IV. The Measure of Motion - Work

• • Introductions
  • II. Dialectics
  • III. Basic Forms of Motion
  • IV. The Measure of Motion - Work
  • V. Heat
  • VI. Electricity
  • VIII: Tidal Friction, Kant and Thomson-Tait On the Rotation of the Earth and Lunar Attraction
  • The Part played by Labour in the Transition from Ape to Man
  • Natural Science and the Spirit World
  • Notes and Fragments
  • Appendices

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"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 mv²/2, 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 mv² 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 pcr 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 mv² tomv.

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 mv² 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 mv², 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; mv² 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. 102); "Thequantity 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 mv² 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 mv²/2, 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 mv²/2) 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 mvand mv² serve to determine two quite distinct processes, but we certainly knew long ago that mv² cannot equal mv, unless v=l. What has to be done is to make it comprehensible why motion should have a twofold measure, a thing which is surely just as unpermissible in natural science as in commerce. Let us, therefore, attempt this in another way.

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 mv². 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, 1x20=m'v', 4x5. On the other hand, if we let each of the weights, after it has been raised, fall freely to the original level, then the one, 1 kg., after falling a distance of 20 m. (the acceleration due to gravity is put in round figures =10 m. instead of 9,81 m.), attains a velocity of 20 m.: the other, 4 kg., after falling a distance of 5 m., attains a velocity of 10 m.

mv²=1 x 20 x 20 =400 =m'v'²=4x10x10=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 mv² 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 mv² 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 mv² after impact is subtracted from the sum of mv² before impact, there is under all circumstances a positive remainder. By this amount (or the half of it, according to the notation adopted) the vis viva is diminished owing both to the mutual penetration and to the change of form of the colliding bodies. The latter is now clear and obvious, but not so the first assertion that the sum of mv remains the same before and after impact. In spite of Suter, vis viva is motion, and if a part of it is lost, motion is lost. Consequently, eithermv here incorrectly expresses the quantity of motion, or the above assertion is untrue. In general the whole theorem has been handed down from a period when there was as yet no inkling of the transformation of motion; when, therefore, a disappearance of mechanical motion was only conceded where there was no other way out. Thus, the equality here of the sum of mv before and after impact was taken as proved by the fact that no loss or gain of this sum had been introduced. If, however, the bodies lose vis viva in internal friction corresponding to their inelasticity, they also lose velocity, and the sum of mv after impact must be smaller than before. For it is surely not possible to neglect the internal friction in calculating mv, When it makes itself felt so clearly in calculating mv².

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 mv² is still found. Here, therefore, mv and mv² conflict, and they do so by the difference of the mechanical motion that has actually disappeared. Moreover, the calculation itself shows that the sum of mv² expresses the quantity of motion correctly, while the sum of mv expresses it incorrectly.

Such are pretty nearly all the cases in which mv is employed in mechanics. Let us now glance at some cases in which mv² is employed.

When a cannon-ball is fired, it uses up in its course an amount of motion that is proportional to mv², 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 mv². Similarly, mv² serves wherever it is necessary to calculate the mechanical force required for overcoming a resistance.

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, mv²/2. 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 mv²/2, 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 mv²/2.

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; mv²/2 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. 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 destroyedand 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 mv². 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=mv²/2 and Helmholtz proposes " to take the magnitude mv²/2 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 vivainto its absolute measure, and remains quite unconscious of the important discovery he has made by his audacious handling, recommending his mv²/2 only because of its convenience as compared with mv²! And it is as a matter of convenience that mechanicians have adopted mv²/2. Only gradually was mv²/2 also proved mathematically. Naumann (Allg. Chemie [General Chemistry], p. 7) gives an algebraical proof, Clausius (Mechanische Wärmetheorie [The Mechanical Theory of Heat], 2nd Cdition, 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 mv²/2 from mv. This does not prevent our two Scotsmen, Thomson and Tait, from asserting (ibid., p. 168): " 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 mv²/2 measures the work, the vis viva must likewise be measured by mv²/2. 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 mv², the other by mv²/2, and finally accepting mv²/2 for both, not from comprehension, but for the sake of simplicity of calculation!