and the pressure being
connected by the law called adiabatic, as happens in the successive
layers of our atmosphere. At the surface itself, the pressure will be
null, and it will be the same with the absolute temperature, that is to
say with the velocity of the molecules.
A question comes here: I have spoken of the adiabatic law, but this law
is not the same for all gases, since it depends upon the ratio of their
two specific heats; for the air and like gases, this ratio is 1.42; but
is it to air that it is proper to liken the milky way? Evidently not, it
should be regarded as a mono-atomic gas, like mercury vapor, like argon,
like helium, that is to say that the ratio of the specific heats should
be taken equal to 1.66. And, in fact, one of our molecules would be for
example the solar system; but the planets are very small personages, the
sun alone counts, so that our molecule is indeed mono-atomic. And even
if we take a double star, it is probable that the action of a strange
star which might approach it would become sufficiently sensible to
deviate the motion of general translation of the system much before
being able to trouble the relative orbits of the two components; the
double star, in a word, would act like an indivisible atom.
However that may be, the pressure, and consequently the temperature, at
the center of the gaseous sphere would be by so much the greater as the
sphere was larger since the pressure increases by the weight of all the
superposed layers. We may suppose that we are nearly at the center of
the milky way, and by observing the mean proper velocity of the stars,
we shall know that which corresponds to the central temperature of our
gaseous sphere and we shall determine its radius.
We may get an idea of the result by the following considerations: make a
simpler hypothesis: the milky way is spherical, and in it the masses are
distributed in a homogeneous manner; thence results that the stars in it
describe ellipses having the same center. If we suppose the velocity
becomes nothing at the surface, we may calculate this velocity at the
center by the equation of vis viva. Thus we find that this velocity is
proportional to the radius of the sphere and to the square root of its
density. If the mass of this sphere was that of the sun and its radius
that of the terrestrial orbit, this velocity would be (it is easy to
see) that of the earth in its orbit. But in the case we have supposed,
the mass of the sun should be distributed in a sphere of radius
1,000,000 times greater, this radius being the distance of the nearest
stars; the density is therefore 10^{18} times less; now, the velocities
are of the same order, therefore it is necessary that the radius be
10^{9} times greater, be 1,000 times the distance of the nearest stars,
which would give about a thousand millions of stars in the milky way.
But you will say these hypothesis differ greatly from the reality;
first, the milky way is not spherical and we shall soon return to this
point, and again the kinetic theory of gases is not compatible with the
hypothesis of a homogeneous sphere. But in making the exact calculation
according to this theory, we should find a different result, doubtless,
but of the same order of magnitude; now in such a problem the data are
so uncertain that the order of magnitude is the sole end to be aimed at.
And here a first remark presents itself; Lord Kelvin's result, which I
have obtained again by an approximative calculation, agrees sensibly
with the evaluations the observers have made with their telescopes; so
that we must conclude we are very near to piercing through the milky
way. But that enables us to answer another question. There are the stars
we see because they shine; but may there not be dark stars circulating
in the interstellar spaces whose existence might long remain unknown?
But then, what Lord Kelvin's method would give us would be the total
number of stars, including the dark stars; as his figure is comparable
to that the telescope gives, this means there is no dark matter, or at
least not so much as of shining matter.
Before going further, we must look at the problem from another angle. Is
the milky way thus constituted truly the image of a gas properly so
called? You know Crookes has introduced the notion of a fourth state of
matter, where gases having become too rarefied are no longer true gases
and become what he calls radiant matter. Considering the slight density
of the milky way, is it the image of gaseous matter or of radiant
matter? The consideration of what is called the _free path_ will furnish
us the answer.
The trajectory of a gaseous molecule may be regarded as formed of
straight segments united by very small arcs corresponding to the
successive impacts. The length of each of these segments is what is
called the free path; of course this length is not the same for all the
segments and for all the molecules; but we may take a mean; this is what
is called the _mean path_. This is the greater the less the density of
the gas. The matter will be radiant if the mean path is greater than the
dimensions of the receptacle wherein the gas is enclosed, so that a
molecule has a chance to go across the whole receptacle without
undergoing an impact; if the contrary be the case, it is gaseous. From
this it follows that the same fluid may be radiant in a little
receptacle and gaseous in a big one; this perhaps is why, in a Crookes
tube, it is necessary to make the vacuum by so much the more complete as
the tube is larger.
How is it then for the milky way? This is a mass of gas of which the
density is very slight, but whose dimensions are very great; has a star
chances of traversing it without undergoing an impact, that is to say
without passing sufficiently near another star to be sensibly deviated
from its route! What do we mean by _sufficiently near_? That is perforce
a little arbitrary; take it as the distance from the sun to Neptune,
which would represent a deviation of a dozen degrees; suppose therefore
each of our stars surrounded by a protective sphere of this radius;
could a straight pass between these spheres? At the mean distance of the
stars of the milky way, the radius of these spheres will be seen under
an angle of about a tenth of a second; and we have a thousand millions
of stars. Put upon the celestial sphere a thousand million little
circles of a tenth of a second radius. Are the chances that these
circles will cover a great number of times the celestial sphere? Far
from it; they will cover only its sixteen thousandth part. So the milky
way is not the image of gaseous matter, but of Crookes' radiant matter.
Nevertheless, as our foregoing conclusions are happily not at all
precise, we need not sensibly modify them.
But there is another difficulty: the milky way is not spherical, and we
have reasoned hitherto as if it were, since this is the form of
equilibrium a gas isolated in space would take. To make amends,
agglomerations of stars exist whose form is globular and to which would
better apply what we have hitherto said. Herschel has already endeavored
to explain their remarkable appearances. He supposed the stars of the
aggregates uniformly distributed, so that an assemblage is a homogeneous
sphere; each star would then describe an ellipse and all these orbits
would be passed over in the same time, so that at the end of a period
the aggregate would take again its primitive configuration and this
configuration would be stable. Unluckily, the aggregates do not appear
to be homogeneous; we see a condensation at the center, we should
observe it even were the sphere homogeneous, since it is thicker at the
center; but it would not be so accentuated. We may therefore rather
compare an aggregate to a gas in adiabatic equilibrium, which takes the
spherical form because this is the figure of equilibrium of a gaseous
mass.
But, you will say, these aggregates are much smaller than the milky way,
of which they even in probability make part, and even though they be
more dense, they will rather present something analogous to radiant
matter; now, gases attain their adiabatic equilibrium only through
innumerable impacts of the molecules. That might perhaps be adjusted.
Suppose the stars of the aggregate have just enough energy for their
velocity to become null when they reach the surface; then they may
traverse the aggregate without impact, but arrived at the surface they
will go back and will traverse it anew; after a great number of
crossings, they will at last be deviated by an impact; under these
conditions, we should still have a matter which might be regarded as
gaseous; if perchance there had been in the aggregate stars whose
velocity was greater, they have long gone away out of it, they have left
it never to return. For all these reasons, it would be interesting to
examine the known aggregates, to seek to account for the law of the
densities, and to see if it is the adiabatic law of gases.
But to return to the milky way; it is not spherical and would rather be
represented as a flattened disc. It is clear then that a mass starting
without velocity from the surface will reach the center with different
velocities, according as it starts from the surface in the neighborhood
of the middle of the disc or just on the border of the disc; the
velocity would be notably greater in the latter case. Now, up to the
present, we have supposed that the proper velocities of the stars, those
we observe, must be comparable to those which like masses would attain;
this involves a certain difficulty. We have given above a value for the
dimensions of the milky way, and we have deduced it from the observed
proper velocities which are of the same order of magnitude as that of
the earth in its orbit; but which is the dimension we have thus
measured? Is it the thickness? Is it the radius of the disc? It is
doubtless something intermediate; but what can we say then of the
thickness itself, or of the radius of the disc? Data are lacking to make
the calculation; I shall confine myself to giving a glimpse of the
possibility of basing an evaluation at least approximate upon a deeper
discussion of the proper motions.
And then we find ourselves facing two hypotheses: either the stars of
the milky way are impelled by velocities for the most part parallel to
the galactic plane, but otherwise distributed uniformly in all
directions parallel to this plane. If this be so, observation of the
proper motions should show a preponderance of components parallel to the
milky way; this is to be determined, because I do not know whether a
systematic discussion has ever been made from this view-point. On the
other hand, such an equilibrium could only be provisory, since because
of impacts the molecules, I mean the stars, would in the long run
acquire notable velocities in the sense perpendicular to the milky way
and would end by swerving from its plane, so that the system would tend
toward the spherical form, the only figure of equilibrium of an isolated
gaseous mass.
Or else the whole system is impelled by a common rotation, and for that
reason is flattened like the earth, like Jupiter, like all bodies that
twirl. Only, as the flattening is considerable, the rotation must be
rapid; rapid doubtless, but it must be understood in what sense this
word is used. The density of the milky way is 10^{23} times less than
that of the sun; a velocity of rotation sqrt(10^{25}) times less than
that of the sun, for it would, therefore, be the equivalent so far as
concerns flattening; a velocity 10^{12} times slower than that of the
earth, say a thirtieth of a second of arc in a century, would be a very
rapid rotation, almost too rapid for stable equilibrium to be possible.
In this hypothesis, the observable proper motions would appear to us
uniformly distributed, and there would no longer be a preponderance of
components parallel to the galactic plane.
They will tell us nothing about the rotation itself, since we belong to
the turning system. If the spiral nebulæ are other milky ways, foreign
to ours, they are not borne along in this rotation, and we might study
their proper motions. It is true they are very far away; if a nebula has
the dimensions of the milky way and if its apparent radius is for
example 20", its distance is 10,000 times the radius of the milky way.
But that makes no difference, since it is not about the translation of
our system that we ask information from them, but about its rotation.
The fixed stars, by their apparent motion, reveal to us the diurnal
rotation of the earth, though their distance is immense. Unluckily, the
possible rotation of the milky way, however rapid it may be relatively,
is very slow viewed absolutely, and besides the pointings on nebulæ can
not be very precise; therefore thousands of years of observations would
be necessary to learn anything.
However that may be, in this second hypothesis, the figure of the milky
way would be a figure of final equilibrium.
I shall not further discuss the relative value of these two hypotheses
since there is a third which is perhaps more probable. We know that
among the irresolvable nebulæ, several kinds may be distinguished: the
irregular nebulæ like that of Orion, the planetary and annular nebulæ,
the spiral nebulæ. The spectra of the first two families have been
determined, they are discontinuous; these nebulæ are therefore not
formed of stars; besides, their distribution on the heavens seems to
depend upon the milky way; whether they have a tendency to go away from
it, or on the contrary to approach it, they make therefore a part of the
system. On the other hand, the spiral nebulæ are generally considered as
independent of the milky way; it is supposed that they, like it, are
formed of a multitude of stars, that they are, in a word, other milky
ways very far away from ours. The recent investigations of Stratonoff
tend to make us regard the milky way itself as a spiral nebula, and this
is the third hypothesis of which I wish to speak.
How can we explain the very singular appearances presented by the spiral
nebulæ, which are too regular and too constant to be due to chance?
First of all, to take a look at one of these representations is enough
to see that the mass is in rotation; we may even see what the sense of
the rotation is; all the spiral radii are curved in the same sense; it
is evident that the _moving wing_ lags behind the pivot and that fixes
the sense of the rotation. But this is not all; it is evident that these
nebulæ can not be likened to a gas at rest, nor even to a gas in
relative equilibrium under the sway of a uniform rotation; they are to
be compared to a gas in permanent motion in which internal currents
prevail.
Suppose, for example, that the rotation of the central nucleus is rapid
(you know what I mean by this word), too rapid for stable equilibrium;
then at the equator the centrifugal force will drive it away over the
attraction, and the stars will tend to break away at the equator and
will form divergent currents; but in going away, as their moment of
rotation remains constant, while the radius vector augments, their
angular velocity will diminish, and this is why the moving wing seems to
lag back.
From this point of view, there would not be a real permanent motion, the
central nucleus would constantly lose matter which would go out of it
never to return, and would drain away progressively. But we may modify
the hypothesis. In proportion as it goes away, the star loses its
velocity and ends by stopping; at this moment attraction regains
possession of it and leads it back toward the nucleus; so there will be
centripetal currents. We must suppose the centripetal currents are the
first rank and the centrifugal currents the second rank, if we adopt the
comparison with a troop in battle executing a change of front; and, in
fact, it is necessary that the composite centrifugal force be
compensated by the attraction exercised by the central layers of the
swarm upon the extreme layers.
Besides, at the end of a certain time a permanent régime establishes
itself; the swarm being curved, the attraction exercised upon the pivot
by the moving wing tends to slow up the pivot and that of the pivot upon
the moving wing tends to accelerate the advance of this wing which no
longer augments its lag, so that finally all the radii end by turning
with a uniform velocity. We may still suppose that the rotation of the
nucleus is quicker than that of the radii.
A question remains; why do these centripetal and centrifugal swarms tend
to concentrate themselves in radii instead of disseminating themselves a
little everywhere? Why do these rays distribute themselves regularly? If
the swarms concentrate themselves, it is because of the attraction
exercised by the already existing swarms upon the stars which go out
from the nucleus in their neighborhood. After an inequality is produced,
it tends to accentuate itself in this way.
Why do the rays distribute themselves regularly? That is less obvious.
Suppose there is no rotation, that all the stars are in two planes at
right angles, in such a way that their distribution is symmetric with
regard to these two planes.
By symmetry, there would be no reason for their going out of these
planes, nor for the symmetry changing. This configuration would give us
therefore equilibrium, but _this would be an unstable equilibrium_.
If on the contrary, there is rotation, we shall find an analogous
configuration of equilibrium with four curved rays, equal to each other
and intersecting at 90°, and if the rotation is sufficiently rapid, this
equilibrium is stable.
I am not in position to make this more precise: enough if you see that
these spiral forms may perhaps some day be explained by only the law of
gravitation and statistical consideration recalling those of the theory
of gases.
What has been said of internal currents shows it is of interest to
discuss systematically the aggregate of proper motions; this may be done
in a hundred years, when the second edition is issued of the chart of
the heavens and compared with the first, that we now are making.
But, in conclusion, I wish to call your attention to a question, that of
the age of the milky way or the nebulæ. If what we think we see is
confirmed, we can get an idea of it. That sort of statistical
equilibrium of which gases give us the model is established only in
consequence of a great number of impacts. If these impacts are rare, it
can come about only after a very long time; if really the milky way (or
at least the agglomerations which are contained in it), if the nebulæ
have attained this equilibrium, this means they are very old, and we
shall have an inferior limit of their age. Likewise we should have of it
a superior limit; this equilibrium is not final and can not last always.
Our spiral nebulæ would be comparable to gases impelled by permanent
motions; but gases in motion are viscous and their velocities end by
wearing out. What here corresponds to the viscosity (and which depends
upon the chances of impact of the molecules) is excessively slight, so
that the present régime may persist during an extremely long time, yet
not forever, so that our milky ways can not live eternally nor become
infinitely old.
And this is not all. Consider our atmosphere: at the surface must reign
a temperature infinitely small and the velocity of the molecules there
is near zero. But this is a question only of the mean velocity; as a
consequence of impacts, one of these molecules may acquire (rarely, it
is true) an enormous velocity, and then it will rush out of the
atmosphere, and once out, it will never return; therefore our atmosphere
drains off thus with extreme slowness. The milky way also from time to
time loses a star by the same mechanism, and that likewise limits its
duration.
Well, it is certain that if we compute in this manner the age of the
milky way, we shall get enormous figures. But here a difficulty presents
itself. Certain physicists, relying upon other considerations, reckon
that suns can have only an ephemeral existence, about fifty million
years; our minimum would be much greater than that. Must we believe that
the evolution of the milky way began when the matter was still dark? But
how have the stars composing it reached all at the same time adult age,
an age so briefly to endure? Or must they reach there all successively,
and are those we see only a feeble minority compared with those
extinguished or which shall one day light up? But how reconcile that
with what we have said above on the absence of a noteworthy proportion
of dark matter? Should we abandon one of the two hypotheses, and which?
I confine myself to pointing out the difficulty without pretending to
solve it; I shall end therefore with a big interrogation point.
However, it is interesting to set problems, even when their solution
seems very far away.
CHAPTER II
FRENCH GEODESY
Every one understands our interest in knowing the form and dimensions of
our earth; but some persons will perhaps be surprised at the exactitude
sought after. Is this a useless luxury? What good are the efforts so
expended by the geodesist?
Should this question be put to a congressman, I suppose he would say:
"I am led to believe that geodesy is one of the most useful of the
sciences; because it is one of those costing us most dear." I shall try
to give you an answer a little more precise.
The great works of art, those of peace as well as those of war, are not
to be undertaken without long studies which save much groping,
miscalculation and useless expense. These studies can only be based upon
a good map. But a map will be only a valueless phantasy if constructed
without basing it upon a solid framework. As well make stand a human
body minus the skeleton.
Now, this framework is given us by geodesic measurements; so, without
geodesy, no good map; without a good map, no great public works.
These reasons will doubtless suffice to justify much expense; but these
are arguments for practical men. It is not upon these that it is proper
to insist here; there are others higher and, everything considered, more
important.
So we shall put the question otherwise; can geodesy aid us the better to
know nature? Does it make us understand its unity and harmony? In
reality an isolated fact is of slight value, and the conquests of
science are precious only if they prepare for new conquests.
If therefore a little hump were discovered on the terrestrial ellipsoid,
this discovery would be by itself of no great interest. On the other
hand, it would become precious if, in seeking the cause of this hump, we
hoped to penetrate new secrets.
Well, when, in the eighteenth century, Maupertuis and La Condamine
braved such opposite climates, it was not solely to learn the shape of
our planet, it was a question of the whole world-system.
If the earth was flattened, Newton triumphed and with him the doctrine
of gravitation and the whole modern celestial mechanics.
And to-day, a century and a half after the victory of the Newtonians,
think you geodesy has nothing more to teach us?
We know not what is within our globe. The shafts of mines and borings
have let us know a layer of 1 or 2 kilometers thickness, that is to say,
the millionth part of the total mass; but what is beneath?
Of all the extraordinary journeys dreamed by Jules Verne, perhaps that
to the center of the earth took us to regions least explored.
But these deep-lying rocks we can not reach, exercise from afar their
attraction which operates upon the pendulum and deforms the terrestrial
spheroid. Geodesy can therefore weigh them from afar, so to speak, and
tell us of their distribution. Thus will it make us really see those
mysterious regions which Jules Verne only showed us in imagination.
This is not an empty illusion. M. Faye, comparing all the measurements,
has reached a result well calculated to surprise us. Under the oceans,
in the depths, are rocks of very great density; under the continents, on
the contrary, are empty spaces.
New observations will modify perhaps the details of these conclusions.
In any case, our venerated dean has shown us where to search and what
the geodesist may teach the geologist, desirous of knowing the interior
constitution of the earth, and even the thinker wishing to speculate
upon the past and the origin of this planet.
And now, why have I entitled this chapter _French Geodesy_? It is
because, in each country, this science has taken, more than all others,
perhaps, a national character. It is easy to see why.
There must be rivalry. The scientific rivalries are always courteous, or
at least almost always; in any case, they are necessary, because they
are always fruitful. Well, in those enterprises which require such long
efforts and so many collaborators, the individual is effaced, in spite
of himself, of course; no one has the right to say: this is my work.
Therefore it is not between men, but between nations that rivalries go
on.
So we are led to seek what has been the part of France. Her part I
believe we are right to be proud of.
At the beginning of the eighteenth century, long discussions arose
between the Newtonians who believed the earth flattened, as the theory
of gravitation requires, and Cassini, who, deceived by inexact
measurements, believed our globe elongated. Only direct observation
could settle the question. It was our Academy of Sciences that undertook
this task, gigantic for the epoch.
While Maupertuis and Clairaut measured a degree of meridian under the
polar circle, Bouguer and La Condamine went toward the Andes Mountains,
in regions then under Spain which to-day are the Republic of Ecuador.
Our envoys were exposed to great hardships. Traveling was not as easy as
at present.
Truly, the country where Maupertuis operated was not a desert and he
even enjoyed, it is said, among the Laplanders those sweet satisfactions
of the heart that real arctic voyagers never know. It was almost the
region where, in our days, comfortable steamers carry, each summer,
hosts of tourists and young English people. But in those days Cook's
agency did not exist and Maupertuis really believed he had made a polar
expedition.
Perhaps he was not altogether wrong. The Russians and the Swedes carry
out to-day analogous measurements at Spitzbergen, in a country where
there is real ice-cap. But they have quite other resources, and the
difference of time makes up for that of latitude.
The name of Maupertuis has reached us much scratched by the claws of
Doctor Akakia; the scientist had the misfortune to displease Voltaire,
who was then the king of mind. He was first praised beyond measure; but
the flatteries of kings are as much to be dreaded as their displeasure,
because the days after are terrible. Voltaire himself knew something of
this.
Voltaire called Maupertuis, my amiable master in thinking, marquis of
the polar circle, dear flattener out of the world and Cassini, and even,
flattery supreme, Sir Isaac Maupertuis; he wrote him: "Only the king of
Prussia do I put on a level with you; he only lacks being a geometer."
But soon the scene changes, he no longer speaks of deifying him, as in
days of yore the Argonauts, or of calling down from Olympus the council
of the gods to contemplate his works, but of chaining him up in a
madhouse. He speaks no longer of his sublime mind, but of his despotic
pride, plated with very little science and much absurdity.
I care not to relate these comico-heroic combats; but permit me some
reflections on two of Voltaire's verses. In his 'Discourse on
Moderation' (no question of moderation in praise and criticism), the
poet has written:
You have confirmed in regions drear
What Newton discerned without going abroad.
These two verses (which replace the hyperbolic praises of the first
period) are very unjust, and doubtless Voltaire was too enlightened not
to know it.
Then, only those discoveries were esteemed which could be made without
leaving one's house.
To-day, it would rather be theory that one would make light of.
This is to misunderstand the aim of science.
Is nature governed by caprice, or does harmony rule there? That is the
question. It is when it discloses to us this harmony that science is
beautiful and so worthy to be cultivated. But whence can come to us this
revelation, if not from the accord of a theory with experiment? To seek
whether this accord exists or if it fails, this therefore is our aim.
Consequently these two terms, which we must compare, are as
indispensable the one as the other. To neglect one for the other would
be nonsense. Isolated, theory would be empty, experiment would be blind;
each would be useless and without interest.
Maupertuis therefore deserves his share of glory. Truly, it will not
equal that of Newton, who had received the spark divine; nor even that
of his collaborator Clairaut. Yet it is not to be despised, because his
work was necessary, and if France, outstripped by England in the
seventeenth century, has so well taken her revenge in the century
following, it is not alone to the genius of Clairauts, d'Alemberts,
Laplaces that she owes it; it is also to the long patience of the
Maupertuis and the La Condamines.
We reach what may be called the second heroic period of geodesy. France
is torn within. All Europe is armed against her; it would seem that
these gigantic combats might absorb all her forces. Far from it; she
still has them for the service of science. The men of that time recoiled
before no enterprise, they were men of faith.
Delambre and Méchain were commissioned to measure an arc going from
Dunkerque to Barcelona. This time there was no going to Lapland or to
Peru; the hostile squadrons had closed to us the ways thither. But,
though the expeditions are less distant, the epoch is so troubled that
the obstacles, the perils even, are just as great.
In France, Delambre had to fight against the ill-will of suspicious
municipalities. One knows that the steeples, which are visible from so
far, and can be aimed at with precision, often serve as signal points to
geodesists. But in the region Delambre traversed there were no longer
any steeples. A certain proconsul had passed there, and boasted of
knocking down all the steeples rising proudly above the humble abode of
the sans-culottes. Pyramids then were built of planks and covered with
white cloth to make them more visible. That was quite another thing:
with white cloth! What was this rash person who, upon our heights so
recently set free, dared to raise the hateful standard of the
counter-revolution? It was necessary to border the white cloth with blue
and red bands.
Méchain operated in Spain; the difficulties were other; but they were
not less. The Spanish peasants were hostile. There steeples were not
lacking: but to install oneself in them with mysterious and perhaps
diabolic instruments, was it not sacrilege? The revolutionists were
allies of Spain, but allies smelling a little of the stake.
"Without cease," writes Méchain, "they threaten to butcher us."
Fortunately, thanks to the exhortations of the priests, to the pastoral
letters of the bishops, these ferocious Spaniards contented themselves
with threatening.
Some years after Méchain made a second expedition into Spain: he
proposed to prolong the meridian from Barcelona to the Balearics. This
was the first time it had been attempted to make the triangulations
overpass a large arm of the sea by observing signals installed upon some
high mountain of a far-away isle. The enterprise was well conceived and
well prepared; it failed however.
The French scientist encountered all sorts of difficulties of which he
complains bitterly in his correspondence. "Hell," he writes, perhaps
with some exaggeration--"hell and all the scourges it vomits upon the
earth, tempests, war, the plague and black intrigues are therefore
unchained against me!"
The fact is that he encountered among his collaborators more of proud
obstinacy than of good will and that a thousand accidents retarded his
work. The plague was nothing, the fear of the plague was much more
redoubtable; all these isles were on their guard against the neighboring
isles and feared lest they should receive the scourge from them. Méchain
obtained permission to disembark only after long weeks upon the
condition of covering all his papers with vinegar; this was the
antisepsis of that time.
Disgusted and sick, he had just asked to be recalled, when he died.
Arago and Biot it was who had the honor of taking up the unfinished work
and carrying it on to completion.
Thanks to the support of the Spanish government, to the protection of
several bishops and, above all, to that of a famous brigand chief, the
operations went rapidly forward. They were successfully completed, and
Biot had returned to France when the storm burst.
It was the moment when all Spain took up arms to defend her independence
against France. Why did this stranger climb the mountains to make
signals? It was evidently to call the French army. Arago was able to
escape the populace only by becoming a prisoner. In his prison, his only
distraction was reading in the Spanish papers the account of his own
execution. The papers of that time sometimes gave out news prematurely.
He had at least the consolation of learning that he died with courage
and like a Christian.
Even the prison was no longer safe; he had to escape and reach Algiers.
There, he embarked for Marseilles on an Algerian vessel. This ship was
captured by a Spanish corsair, and behold Arago carried back to Spain
and dragged from dungeon to dungeon, in the midst of vermin and in the
most shocking wretchedness.
If it had only been a question of his subjects and his guests, the dey
would have said nothing. But there were on board two lions, a present
from the African sovereign to Napoleon. The dey threatened war.
The vessel and the prisoners were released. The port should have been
properly reached, since they had on board an astronomer; but the
astronomer was seasick, and the Algerian seamen, who wished to make
Marseilles, came out at Bougie. Thence Arago went to Algiers, traversing
Kabylia on foot in the midst of a thousand perils. He was long detained
in Africa and threatened with the convict prison. Finally he was able to
get back to France; his observations, which he had preserved and
safeguarded under his shirt, and, what is still more remarkable, his
instruments had traversed unhurt these terrible adventures. Up to this
point, not only did France hold the foremost place, but she occupied the
stage almost alone.
In the years which follow, she has not been inactive and our
staff-office map is a model. However, the new methods of observation and
calculation have come to us above all from Germany and England. It is
only in the last forty years that France has regained her rank. She owes
it to a scientific officer, General Perrier, who has successfully
executed an enterprise truly audacious, the junction of Spain and
Africa. Stations were installed on four peaks upon the two sides of the
Mediterranean. For long months they awaited a calm and limpid
atmosphere. At last was seen the little thread of light which had
traversed 300 kilometers over the sea. The undertaking had succeeded.
To-day have been conceived projects still more bold. From a mountain
near Nice will be sent signals to Corsica, not now for geodesic
determinations, but to measure the velocity of light. The distance is
only 200 kilometers; but the ray of light is to make the journey there
and return, after reflection by a mirror installed in Corsica. And it
should not wander on the way, for it must return exactly to the point of
departure.
Ever since, the activity of French geodesy has never slackened. We have
no more such astonishing adventures to tell; but the scientific work
accomplished is immense. The territory of France beyond the sea, like
that of the mother country, is covered by triangles measured with
precision.
We have become more and more exacting and what our fathers admired does
not satisfy us to-day. But in proportion as we seek more exactitude, the
difficulties greatly increase; we are surrounded by snares and must be
on our guard against a thousand unsuspected causes of error. It is
needful, therefore, to create instruments more and more faultless.
Here again France has not let herself be distanced. Our appliances for
the measurement of bases and angles leave nothing to desire, and, I may
also mention the pendulum of Colonel Defforges, which enables us to
determine gravity with a precision hitherto unknown.
The future of French geodesy is at present in the hands of the
Geographic Service of the army, successively directed by General Bassot
and General Berthaut. We can not sufficiently congratulate ourselves
upon it. For success in geodesy, scientific aptitudes are not enough; it
is necessary to be capable of standing long fatigues in all sorts of
climates; the chief must be able to win obedience from his collaborators
and to make obedient his native auxiliaries. These are military
qualities. Besides, one knows that, in our army, science has always
marched shoulder to shoulder with courage.
I add that a military organization assures the indispensable unity of
action. It would be more difficult to reconcile the rival pretensions of
scientists jealous of their independence, solicitous of what they call
their fame, and who yet must work in concert, though separated by great
distances. Among the geodesists of former times there were often
discussions, of which some aroused long echoes. The Academy long
resounded with the quarrel of Bouguer and La Condamine. I do not mean to
say that soldiers are exempt from passion, but discipline imposes
silence upon a too sensitive self-esteem.
Several foreign governments have called upon our officers to organize
their geodesic service: this is proof that the scientific influence of
France abroad has not declined.
Our hydrographic engineers contribute also to the common achievement a
glorious contingent. The survey of our coasts, of our colonies, the
study of the tides, offer them a vast domain of research. Finally I may
mention the general leveling of France which is carried out by the
ingenious and precise methods of M. Lallemand.
With such men we are sure of the future. Moreover, work for them will
not be lacking; our colonial empire opens for them immense expanses illy
explored. That is not all: the International Geodetic Association has
recognized the necessity of a new measurement of the arc of Quito,
determined in days of yore by La Condamine. It is France that has been
charged with this operation; she had every right to it, since our
ancestors had made, so to speak, the scientific conquest of the
Cordilleras. Besides, these rights have not been contested and our
government has undertaken to exercise them.
Captains Maurain and Lacombe completed a first reconnaissance, and the
rapidity with which they accomplished their mission, crossing the
roughest regions and climbing the most precipitous summits, is worthy of
all praise. It won the admiration of General Alfaro, President of the
Republic of Ecuador, who called them 'los hombres de hierro,' the men of
iron.
The final commission then set out under the command of
Lieutenant-Colonel (then Major) Bourgeois. The results obtained have
justified the hopes entertained. But our officers have encountered
unforeseen difficulties due to the climate. More than once, one of them
has been forced to remain several months at an altitude of 4,000 meters,
in the clouds and the snow, without seeing anything of the signals he
had to aim at and which refused to unmask themselves. But thanks to
their perseverance and courage, there resulted from this only a delay
and an increase of expense, without the exactitude of the measurements
suffering therefrom.
GENERAL CONCLUSIONS
What I have sought to explain in the preceding pages is how the
scientist should guide himself in choosing among the innumerable facts
offered to his curiosity, since indeed the natural limitations of his
mind compel him to make a choice, even though a choice be always a
sacrifice. I have expounded it first by general considerations,
recalling on the one hand the nature of the problem to be solved and on
the other hand seeking to better comprehend that of the human mind,
which is the principal instrument of the solution. I then have explained
it by examples; I have not multiplied them indefinitely; I also have had
to make a choice, and I have chosen naturally the questions I had
studied most. Others would doubtless have made a different choice; but
what difference, because I believe they would have reached the same
conclusions.
There is a hierarchy of facts; some have no reach; they teach us nothing
but themselves. The scientist who has ascertained them has learned
nothing but a fact, and has not become more capable of foreseeing new
facts. Such facts, it seems, come once, but are not destined to
reappear.
There are, on the other hand, facts of great yield; each of them teaches
us a new law. And since a choice must be made, it is to these that the
scientist should devote himself.
Doubtless this classification is relative and depends upon the weakness
of our mind. The facts of slight outcome are the complex facts, upon
which various circumstances may exercise a sensible influence,
circumstances too numerous and too diverse for us to discern them all.
But I should rather say that these are the facts we think complex, since
the intricacy of these circumstances surpasses the range of our mind.
Doubtless a mind vaster and finer than ours would think differently of
them. But what matter; we can not use that superior mind, but only our
own.
The facts of great outcome are those we think simple; may be they really
are so, because they are influenced only by a small number of
well-defined circumstances, may be they take on an appearance of
simplicity because the various circumstances upon which they depend obey
the laws of chance and so come to mutually compensate. And this is what
happens most often. And so we have been obliged to examine somewhat more
closely what chance is.
Facts where the laws of chance apply become easy of access to the
scientist who would be discouraged before the extraordinary complication
of the problems where these laws are not applicable. We have seen that
these considerations apply not only to the physical sciences, but to the
mathematical sciences. The method of demonstration is not the same for
the physicist and the mathematician. But the methods of invention are
very much alike. In both cases they consist in passing up from the fact
to the law, and in finding the facts capable of leading to a law.
To bring out this point, I have shown the mind of the mathematician at
work, and under three forms: the mind of the mathematical inventor and
creator; that of the unconscious geometer who among our far distant
ancestors, or in the misty years of our infancy, has constructed for us
our instinctive notion of space; that of the adolescent to whom the
teachers of secondary education unveil the first principles of the
science, seeking to give understanding of the fundamental definitions.
Everywhere we have seen the rôle of intuition and of the spirit of
generalization without which these three stages of mathematicians, if I
may so express myself, would be reduced to an equal impotence.
And in the demonstration itself, the logic is not all; the true
mathematical reasoning is a veritable induction, different in many
regards from the induction of physics, but proceeding like it from the
particular to the general. All the efforts that have been made to
reverse this order and to carry back mathematical induction to the rules
of logic have eventuated only in failures, illy concealed by the
employment of a language inaccessible to the uninitiated. The examples I
have taken from the physical sciences have shown us very different cases
of facts of great outcome. An experiment of Kaufmann on radium rays
revolutionizes at the same time mechanics, optics and astronomy. Why?
Because in proportion as these sciences have developed, we have the
better recognized the bonds uniting them, and then we have perceived a
species of general design of the chart of universal science. There are
facts common to several sciences, which seem the common source of
streams diverging in all directions and which are comparable to that
knoll of Saint Gothard whence spring waters which fertilize four
different valleys.
And then we can make choice of facts with more discernment than our
predecessors who regarded these valleys as distinct and separated by
impassable barriers.
It is always simple facts which must be chosen, but among these simple
facts we must prefer those which are situated upon these sorts of knolls
of Saint Gothard of which I have just spoken.
And when sciences have no direct bond, they still mutually throw light
upon one another by analogy. When we studied the laws obeyed by gases we
knew we had attacked a fact of great outcome; and yet this outcome was
still estimated beneath its value, since gases are, from a certain point
of view, the image of the milky way, and those facts which seemed of
interest only for the physicist, ere long opened new vistas to astronomy
quite unexpected.
And finally when the geodesist sees it is necessary to move his
telescope some seconds to see a signal he has set up with great pains,
this is a very small fact; but this is a fact of great outcome, not only
because this reveals to him the existence of a small protuberance upon
the terrestrial globe, that little hump would be by itself of no great
interest, but because this protuberance gives him information about the
distribution of matter in the interior of the globe, and through that
about the past of our planet, about its future, about the laws of its
development.
* * * * *
INDEX
aberration of light, 315, 496
Abraham, 311, 490-1, 505-7, 509, 515-6
absolute motion, 107
orientation, 83
space, 85, 93, 246, 257, 353
acceleration, 94, 98, 486, 509
accidental constant, 112
errors, 171, 402
accommodation of the eyes, 67-8
action at a distance, 137
addition, 34
aim of mathematics, 280
alchemists, 11
Alfaro, 543
algebra, 379
analogy, 220
analysis, 218-9, 279
analysis situs, 53, 239, 381
analyst, 210, 221
ancestral experience, 91
Andrade, 93, 104, 228
Andrews, 153
angle sum of triangle, 58
Anglo-Saxons, 3
antinomies, 449, 457, 477
Arago, 540-1
Aristotle, 205, 292, 460
arithmetic, 34, 379, 441, 463
associativity, 35
assumptions, 451, 453
astronomy, 81, 289, 315, 512
Atwood, 446
axiom, 60, 63, 65, 215
Bacon, 128
Bartholi, 503
Bassot, 542
beauty, 349, 368
Becquerel, 312
Beltrami, 56, 58
Bergson, 321
Berkeley, 4
Berthaut, 542
Bertrand, 156, 190, 211, 395
Betti, 239
Biot, 540
bodies, solid, 72
Boltzmann, 304
Bolyai, 56, 201, 203
Borel, 482
Bouguer, 537, 542
Bourgeois, 543
Boutroux, 390, 464
Bradley, 496
Briot, 298
Brownian movement, 152, 410
Bucherer, 507
Burali-Forti, 457-9, 477, 481-2
Caen, 387-8
Calinon, 228
canal rays, 491-2
canals, semicircular, 276
Cantor, 11, 448-9, 457, 459, 477
Cantorism, 381, 382, 480, 484
capillarity, 298
Carlyle, 128
Carnot's principle, 143, 151, 300, 303-5, 399
Cassini, 537
cathode rays, 487-92
cells, 217
center of gravity, 103
central forces, 297
Chaldeans, 290
chance, 395, 408
change of position, 70
state, 70
chemistry of the stars, 295
circle-squarers, 11
Clairaut, 537-8
Clausius, 119, 123, 143
color sensation, 252
Columbus, 228
commutativity, 35-6
compensation, 72
complete induction, 40
Comte, 294
Condorcet, 411
contingence, 340
continuity, 173
continuum, 43
amorphous, 238
mathematical, 46
physical, 46, 240
tridimensional, 240
convention, 50, 93, 106, 125, 173, 208, 317, 440, 451
convergence, 67-8
coordinates, 244
Copernicus, 109, 291, 354
Coulomb, 143, 516
Couturat, 450, 453, 456, 460, 462-3, 467, 472-6
creation, mathematical, 383
creed, 1
Crémieu, 168-9, 490
crisis, 303
Crookes, 195, 488, 527-8
crude fact, 326, 330
Curie, 312-3, 318
current, 186
curvature, 58-9
curve, 213, 346
curves without tangents, 51
cut, 52, 256
cyclones, 353
d'Alembert, 538
Darwin, 518-9
De Cyon, 276, 427
Dedekind, 44-5
Defforges, 542
definitions, 430, 453
deformation, 73, 415
Delage, 277
Delambre, 539
Delbeuf, 414
Descartes, 127
determinism, 123, 340
dictionary, 59
didymium, 333
dilatation, 76
dimensions, 53, 68, 78, 241, 256, 426
direction, 69
Dirichlet, 213
dispersion, 141
displacement, 73, 77, 247, 256
distance, 59, 292
distributivity, 36
Du Bois-Reymond, 50
earth, rotation of, 326, 353
eclipse, 326
electricity, 174
electrified bodies, 117
electrodynamic attraction, 308
induction, 188
mass, 311
electrodynamics, 184, 282
electromagnetic theory of light, 301
electrons, 316, 492-4, 505-8, 510, 512-4
elephant, 217, 436
ellipse, 215
Emerson, 203
empiricism, 86, 271
Epimenides, 478-9
equation of Laplace, 283
Erdély, 203
errors, accidental, 171, 402
law of, 119
systematic, 171, 402
theory of, 402, 406
ether, 145, 351
ethics, 205
Euclid, 62, 86, 202-3, 213
Euclidean geometry, 65, 235-6, 337
Euclid's postulate, 83, 91, 124, 353, 443, 453, 468, 470-1
experience, 90-1
experiment, 127, 317, 336, 446
fact, crude, 326, 330
in the rough, 327
scientific, 326
facts, 362, 371
Fahrenheit, 238
Faraday, 150, 192
Faye, 536
Fechner, 46, 52
Fehr, 383
finite, 57
Fitzgerald, 415-6, 500-1, 505
Fizeau, 146, 149, 309, 498, 504
Flammarion, 400, 406-7
flattening of the earth, 353
force, 72, 98, 444
direction of, 445
-flow, 284
forces, central, 297
equivalence of, 445
magnitude of, 445
Foucault's pendulum, 85, 109, 353
four dimensions, 78
Fourier, 298-9
Fourier's problem, 317
series, 286
Franklin, 513-4
Fresnel, 132, 140, 153, 174, 176, 181, 351, 498
Fuchsian, 387-8
function, 213
continuous, 218, 288
Galileo, 97, 331, 353-4
gaseous pressure, 141
gases, theory of, 400, 405, 523
Gauss, 384-5, 406
Gay-Lussac, 157
generalize, 342
geodesy, 535
geometer, 83, 210, 438
geometric space, 66
geometry, 72, 81, 125, 207, 380, 428, 442, 467
Euclidean, 65, 93
fourth, 62
non-Euclidean, 55
projective, 201
qualitative, 238
rational, 5, 467
Riemann's, 57
spheric, 59
Gibbs, 304
Goldstein, 492
Gouy, 152, 305, 410
gravitation, 512
Greeks, 93, 368
Hadamard, 459
Halsted, 3, 203, 464, 467
Hamilton, 115
helium, 294
Helmholtz, 56, 115, 118, 141, 190, 196
Hercules, 449
Hermite, 211, 220, 222, 285
Herschel, 528
Hertz, 102, 145, 194-5, 427, 488, 498, 502, 504, 510
Hertzian oscillator, 309, 317
Hilbert, 5, 11, 203, 433, 450-1, 464-8, 471, 475-7, 484
Himstedt, 195
Hipparchus, 291
homogeneity, 74, 423
homogeneous, 67
hydrodynamics, 284
hyperbola, 215
hypotheses, 6, 15, 127, 133
hysteresis, 151
identity of spaces, 268
of two points, 259
illusions, optical, 202
incommensurable numbers, 44
induction, complete, 40, 452-3, 467-8
electromagnetic, 188
mathematical, 40, 220
principle of, 481
inertia, 93, 486, 489, 507
infinite, 448
infinitesimals, 50
inquisitor, 331
integration, 139
interpolation, 131
intuition, 210, 213, 215
invariant, 333
Ionians, 127
ions, 152
irrational number, 44
irreversible phenomena, 151
isotropic, 67
Japanese mice, 277, 427
Jevons, 451
John Lackland, 128
Jules Verne, 111, 536
Jupiter, 131, 157, 231, 289
Kant, 16, 64, 202-3, 450-1, 471
Kauffman, 311, 490-1, 495, 506-7, 522, 545
Kazan, 203
Kelvin, 145, 523-4, 526-7
Kepler, 120, 133, 153, 282, 291-2
Kepler's laws, 136, 516
kinematics, 337
kinetic energy, 116
theory of gases, 141
Kirchhoff, 98-9, 103-5
Klein, 60, 211, 287
knowledge, 201
König, 144, 477
Kovalevski, 212, 286
Kronecker, 44
Lacombe, 543
La Condamine, 535, 537-8, 542-3
Lagrange, 98, 151, 179
Laisant, 383
Lallamand, 543
Langevin, 509
Laplace, 298, 398, 514-5, 518, 522, 538
Laplace's equation, 283, 287
Larmor, 145, 150
Lavoisier's principle, 301, 310, 312
law, 207, 291, 395
Leibnitz, 32, 450, 471
Le Roy, 28, 321-6, 332, 335, 337, 347-8, 354, 468
Lesage, 517-21
Liard, 440
Lie, 62-3, 212
light sensations, 252
theory of, 351
velocity of, 232, 312
Lindemann, 508
line, 203, 243
linkages, 144
Lippmann, 196
Lobachevski, 29, 56, 60, 62, 83, 86, 203
Lobachevski's space, 239
local time, 306-7, 499
logic, 214, 435, 448, 460-2, 464
logistic, 457, 472-4
logisticians, 472
Lorentz, 147, 149, 196-7, 306, 308, 311, 315, 415-6, 492, 498-502,
504-9, 512, 514-6, 521
Lotze, 264
luck, 399
Lumen, 407-8
MacCullagh, 150
Mach, 375
Mach-Delage, 276
magnetism, 149
magnitude, 49
Mariotte's law, 120, 132, 157, 342, 524
Maros, 203
mass, 98, 312, 446, 486, 489, 494, 515
mathematical analysis, 218
continuum, 46
creation, 383
induction, 40, 220
physics, 136, 297, 319
mathematics, 369, 448
matter, 492
Maupertuis, 535, 537-8
Maurain, 543
Maxwell, 140, 152, 175, 177, 181, 193, 282-3, 298, 301, 304-5, 351,
503, 524-5
Maxwell-Bartholi, 309, 503-4, 519, 521
Mayer, 119, 123, 300, 312, 318
measurement, 49
Méchain, 539-40
mechanical explanation, 177
mass, 312
mechanics, 92, 444, 486, 496, 512
anthropomorphic, 103
celestial, 279
statistical, 304
Méray, 211
metaphysician, 221
meteorology, 398
mice, 277
Michelson, 306, 309, 311, 316, 498, 500-1
milky way, 523-30
Mill, Stuart, 60-1, 453-4
Monist, 4, 89, 464
moons of Jupiter, 233
Morley, 309
motion of liquids, 283
of moon, 28
of planets, 341
relative, 107, 487
without deformation, 236
multiplication, 36
muscular sensations, 69
Nagaoka, 317
nature, 127
navigation, 289
neodymium, 333
neomonics, 283
Neumann, 181
Newton, 85, 96, 98, 109, 153, 291, 370, 486, 516, 536, 538
Newton's argument, 108, 334, 343
law, 111, 118, 132, 136, 149, 157, 233, 282, 292, 512, 514-5,
518, 525
principle, 146, 300, 308-9, 312
no-class theory, 478
nominalism, 28, 125, 321, 333, 335
non-Euclidean geometry, 55, 59, 388
language, 127
space, 55, 235, 237
straight, 236, 470
world, 75
number, 31
big, 88
imaginary, 283
incommensurable, 44
transfinite, 448-9
whole, 44, 469
objectivity, 209, 347, 349, 408
optical illusions, 202
optics, 174, 496
orbit of Saturn, 341
order, 385
orientation, 83
osmotic, 141
Padoa, 463
Panthéon, 414
parallax, 470
parallels, 56, 443
Paris time, 233
parry, 419-22, 427
partition, 45
pasigraphy, 456-7
Pasteur, 128
Peano, 450, 456-9, 463, 472
Pender, 490
pendulum, 224
Perrier, 541
Perrin, 195
phosphorus, 333, 468, 470-1
physical continuum, 46
physics, 127, 140, 144, 279, 297
physics of central forces, 297
of the principles, 299
Pieri, 11, 203
Plato, 292
Poincaré, 473
point, 89, 244
Poncelet, 215
postulates, 382
potential energy, 116
praseodymium, 333
principle, 125, 299
Carnot's, 143, 151, 300, 303-5, 399
Clausius', 119, 123, 143
Hamilton's, 115
Lavoisier's, 300, 310
Mayer's, 119, 121, 123, 300, 312, 318
Newton's, 146, 300, 308-9, 312
of action and reaction, 300, 487, 502
of conservation of energy, 300
of degradation of energy, 300
of inertia, 93, 486, 507
of least action, 118, 300
of relativity, 300, 305, 498, 505
Prony, 445
psychologist, 383
Ptolemy, 110, 291, 353-4
Pythagoras, 292
quadrature of the circle, 161
qualitative geometry, 238
space, 207
time, 224
quaternions, 282
radiometer, 503
radium, 312, 318, 486-7
Rados, 201
Ramsay, 313
rational geometry, 5, 467
reaction, 502
reality, 217, 340, 349
Réaumur, 238
recurrence, 37
Regnault, 170
relativity, 83, 305, 417, 423, 498, 505
Richard, 477-8, 480-1
Riemann, 56, 62, 145, 212, 239, 243, 381, 432
surface, 211, 287
Roemer, 233
Röntgen, 511, 520
rotation of earth, 225, 331, 353
roulette, 403
Rowland, 194-7, 305, 489
Royce, 202
Russell, 201, 450, 460-2, 464-7, 471-4, 477-82, 484-5
St. Louis exposition, 208, 320
Sarcey, 442
Saturn, 231, 317
Schiller, 202
Schliemann, 19
science, 205, 321, 323, 340, 354
Science and Hypothesis, 205-7, 220, 240, 246-7, 319, 353, 452
semicircular canals, 276
series, development in, 287
Fourier's, 286
Sirius, 226, 229
solid bodies, 72
space, 55, 66, 89, 235, 256
absolute, 85, 93
amorphous, 417
Bolyai, 56
Euclidean, 65
geometric, 66
Lobachevski's, 239
motor, 69
non-Euclidean, 55, 235, 237
of four dimensions, 78
perceptual, 66, 69
tactile, 68, 264
visual, 67, 252
spectra, 316
spectroscope, 294
Spencer, 9
sponge, 219
Stallo, 10
stars, 292
statistical mechanics, 304
straight, 62, 82, 236, 433, 450, 470
Stratonoff, 531
surfaces, 58
systematic errors, 171
tactile space, 68, 264
Tait, 98
tangent, 51
Tannery, 43
teaching, 430, 437
thermodynamics, 115, 119
Thomson, 98, 488
thread, 104
time, 223
equality, 225
local, 306, 307
measure of, 223-4
Tisserand, 515-6
Tolstoi, 354, 362, 368
Tommasina, 519
Transylvania, 203
triangle, 58
angle sum of, 58
truth, 205
Tycho Brahe, 133, 153, 228
unity of nature, 130
universal invariant, 333
Uriel, 203
van der Waals, 153
Vauban, 210
Veblen, 203
velocity of light, 232, 312
Venus of Milo, 201
verification, 33
Virchow, 21
visual impressions, 252
space, 67, 252
Volga, 203
Voltaire, 537-8
Weber, 117, 515-6
Weierstrass, 11, 212, 432
Whitehead, 472, 481-2
whole numbers, 44
Wiechert, 145, 488
x-rays, 152, 511, 520
Zeeman effect, 152, 196, 317, 494
Zeno, 382
Zermelo, 477, 482-3
zigzag theory, 478
zodiac, 398, 404
* * * * *
Transcriber's Note: The Greek alphabets are represented within square
brackets. For example, [alpha] stands for first Greek alphabet alpha.
Square root of a number is represented using the symbol (sqrt). That
is to say, sqrt(25) stands for square root of 25. The superscript is
shown with carat (^) symbol, example, 10^{5} stands for 5th power of
10. Similarly, subscript is represented by underscore (_) symbol. For
instance, n_{3} stands for letter n with subscript 3.