Engels' Dialectics of Nature

"On the other hand, I have always found hitherto that the
basic concepts in this field (*i.e.* “the basic physical concepts
of work and their unalterability”) seem very difficult to grasp for persons
who have not gone through the school of mathematical mechanics, in spite
of all zeal, all intelligence, and even a fairly high degree of scientific
knowledge. Moreover, it cannot be denied that they are abstractions of
a quite peculiar kind. It was not without difficulty that even such an
intellect as that of I. Kant succeeded in understanding them, as is proved
by his polemic against Leibniz on this subject."

So says Helmholtz (*Pop. wiss. Vorträge* [*Popular Scientific
Lectures*], II, Preface).

According to this, we are venturing now into a very dangerous field, the more so since we cannot very well take the liberty of guiding the reader "through the school of mathematical mechanics.” Perhaps, however, it will turn out that, where it is a question of concepts, dialectical thinking will carry us at least as far as mathematical calculation.

Galileo discovered, on the one hand, the law of falling, according to
which the distances traversed by falling bodies are proportional to the
squares of the times taken in falling. On the other hand, as we shall see,
he put forward the not quite compatible law that the magnitude of motion
of a body (its *impeto* or *momento*) is determined by the mass
and the velocity in such a way that for constant mass it is proportional
to the velocity. Descartes adopted this latter law and made the product
of the mass and the velocity of the moving body quite generally into the
measure of its motion.

Huyghens had already found that, on elastic impact, the sum of the products of the masses, multiplied by the squares of their velocities, remains the same before and after impact, and that an analogous law holds good in various other cases of motion to a system of connected bodies.

Leibniz was the first to realise that the Cartesian measure of motion
was in contradiction to the law of falling. On the other hand, it could
not be denied that in many cases the Cartesian measure was correct. Accordingly,
Leibniz divided moving forces into dead forces and live forces. The dead
were the “pushes” or “pulls” of resting bodies, and their measure the product
of the mass and the velocity with which the body would move if it were
to pass from a state of rest to one of motion. On the other hand, he put
forward as the measure of *vis viva*, of the real motion of a body,
the product of the mass and the square of the velocity. This new measure
of motion he derived directly from the law of falling.

"The same force is required,” so Leibniz concluded, “to raise
a body of four pounds in weight one foot as to raise a body of one pound
in weight four feet; but the distances are proportional to the square of
the velocity, for when a body has fallen four feet, it attains twice the
velocity reached on falling only one foot. However, bodies on falling acquire
the force for rising to the same height as that from which they fell; hence
the forces are proportional to the square of the velocity.” (Suter,*
Geschichte der Mathematik* [*History of Mathematics*], II, p. 367.)

But he showed further that the measure of motion *mv* is in contradiction
to the Cartesian law of the constancy of the quantity of motion, for if
it was really valid the force (*i.e.* the quantity of motion) in nature
would continually increase or diminish. He even devised an apparatus (1690,
*Acta Eruditorum*) which, if the measure *mv* were correct, would
be bound to act as a *perpetuum mobile* with continual gain of force,
which, however, would be absurd. Recently, Helmholtz has again frequently
employed this kind of argument.

The Cartesians protested with might and main and there developed a famous
controversy lasting many years, in which Kant also participated in his
very first work (*Gedanken von der wahren Schätzung der lebendigen
Kräfte* [*Thoughts on the True Estimation of Live Forces*],
1746), without, however, seeing clearly into the matter. Mathematicians
to-day look down with a certain amount of scorn on this “barren” controversy
which “dragged out for more than forty years and divided the mathematicians
of Europe into two hostile camps, until at last d'Alembert by his *Traité
de dynamique* (1743), as it were by a final verdict, put an end to the
*useless verbal dispute*, for it was nothing else.” (Suter, *ibid*.,
p. 366.)

It would, however, seem that a controversy could not rest entirely on
a useless verbal dispute when it had been initiated by a Leibniz against
a Descartes, and had occupied a man like Kant to such an extent that he
devoted to it his first work, a fairly large volume. And in point of fact,
how is it to be understood that motion has two contradictory measures,
that on one occasion it is proportional to the velocity, and on another
to the square of the velocity? Suter makes it very easy for himself; he
says both sides were right and both were wrong; “nevertheless, the expression
'*vis viva*' has endured up to the present day; *only it no longer
serves as the measure of force*, but is merely a term that was once
adopted for the product of the mass and half the square of the velocity,
a product so full of significance in mechanics.” Hence, *mv* remains
the measure of motion, and *vis viva* is only another expression for
mv^{2}/2, concerning which formula we learn indeed that it is of
great significance for mechanics, but now most certainly do not know what
significance it has.

Let us, however, take up the salvation-bringing *Traité de
dynamique* and look more closely at d'Alembert's “final verdict"; it
is to be found in the *preface*. In the text, it says, the whole question
does not occur, on account of *l'inutilité parfaite dont elle
est pour la mécanique*. This is quite correct for *purely mathematical*
mechanics, in which, as in the case of Suter above, words used as designations
are only other expressions, or names, for algebraic formulae, names in
connection with which it is best not to think at all. Nevertheless, since
such important people have concerned themselves with the matter, he desires
to examine it briefly in the preface. Clearness of thought demands that
by the force of moving bodies one should understand only their property
of overcoming obstacles or resisting them. Hence, force is to be measured
neither by *mv*^{2} nor by XXX, but solely by the obstacles
and the resistance they offer.

Now, there are, he says, three kinds of obstacles: (1) insuperable obstacles which totally destroy the motion, and for that very reason cannot be taken into account here; (2) obstacles whose resistance suffices to arrest the motion and to do so instantaneously: the case of equilibrium; (3) obstacles which only gradually arrest the motion: the case of retarded motion.

"Or tout le monde convient qu'il y a équilibre entre deux corps,
quand les produits de leurs masses par leurs vitesses virtuelles, c'est
à dire par les vitesses avec lesquelles ils tendent à se
mouvoir, sont égaux de part et d'autre. Donc dans l'équilibre
le produit de la masse par la vitesse, ou, ce qui est la même chose,
la quantité de mouvement, peut représenter la force. Tout
le monde convient aussi que dans le mouvement retardé, le nombre
des obstacles vaincus est comme le carré de la vitesse, en sorte
qu'un corps qui a fermé un ressort, par exemple, avec une certaine
vitesse, pourra, avec une vitesse double, fermer ou tout à la fois,
ou successivement, non pas deux, mais quatre ressorts semblables au premier,
neuf avec une vitesse triple, et ainsi du reste. D'où les partisans
des forces vives [the Leibnizians] concluent que la force des corps qui
se meuvent actuellement, est en général comme le produit
de la masse par le carré de la vitesse. Au fond, quel inconvénient
pourrait-il y avoir, à ce que la mesure des forces fût différente
dans l'équilibre et dans le mouvement retardé, puisque, si
on veut ne raisonner que d'après des idées claires, on doit
n'entendre par le mot *force* que l'effet produit en surmontant l'obstacle
ou en lui résistant?” (Preface, pp. 19-20, of the original edition.)

D'Alembert, however, is far too much of a philosopher not to realise
that the contradiction of a twofold measure of one and the same force is
not to be got over so easily. Therefore, after repeating what is basically
only the same thing as Leibniz had already said - for his *équilibre*
is precisely the same thing as the “dead pressure” of Leibniz - he suddenly
goes over to the side of the Cartesians and finds the following expedient:
the product *mv* can serve as a measure of force, even in the case
of delayed motion,

"si dans ce dernier cas on mesure la force, non par la quantité
absolue des obstacles, mais par la somme des résistances de ces
mêmes obstacles. Car on ne saurait douter que cette somme des résistances
ne soit proportionelle à la quantité du mouvement *mv*,
puisque, de l'aveu de tout le monde, la quantité du mouvement que
le corps perd à chaque instant, est proportionelle au produit de
la résistance par la durée infiniment petite de l'instant,
et que la somme de ces produits est evidemment la résistance totale."

This latter mode of calculation seems to him the more natural one, “car un obstacle n'est tel qu'en tant qu'il résiste et c'est, à proprement parler, la somme des résistances qui est 1'obstacle vaincu; d'ailleurs, en estimant ainsi la force, on a l'avantage d'avoir pour l'équilibre et pour le mouvement retardé une mesure commune.” Still, everyone can take that as he likes. And so, believing he has solved the question, by what, as Suter himself acknowledges, is a mathematical blunder, he concludes with unkind remarks on the confusion reigning among his predecessors, and asserts that after the above remarks there is possible only a very futile metaphysical discussion or a still more discreditable purely verbal dispute.

D'Alembert's proposal for reaching a reconciliation amounts to the following calculation:

A mass 1, with velocity 1, compresses 1 spring in unit time.

A mass 1, with velocity 2, compresses 4 springs, but requires two units
of time; *i.e.* only 2 springs per unit of time.

A mass 1, with velocity 3, compresses 9 springs in three units of time,
*i.e.* only 3 springs per unit of time.

Hence if we divide the effect by the time required for it, we again
come from mv^{2} to *mv*.

This is the same argument that Catelan in particular had already employed
against Leibniz; it is true that a body with velocity 2 rises against gravity
four times as high as one with velocity 1, but it requires double the time
for it; consequently the quantity of motion must be divided by the time,
and =2, not =4. Curiously enough, this is also Suter's view, who indeed
deprived the expression “*vis viva*” of all logical meaning and left
it only a mathematical one. But this is natural. For Suter it is a question
of saving the formula *mv* in its significance as sole measure of
the quantity of motion; hence logically mv^{2} is sacrificed in
order to arise again transfigured in the heaven of mathematics.

However, this much is correct: Catelan's argument provides one of the
bridges connecting *mv* with mv^{2}, and so is of importance.

The mechanicians subsequent to d'Alembert by no means accepted his verdict,
for his final verdict was indeed in favour of *mv* as the measure
of motion. They adhered to his expression of the distinction which Leibniz
had already made between dead and live forces: *mv* is valid for equilibrium,
*i.e.* for statics; mv^{2} is valid for motion against resistance, *i.e.* for dynamics. Although on the whole correct, the distinction
in this form has, however, logically no more meaning than the famous pronouncement of the junior officer: on duty always “to me,” off duty always “me."
It is accepted tacitly, it just exists. We cannot alter it, and if a contradiction lurks in this double measure, how can we help it?

Thus, for instance, Thomson and Tait say (*A Treatise on Natural Philosophy*, Oxford, 1867, p. 162); “The *quantity of motion* or the *momentum*
of a rigid body moving without rotation is proportional to its mass and
velocity conjointly. Double mass or double velocity would correspond to
double quantity of motion.” And immediately below that they say: “The
*vis viva* or *kinetic energy* of a moving body is proportional
to the mass and the square of the velocity conjointly.”

The two contradictory measures of motion are put side by side in this very glaring form. Not so much as the slightest attempt is made to explain the contradiction, or even to disguise it. In the book by these two Scotsmen, thinking is forbidden, only calculation is permitted. No wonder that at least one of them, Tait, is accounted one of the most pious Christians of pious Scotland.

In Kirchhoff's *Vorlesungen über mathematische Mechanik* [*Lectures
on Mathematical Mechanics*] the formulae *mv* and mv^{2}
do not occur at all *in this form*.

Perhaps Helmholtz will aid us. In his *Erhaltung der Kraft *[*Conservation
of Force*] he proposes to express *vis viva* by mv^{2}/2,
a point to which we shall return later. Then, on page 20 *et seq.,*
he enumerates briefly the cases in which so far the principle of the conservation
of *vis viva* (hence of mv^{2}/2) has been recognised and
made use of. Included therein under No. 2 is

“the transference of motion by incompressible solid and fluid
bodies, in so far as friction or impact of inelastic materials does not
occur. For these cases our general principle is usually expressed in the
rule that motion propagated and altered by mechanical powers always decreases
in intensity of force in the same proportion as it increases in velocity.
If, therefore, we imagine a weight *m* being raised with velocity
*c* by a machine in which a force for performing work is produced
uniformly by some process or other, then with a different mechanical arrangement
the weight *nm* could be raised, but only with velocity *c/n*,
so that in both cases the quantity of tensile force produced by the machine
in unit time is represented by *mgc*, where *g* is the intensity
of the gravitational force.”

Thus, here too we have the contradiction that an “intensity of force," which decreases and increases in simple proportion to the velocity, has to serve as proof for the conservation of an intensity of force which decreases and increases in proportion to the square of the velocity.

In any case, it becomes evident here that *mv* and *mv ^{2}* serve to determine two quite distinct processes, but we certainly knew
long ago that

By *mv*, then, one measures “a motion propagated and altered by
mechanical powers”; hence this measure holds good for the lever and all
its derivatives, for wheels, screws, etc., in short, for all machinery
for the transference of motion. But from a very simple and by no means
new consideration it becomes evident that in so far as *mv* applies
here, so also does *mv ^{2}*. Let us take any mechanical contrivance
in which the sums of the lever-arms on the two sides are related to each
other as 4:1, in which, therefore, a weight of 1 kg. holds a weight of
4 kg. in equilibrium. Hence, by a quite insignificant additional force
on one arm of the lever we can raise 1 kg. by 20 m.; the same additional
force, when applied to the other arm of the lever, raises 4 kg. a distance
of 5 m., and the preponderating weight sinks in the same time that the
other weight requires for rising. Mass and velocity are inversely proportional
to one another;

*mv ^{2}*=1 x 20 x 20 =400 =

On the other hand the times of fall are different: the 4 kg. traverse their 5 m. in 1 second, the 1 kg. traverses its 20 m. in 2 seconds. Friction and air resistance are, of course, neglected here.

But after each of the two bodies has fallen from its, height, its motion
ceases. Therefore, *mv* appears here as the measure of simple transferred,
hence lasting, mechanical motion, and *mv ^{2}* as the measure
of the vanished mechanical motion.

Further, the same thing applies to the impact of perfectly elastic bodies:
the sum of both *mv* and of mv^{2} is unaltered before and
after impact. Both measures have the same validity.

'This is not the case on impact of inelastic bodies. Here, too, the
current elementary textbooks (higher mechanics is hardly concerned at all
with such trifles) teach that before and after impact the sum of *mv*
remains the same. On the other hand a loss of *vis viva* occurs, for
if the sum of *mv ^{2} *after impact is subtracted from the
sum of

But this does not matter. Even if we admit the theorem, and calculate
the velocity after falling, on the assumption that the sum of *mv*
has remained the same, this decrease of the sum of *mv ^{2}*
is still found. Here, therefore,

Such are pretty nearly all the cases in which *mv* is employed
in mechanics. Let us now glance at some cases in which *mv ^{2}*
is employed.

When a cannon-ball is fired, it uses up in its course an amount of motion
that is proportional to *mv ^{2}*, irrespective of whether
it encounters a solid target or comes to a standstill owing to air resistance
and gravitation. If a railway train runs into a stationary one, the violence
of the collision, and the corresponding destruction, is proportional to
its

But what is the meaning of this convenient phrase, so current in mechanics: overcoming a resistance?

If we overcome the resistance of gravity by raising a weight, there
disappears a quantity of motion, a quantity of mechanical force, equal
to that produced anew by the direct or indirect fall of the raised weight
from the height reached back to its original level. The quantity is measured
by half the product of the mass and the final velocity after falling, mv^{2}/2.
What then occurred on raising the weight? Mechanical motion, or force,
disappeared as such. But it has not been annihilated; it has been converted
into mechanical force of tension, to use Helmholtz's expression; into potential
energy, as the moderns say; into ergal as Clausius calls it; and this can
at any moment, by any mechanically appropriate means, be reconverted into
the same quantity of mechanical motion as was necessary to produce it.
The potential energy is only the negative expression of the *vis viva*
and *vice versa*.

A 24-lb. cannon-ball moving with a velocity of 400 m. per second strikes
the one-metre thick armour-plating of a warship and under these conditions
has apparently no effect on the armour. Consequently an amount of mechanical
motion has vanished equal to mv^{2}/2, *i.e.* (since 24 lbs.
=12 kg.) =12 X 400 X 400 X 1/2= 960,000 kilogram-metres. Wat has become
of it? A small portion has been expended in the concussion and molecular
alteration of the armour-plate. A second portion goes in smashing the cannon-ball
into innumerable fragments. But the greater part has been converted into
heat and raises the temperature of the cannon-hall to red heat. When the
Prussians, in passing over to Alsen in 1864, brought their heavy batteries
into play against the armoured sides of the Rolf Krake, after each hit
they saw in the darkness the flare produced by the suddenly glowing shot.
Even earlier, Whitworth had proved by experiment that explosive shells
need no detonator when used against armoured warships; the glowing metal
itself ignites the charge. Taking the mechanical equivalent of the unit
of heat as 424 kilogram-metres, the quantity of heat corresponding to the
above-mentioned amount of mechanical motion is 2,264 units. The specific
heat of iron=0.1140; that is to say, the amount of heat that raises the
ternperature of 1 kg. of water by 1º C. (which serves as the unit
of heat) suffices to raise the temperature of 1/0.1140 = 8.772 kg. of iron
by 1º C. Therefore the 2,264 heat-units mentioned above raise the
temperature of 1 kg. of iron by 8.772 X 2,264 =19,860º C. or 19,860
kg. of iron by 1º C. Since this quantity of heat is distributed uniformly
in the armour and the shot, the latter has its temperature raised by 19,860/2X12=828º,
amounting to quite a good glowing heat. But since the foremost, striking
end of the shot receives at any rate by far the greater part of the heat,
certainly double that of the rear half, the former would be raised to a
temperature of 1,104º C. and the latter to 552º C., which would
fully suffice to explain the glowing effect even if we make a big deduction
for the actual mechanical work performed on impact.

Mechanical motion also disappears in friction, to reappear as heat; it is well known that, by the most accurate possible measurement of the two processes, Joule in Manchester and Codling in Copenhagen were the first to make an approximate experimental measurement of the mechanical equivalent of heat.

The same thing applies to the production of an electric current in a
magneto-electrical machine by means of mechanical force, *e.g.* from
a steam engine. The quantity of so-called electromotive force produced
in a given time is proportional to the quantity of mechanical motion used
up in the same period, being equal to it if expressed in the same units.
We can imagine this quantity of mechanical motion being produced, not by
a steam engine, but by a weight falling in accordance with the pressure
of gravity. The mechanical force that this is capable of supplying is measured
by the *vis viva* that it would obtain on falling freely through the
same distance, or by the force required to raise it again to the original
height; in both cases mv^{2}/2.

Hence we find that while it is true that mechanical motion has a two-fold
measure, each of these measures holds good for a very definitely demarcated
series of phenomena. If already existing mechanical motion is transferred
in such a way that it remains as mechanical motion, the transference takes
place in proportion to the product of the mass and the velocity. If, however,
it is transferred in such a way that. it disappears as mechanical motion
in order to reappear in the form of potential energy, heat, electricity,
etc., in short, if it is converted into another form of motion, then the
quantity of this new form of motion is proportional to the product of the
originally moving mass and the square of the velocity. In short, *mv*
is mechanical motion measured as mechanical motion; mv^{2}/2 is
mechanical motion measured by its capacity to become converted into a definite
quantity of another form of motion. And, as we have seen, these two measures,
because different, do not contradict one another.

It becomes clear from this that Leibniz's quarrel with the Cartesians was by no means a mere verbal dispute, and that d'Alembert's verdict in point of fact settled nothing at all. D'Alembert. might have spared himself his tirades on the unclearness of his predecessors, for he was just as unclear as they were. In fact, as long as it was not known what becomes of the apparently annihilated mechanical motion. the absence of clarity was inevitable. And as long as mathematical mechanicians like Suter remain obstinately shut in by the four walls of their special science, they are bound to remain just as unclear as d'Alembert and to put us off with empty and contradictory phrases.

But how does modern mechanics express this conversion of mechanical
motion into another form of motion, proportional in quantity to the former?
It has *performed work*, and indeed a definite amount of work.

But this does not exhaust the concept of work in the physical sense
of the word. If, as in a steam or heat engine, heat is converted into mechanical
motion,*i.e.* molecular motion is converted into mass motion, if heat
breaks up a chemical compound, if it becomes converted into electricity
in a thermopile, if an electric current sets free the elements of water
from dilute sulphuric acid, or, conversely, if the motion (alias energy)
produced in the chemical process of a current-producing cell takes the
form of electricity and this in the circuit once more becomes converted
into heat - in all these processes the form of motion that initiates the
process, and which is converted by it into another form, performs work,
and indeed a quantity of work corresponding to its own quantity.

Work, therefore, is change of form of motion regarded in its quantitative aspect.

But how so? If a raised weight remains suspended and at rest, is its
potential energy during the period of rest also a form of motion? Certainly.
Even Tait arrives at the conviction that potential energy is subsequently
resolved into a form of actual motion (*Nature*, XIV p.459). And,
apart from that, Kirchhoff goes much further in saying (*Mathematical
Mechanics*, p. 32) “Rest is a special case of motion,” and thus proves
that he can not only calculate but can also think dialectically.

Hence, by a consideration of the two measures of rnechanical motion,
we arrive incidentally, easily, and almost as a matter of course, at the
concept of work, which was described to us as being so difficult to comprehend
without mathematical mechanics. At any rate, we now know more about it
than from Helmholtz's lecture *On the Conservation of Force*(1862),
which was intended precisely “to make as clear as possible the fundamental
physical concepts of work and their invariability.” All that we learn there
about work is: that it is something which is expressed in foot-pounds or
in units of heat, and that the number of these foot-pounds or units of
heat is invariable for a definite quantity of work; and, further, that
besides mechanical forces and heat, chemical and electric forces can perform
work, but that all these forces exhaust their capacity for work in the
measure that they actually result in work. We learn also that it follows
from this that the sum of all effective quantities of force in nature as
a whole remains eternally and invariably the same throughout all the changes
taking place in nature. The concept of work is neither developed, nor even
defined.[1]
And it is precisely the quantitative invariability of the magnitude of
work which prevents him from realising that the qualitative alteration,
the change of form, is the basic condition for all physical work. And so
Helmholtz can go so far as to assert that “friction and inelastic impact
are processes in which *mechanical work is destroyed* and heat is produced
instead.” (*Pop. Vorträge* [*Popular Lectures*], II, p.
166.) Just the contrary. Here mechanical work is not *destroyed*,
here mechanical work is *performed*. It is mechanical *motion*
that is apparently destroyed. But mechanical motion *can* never perform
even a millionth part of a kilogram-metre of work, without apparently being
destroyed as such, without becoming converted into another form of motion.

But, as we have seen, the capacity for work contained in a given quantity
of mechanical motion is what is known as its *vis viva,* and until
recently was measured by mv^{2}. And here a new contradiction arose.
Let us listen to Helmholtz (*Conservation of Force,* p. 9).

We read there that the magnitude of work can be expressed by a weight
*m* being raised to a height *h*, when, if the force of gravity
is put as *g*, the magnitude of work =*mgh*. For the body *m*
to rise freely to the vertical height *h*, it requires a velocity
v= (square root of) 2gh, and it attains the same velocity on falling. Consequently,
*mgh*=mv^{2}/2 and Helmholtz proposes “to take the magnitude
mv^{2}/2 as the quantity of *vis viva*, whereby it becomes
identical with the measure of the magnitude of work. From the viewpoint
of how the concept of *vis viva* has been applied hitherto... this
change has no significance, but it will offer essential advantages in the
future.”

It is scarcely to be believed. In 1847, Helmholtz was so little clear
about the mutual relations of *vis viva* and work, that he totally
fails to notice how he transforms the former proportional measure of *vis
viva* into its absolute measure, and remains quite unconscious of the
important discovery he has made by his audacious handling, recommending
his mv^{2}/2 only because of its convenience as compared with mv^{2}!
And it is as a matter of convenience that mechanicians have adopted mv^{2}/2.
Only gradually was mv^{2}/2 also proved mathematically. Naumann
(*Allg. Chemie* [*General Chemistry*], p. 7) gives an algebraical
proof, Clausius (*Mechanische Wärmetheorie* [*The Mechanical
Theory of Heat*], 2nd Edition, p. 18), an analytical one, which is then
to be met with in another form and a different method of deduction in Kirchhoff
(*ibid*., p. 27) Clerk Maxwell (*ibid*., p. 88) gives an elegant
algebraical proof of the deduction of mv^{2}/2 from *mv*.
This does not prevent our two Scotsmen, Thomson and Tait, from asserting
(*ibid*., p. 164): “The *vis viva* or kinetic energy of a moving
body is proportional to the mass and the square of the velocity conjointly.
If we adopt the same units of mass as above (namely, unit of mass moving
with unit velocity) there is a *particular advantage* in defining
kinetic energy as *half* the product of the mass and the square of
the velocity.” Here, therefore, we find that not only the ability to think,
but also to calculate, has come to a standstill in the two foremost mechanicians
of Scotland. The particular advantage, the convenience of the formula,
accomplishes everything in the most beautiful fashion.

For us, who have seen that *vis viva* is nothing but the capacity
of a given quantity of mechanical motion to perform work, it is obvious
on the face of it that the expression in mechanical terms of this capacity
for work and the work actually performed by the latter must be equal to
each other; and that, consequently, if mv^{2}/2 measures the work,
the *vis viva* must likewise be measured by mv^{2}/2. But
that is what happens in science. Theoretical mechanics arrives at the concept
of *vis viva*, the practical mechanics of the engineer arrives at
the concept of work and forces it on the theoreticians. And, immersed in
their calculations, the theoreticians have become so unaccustomed to thinking
that for years they fail to recognise the connection between the two concepts,
measuring one of them by mv^{2}, the other by mv^{2}/2,
and finally accepting mv^{2}/2 for both, not from comprehension,
but for the sake of simplicity of calculation!
[2]

1. We get no
further by consulting Clerk Maxwell. The latter says (*Theory of Heat*,
4th edition, London, 1875, p. 87): “Work is done when resistance is overcome,"
and on p. 183, “The energy of a body is its capacity for doing work."
That is all that we learn about it. [*Note by F. Engels.*]

2. The word
“work” and the corresponding idea is derived from English engineers. But
in English practical work is called “work,” while work in the economic
sense is called “labour.” Hence, physical work also is termed “work,”
thereby excluding all confusion with work in the economic sense. This is
not the case in German; therefore it has been possible in recent pseudo-scientific
literature to make various peculiar applications of work in the physical
sense to economic conditions of labour and *vice versa*. But we have
also the word “*Werk*” which, like the English word “work,” is excellently
adapted for signifying physical work. Economics, however, being a sphere
far too remote from our natural scientists, they will scarcely decide to
introduce it to replace the word *Arbeit*, which has already obtained
general currency - unless, perhaps, when it is too late. Only Clausius
has made the attempt to retain the expression “*Werk*,” at least
alongside the expression “*Arbeit.*” [*Note by F. Engels.*]