About the Work
This book is a philosophical and historical inquiry into the principle of the conservation of energy, tracing its intellectual lineage from antiquity to the nineteenth century. The author, writing from the perspective of a physicist deeply engaged with the history of his discipline, presents his investigation as a corrective to the ahistorical manner in which scientific principles are often taught and understood. The work is structured as a sustained essay, combining close readings of original scientific texts with broader epistemological reflections on the nature of scientific concepts, the role of history in science education, and the relationship between physical theories and the human mind that constructs them.
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The author explicitly situates his own intellectual development within the work, noting that his views arose largely independently of his contemporaries, rooted instead in historical study. He acknowledges that his ideas, which date from the early 1860s, were often met with incomprehension by colleagues who lacked historical training. The book is thus both a contribution to the history of science and a methodological statement about how science ought to be practiced and understood. It is not a textbook of physics, nor a purely antiquarian chronicle, but rather a philosophical argument conducted through historical evidence, aimed at clarifying the logical status and proper interpretation of one of the most celebrated theorems of modern science.
The author’s approach is deliberately anti-metaphysical, though he is careful to define what he means by this. He argues that concepts become “metaphysical” only when we forget how we came to hold them; the remedy is to keep in view the path by which our ideas were formed. The book is therefore an exercise in intellectual genealogy, seeking to recover the concrete, historical circumstances in which the conservation principle emerged, and in doing so, to reveal its true logical character. The work is addressed not only to physicists but to anyone concerned with the nature of scientific knowledge, and it draws freely on examples from mechanics, thermodynamics, electricity, and even music to illustrate its points.
Context
The book emerges from a specific moment in the history of physics when the conservation of energy had become a central organizing principle, yet its foundations remained philosophically obscure. The author writes against a backdrop of intense debate about the nature of physical explanation, particularly the growing dominance of the mechanical worldview, which sought to reduce all phenomena to the motions of atoms and molecules. He positions himself as a critical observer of this trend, sympathetic to the achievements of the mechanical program but deeply skeptical of its claims to ultimate truth.
The author’s own intellectual formation is relevant to understanding his approach. He acknowledges the influence of Kant and Herbart on his early thinking, though he insists that his mature views arose independently, through historical study rather than philosophical speculation. He describes the isolation he felt in the 1860s, when his ideas were too strange for fruitful discussion with colleagues, and he notes the rare exception of Josef Popper, with whom he could converse without horrified opposition. By the time of writing, however, the situation had changed: “renowned investigators” had entered the field, and the author offers his contribution as a way of classifying the questions at issue.
The book is also a response to a pedagogical crisis. The author observes that the historical method has unique value in teaching, because it reveals the contingency and variability of scientific views. He invokes Heraclitus’s image of the river: one cannot step into the same stream twice, and likewise science is never finished or fixed. Textbooks, he argues, have failed to capture this dynamic quality, presenting science as a static collection of results rather than a living, evolving enterprise. The author advocates for what he calls a “classical education” for the scientist—not the traditional study of Greek and Latin, but knowledge of the historical development of one’s own science. This, he insists, is the true antidote to dogmatism and the foundation of genuine scientific understanding.
The scientific context is equally important. The author engages with a rich tradition of mechanical and dynamical thought, from Stevinus and Galileo through Huygens and d’Alembert to the nineteenth-century founders of thermodynamics. He is concerned with the relationship between the principle of excluded perpetual motion and the more formal statement of energy conservation, and he argues that the former is historically and logically prior. He also addresses the contemporary debates about the nature of heat, electricity, and matter, challenging the assumption that the mechanical conception of physics is the only possible one. His reflections on the conventionality of scientific measurement and the arbitrary division of phenomena into “substances” and “forces” place him in dialogue with the emerging positivist tradition, though he resists easy categorization.
Structure of the Book
The book opens with a preface in which the author explains the origins and aims of his investigation. He describes his intellectual isolation, his debt to historical study, and his hope that his contribution will help clarify the questions surrounding the conservation principle. He also offers a defense of the historical method in science education, arguing that knowledge of the development of scientific ideas is essential for genuine understanding.
The main body of the work is devoted to a historical survey of the principle of conservation of work, as the author prefers to call it. He begins with Simon Stevinus, whose work on the equilibrium of bodies on inclined planes in the early seventeenth century contains one of the earliest explicit appeals to the impossibility of perpetual motion. The author examines Stevinus’s famous cord-and-balls argument and his hydrostatical principle that water preserves its place, showing how these served as foundations for deriving laws of equilibrium. He then turns to Galileo, analyzing the pendulum experiments described in the Dialogues and their application to inclined planes and the law of inertia. Galileo’s insight that a body’s momentum acquired in descent is sufficient to carry it to the same height on any path is presented as a crucial step toward the conservation principle.
The narrative continues with Huygens, who generalized Galileo’s principle and used it to solve the problem of the center of oscillation. The author examines Huygens’s hypotheses about inertial motion and his assertion that the common center of gravity of a system of weights cannot rise above its initial position. He then traces the subsequent development of these ideas through James Bernoulli, d’Alembert, and the Bernoullis, showing how the principle of virtual velocities emerged as a generalization of the excluded perpetual motion. The author pays particular attention to Lagrange’s treatment of these matters, noting both his historical inaccuracies and the tacit reliance of his proofs on the very principle he sought to demonstrate.
The historical survey is followed by a series of philosophical reflections on the nature and scope of the conservation principle. The author examines the application of the principle beyond mechanics, first to heat through the work of S. Carnot and Clausius, and then to electricity through the investigations of Neumann and Helmholtz. He introduces a crucial distinction between the “quantity” of a substance and its “capacity for work,” arguing that our choice of what to measure is historically conditioned and conventional. This leads to a broader critique of the mechanical worldview and its assumption that all physical phenomena can be reduced to spatial motions of molecules. The author concludes with a statement of the principle in abstract form: for one group of phenomena to be a source of work on another, the first group must itself be undergoing continual variation. This, he argues, is the true logical root of the conservation principle, which is nothing other than a special form of the law of causality.
The book closes with a series of notes in which the author addresses specific points in greater detail. These include a discussion of the law of inertia and its relativity to a frame of reference, a proposed redefinition of mass and force, a generalization of the second law of thermodynamics, and reflections on the spatial representation of atoms and the economy of thought in science. The notes reveal the author’s engagement with contemporary debates and his willingness to challenge orthodox positions.
Inside the Book
The Author’s Apology and Method
The author begins by clarifying the provenance of his ideas. Except for his reading of Kant and Herbart, his considerations arose independently, grounded in historical studies rather than in the prevailing theoretical debates of his colleagues. He confesses that discussions with able colleagues often failed because they attributed his “strange” views to confusion, since historical studies were not widely cultivated among physicists. These thoughts date from 1862 and were ill-suited for collegial debate; he published little on them, only short notices in journals rarely read by physicists. In a footnote, he reveals that he knew only one man, Josef Popper, with whom he could discuss these views without horrified opposition, and that Popper and he arrived at similar conclusions on many points of physics independently.
Now that renowned investigators have entered this province, he offers his contribution to classifying the questions. He protests against the investigation being considered metaphysical. Concepts are called metaphysical, he argues, when we forget how we reached them; one cannot lose one’s footing if one keeps in view the path by which one came. The pamphlet contains straightforward reflections on facts belonging to natural science and history.
He then argues for the value of the historical method in teaching. If history taught only the variability of views, it would already be invaluable. Heraclitus’s words apply to science: “One cannot go up the same stream twice.” Attempts to fix the moment via textbooks have failed; we should grow accustomed to science being unfinished and variable. Whoever knows only one view does not believe another stood in its place or will succeed it; he neither doubts nor tests. The value of classical education is not eight years of declining and conjugating, but knowing another eminent nation’s viewpoint so as to put ourselves in a different position. The essence of classical education is historical education. But if this is correct, our idea is too narrow: not the Greeks alone concern us, but all cultured people of the past. For the investigator of nature, there is a special classical education: knowledge of the historical development of his own science. He urges his readers not to let go of history’s guiding hand, for history has made all, and history can alter all. He hopes his historical investigation will not prove too tedious.
On the History of the Theorem of the Conservation of Work
The next section is titled “On the History of the Theorem of the Conservation of Work.” The law of conservation of energy holds a prominent place in modern science, so the question of its validity obtrudes itself. The author calls it the law of conservation of work because that name is understood by all and prevents wrong ideas. He recalls Faraday’s misunderstandings on the “law of the conservation of force” and a controversy rich in obscurities. One should say “law of the conservation of force” only when, with J. R. Mayer, one calls “force” what Euler called “effort” and Poncelet “travail.” Mayer, who did not get his concepts from the schools, cannot be faulted for using his own names.
The theorem is usually expressed in two forms: first, ½Σmv² − ½Σmv₀² = Σ(Xdx + Ydy + Zdz); second, it is impossible to create work out of nothing, or to construct a perpetual mobile. This theorem is considered the flower of the mechanical view of the world, the highest and most general theorem of natural science. The author will show two things. Firstly, the theorem in the second form is not as new as is believed; almost all eminent investigators had a confused idea of it, and since Stevinus and Galileo it has served as the foundation of important extensions of physical science. Secondly, the theorem does not stand or fall with the mechanical view of the world; its logical root is deeper in our mind.
For the first part, proof must come from original sources. Lagrange, in his historical introductions to the Mécanique analytique, repeatedly refers to the theorem’s development, but consulting the originals shows the theorem did not play the part Lagrange’s exposition suggests. The facts, except for a few, coincide with Lagrange’s, but the author derives another view from passages given in full.
Stevinus and the Principle of Excluded Perpetual Motion
The author emphasizes some key points. Simon Stevinus, in Hypomnemata mathematica, Tom. IV, De statica, 1605, treats the equilibrium of bodies on inclined planes. Over a triangular prism ABC, with side AC horizontal, an endless cord with fourteen equally spaced, equal-weight balls is slung. Stevinus concludes that the four balls on AB hold in equilibrium the two balls on BC, since if equilibrium were disturbed, the cord would move forever—perpetual motion. Stevinus’s Latin quote says the ring would return to its original position, the eight left globes would again be heavier than the six right, they would sink again, and the globes would keep up continuous unending motion, which is false. From this principle, Stevinus derives the laws of equilibrium on inclined planes and other consequences.
In the “Hydrostatics” chapter, Stevinus sets up the principle: “Aquam datam, datum sibi intra aquam locum servare”—a given mass of water preserves its place within water. This is demonstrated by noting that if A sinks to D, the water succeeding A will also flow to D, A will be forced out, and the body of water will set up a perpetual motion, which is absurd. From this, all the principles of hydrostatics are deduced. Stevinus also first develops the thought that equilibrium is not destroyed by rigid connexions. The principle of the conservation of the centre of gravity is now deduced from d’Alembert’s principle with that remark. A modern reproduction would change the demonstration slightly: we can imagine uniform unending motion if hindrances are removed, but would protest accelerated motion or uniform motion with resistances. The string of balls could be replaced by a heavy homogeneous cord of infinite flexibility. This does not affect the historical value of Stevinus’s thoughts: he deduces simpler truths from the principle of impossible perpetual motion.
Galileo’s Pendulum and the Law of Inertia
The passage then presents Galileo’s thought experiment with a lead ball weighing two or three ounces, hanging from a vertical wall by a fine thread AB, four or five feet long. A horizontal line DC is marked perpendicular to the vertical AB. When the thread takes position AC and the ball is released, it descends through arc CB and rises through arc BD almost to the level of line CD, prevented from reaching it exactly by air and thread resistance. Galileo concludes that the impetus at point B from descent through arc CB suffices to urge the ball through a similar arc BD to the same height.
After repeating this experiment, Galileo proposes driving a nail into the wall at point E or F, projecting five or six inches, so that when the thread returns to position AB, it strikes the nail. The ball is then compelled to move up arc BG described about centre E. The ball rises to the horizontal line at point G. If the nail is placed lower at F, the ball describes arc BJ, always terminating its ascent precisely at line CD. If the nail is so low that the thread below it cannot reach height CD, the thread winds about the nail.
Galileo’s reasoning is as follows. Since arcs CB and DB are equal and similarly situated, the momentum acquired descending arc CB equals that acquired descending arc DB. The momentum acquired at B by descent through CB can drive the ball up arc BD; therefore momentum from descent DB equals that driving the ball from B to D. Generally, every momentum acquired descending an arc equals that causing ascent through the same arc. All momenta causing ascent of arcs BD, BG, BJ are equal, since they are produced by the same momentum from descent CB, so all momenta acquired descending arcs DB, GB, JB are equal.
Galileo applies the pendulum remark to the inclined plane. A body starting from rest at A descending inclined plane AB acquires velocity proportional to time increments. Velocity at B is the greatest of acquired velocities, immutably impressed once causes of new acceleration or retardation are removed. On the horizontal plane GH, equable motion continues ad infinitum.
Huygens, the Centre of Oscillation, and the Bernoullis
Huygens sharpened the law of inertia and generalized Galileo’s principle of heights of ascent. He used it to solve the centre of oscillation problem and recognized it as identical with the principle of excluded perpetual motion. His hypotheses state that without gravity or atmospheric resistance, a body would continue forever with equable velocity in a straight line. In his Horologium oscillatorium, part four, he asserts that weights set in motion by gravity cannot have their common centre of gravity rise higher than its initial position—meaning simply that heavy bodies do not move upward. He notes that perpetual motion is utterly impossible by mechanical means; Mach suggests a possible Jesuitical mental reservation in the phrase “mechanical means.”
Huygens’s Proposition IV states that a pendulum of several weights, after completing part of an oscillation, with the individual weights’ connexions dissolved and velocities turned upward, will have its common centre of gravity carried to the same altitude as before the oscillation began.
Lagrange characterized Huygens’s principle as precarious; James Bernoulli in 1681 reduced the theory of the centre of oscillation to the lever laws. This problem occupied seventeenth- and eighteenth-century inquirers, leading to d’Alembert’s 1743 principle in the Traité de dynamique, previously used differently by Euler and Hermann. The Huygenian principle became the foundation for the “law of conservation of living force” enunciated by John and Daniel Bernoulli, used in Daniel’s Hydrodynamics; their theorems differ from Lagrange’s expression in the Analytical Mechanics only in form.
Torricelli’s law of efflux assumed that liquid flowing from a vessel’s basal orifice cannot by its efflux velocity ascend higher than its level in the vessel.
The Principle of Virtual Velocities
The principle of virtual velocities was first enunciated by Stevinus, not Galileo as Lagrange asserted, in his Trochleostatica: “As the space of the body acting is to the space of the body acted upon, so is the power of the body acted upon to the power of the body acting.” Galileo recognized its truth in simple machines and deduced liquid equilibrium laws from it. Torricelli traced it to the properties of the centre of gravity: equilibrium in a simple machine requires that the common centre of gravity not sink; conversely, if it cannot sink, equilibrium obtains. John Bernoulli in 1717 first perceived its universal import for all systems, stated in a letter to Varignon. Lagrange gave a general demonstration using pulleys and weights, but this rests on Huygens’s and Torricelli’s remarks—fundamentally on the impossibility of perpetual motion.
Poinsot observed that seeking a demonstration of the principle of virtual velocities brought back all the difficulties; Lagrange’s work offered nothing clearer than an analytical procedure. A general demonstration would establish mechanics on an entirely different basis. Mach notes that mathematicians find it uncomfortable that dynamics is historically founded on statics, when statical theorems should derive from dynamical principles.
Gauss, Lagrange, and Carnot
Mach reports that Gauss, in presenting his principle of least constraint in Crelle’s Journal, expressed a desire that while science develops gradually from the easy to the difficult, the simple to the complex, and the special to the general, once the mind reaches a higher viewpoint it demands the reverse—that all statics appear as a special case of mechanics. Gauss’s own principle possesses universality but is not immediately intelligible, and Gauss deduced it using d’Alembert’s principle, which left matters unchanged.
Mach describes how Lagrange’s proof of the principle of virtual motion impressed him differently as a student versus after historical research. Initially it seemed insipid due to the pulleys and cords that did not fit the mathematical view. After studying the history of science, Mach finds it a beautiful demonstration. The principle of excluded perpetual motion accomplishes nearly everything that displeased Lagrange, yet Lagrange had to employ it tacitly in his own demonstration.
S. Carnot, in his Réflexions sur la puissance motrice du feu of 1824, established that whenever work is performed by heat, a quantity of heat passes from a warmer to a colder body, absent permanent alteration of the acting body. The quantity of heat flowing from temperature t to temperature t₁ for a definite performance of work cannot depend on the chemical nature of the bodies, only on the temperatures. Otherwise a combination generating work from nothing could be imagined. Mach calls this the first extra-mechanical application of the principle of excluded perpetual motion. Carnot considered the quantity of heat invariable.
Clausius later found that with the performance of work, heat not merely flows from t to t₁, but part proportional to the work performed is lost. He derived that Q, the heat transformed into work, relates to Q₁, the heat flowing from absolute temperature T to T₁.
The Mechanical Theory of Heat and Its Limits
Mayer, Helmholtz, and W. Thomson confirmed that heat vanishes with work and is formed with the expenditure of mechanical work; experiments by Rumford, Joule, Favre, Silbermann and others supported this. The conclusion spread that heat consists in mechanical motion. Mach quotes J. R. Mayer: just as from the connexion between falling tendency and motion one cannot conclude that the essence of falling is motion, neither does this hold for heat—rather one might conclude that motion must cease to be motion to become heat.
Neumann made the second extra-mechanical application for the laws of electrical induction—perhaps the most talented work of this kind. Helmholtz attempted to carry the conservation of work through all physics in two ways: from the theorem that work cannot be created from nothing, or by considering physical processes as molecular processes produced by central forces alone, that is, forces with a potential.
Mach quotes Wundt’s axioms from his 1866 tract Die physikalischen Axiome: all causes in nature are motional causes; every motional cause lies outside the object moved; all motional causes act along the straight line of junction; the effect of every cause persists; to every effect corresponds an equal counter-effect; every effect is equivalent to its cause. Mach regards these as mechanical only, not physical.
The principle of excluded perpetual motion cannot be founded on mechanics, since its validity was felt before mechanics existed. The mechanical conception of physics suffers from doubtful anticipation and one-sidedness.
Analogies and the Conventionality of Substance
Mach offers analogies: water performs work by falling from a higher to a lower level, its quantity remaining constant; wood swelling with dampness bursts rocks, as the Egyptians used it; electricity flows from higher to lower potential, its quantity constant; vis viva transfers from a higher to a lower velocity-level, and vis viva decreases. The last analogy breaks down completely.
There is no mechanical equivalent of quantity of electricity, but there is one of electrical energy; there is no mechanical equivalent of water, but of weight times distance of descent. With Leyden jars, we assume electricities unite rather than disappear. The difference in treating heat versus electricity is purely historical, wholly conventional, and wholly indifferent.
In 1785, Coulomb’s torsion balance showed that with two similarly electrified balls A and B at distance r with repulsion p, bringing ball C into contact with B and measuring repulsions yields the sum again as p—something remained constant. In 1838, Riess’s thermoelectrometer showed that the heat produced by jar discharge is proportional to q²/s, the energy, not to the quantity of electricity by Coulomb’s measure. A partial discharge loses capacity for work, though Coulomb’s measure shows the quantity unaltered.
If Riess’s thermometer had been invented before Coulomb’s torsion balance, the “quantity” of electricity in a jar would naturally be measured by the heat it produces. In that case, this quantity would decrease when work is performed, making electricity appear to be a motion, not a substance. Our current notion of electricity as a substance is therefore purely historical, accidental, and conventional.
The same logic applies to other physical things. Water does not disappear when work is done because we measure its quantity with scales. But if we measured water’s “quantity” by its capacity for work, using a mill, that quantity would also disappear as work is performed. Many substances are not as easily measured by scales as water, leaving other modes of measurement.
For heat, the historically established measure of “quantity” is accidentally its work-value. Thus its quantity disappears when work is done. However, this does not prove that heat is not a substance, nor that it is one. In Black’s case, the quantity of heat remains constant because the heat passes into no other form of energy. If someone today wished to think of heat as a substance, they could assume that what we call quantity of heat is the energy of a substance whose quantity remains unaltered but whose energy changes. It might be better to say “energy of heat” instead of “quantity of heat.”
This reflection makes the peculiar character of the second principal theorem of the mechanical theory of heat vanish. The author has shown elsewhere that this theorem can be applied to electrical phenomena by putting “potential” instead of “quantity of heat” and “potential function” instead of “absolute temperature.” Astonishment at the discovery that heat is motion is misplaced, as that discovery was never made; it is irrelevant for scientific purposes whether we think of heat as a substance.
Substance, Oxygen, and the Limits of the Mechanical View
A physicist deceiving himself with his own notation would behave like musicians who think a piece in six flats must sound different from one in six sharps, having forgotten how notation arose. The author states that heat is a substance just as much as oxygen is, and is not a substance just as little as oxygen is. Substance is possible phenomenon, a convenient word for a gap in our thoughts. The concept “soul” is irrelevant to investigators, but matter is an abstraction of exactly the same kind. We know as much about the soul as we do of matter.
When exploding oxygen and hydrogen in a eudiometer-tube, the phenomena of oxygen and hydrogen vanish, replaced by water. We say water consists of oxygen and hydrogen, but these are merely thoughts or names kept ready to describe phenomena not present but which will reappear when we decompose water. Oxygen is the same case as latent heat: both can appear when they cannot yet be remarked. If latent heat is not a substance, oxygen need not be one. The indestructibility of matter cannot be urged against this; rather, we should say “conservation of weight,” which is a pure fact with nothing to do with theory.
The author maintains that in investigating nature, we deal only with knowledge of the connexion of appearances. What we represent behind appearances exists only in our understanding, having the value of a memoria technica or formula, whose form is arbitrary and varies with our culture. If we hold onto the new laws connecting heat and work, it does not matter how we think of heat itself. But if a way of presentation is so inflexible that it hinders following phenomena, it should not be used.
This happens in the mechanical conception of physics, which reduces all phenomena to equilibrium and movement of molecules and atoms. According to Wundt, all changes of nature are mere changes of place, and all causes are motional causes. Wundt holds that change of place is the only change in which a thing remains identical with itself; qualitative change would imply annihilation and creation, irreconcilable with the identity of the object and the indestructibility of matter. But the Eleatics encountered the same difficulties with motion—can we not imagine a thing destroyed in one place and an exactly similar thing created in another? It is a bad sign that the mechanical view supports itself on such preposterous, thousands-of-years-old things. Ideas of matter made at a lower stage of culture must be given up if unsuitable for higher knowledge.
Suppose all physical events reduce to spatial motions of molecules. We then suppose that things which can never be seen or touched, existing only in imagination, can have the properties of tangible things—imposing the limitations of the visible and tangible on creations of thought. Other senses have analogous forms of perception, such as the tone-series for hearing, corresponding to a one-dimensional space. We do not think of all things as sounding or figure molecular events musically, though we are as justified in this as in thinking spatially. This shows an unnecessary restriction.
The author demonstrates a drawback of this limitation with a table. A system of n points is determined in a space of r dimensions if e distances between pairs of points are given. The table gives e for different values of r: for r=1, e₁=2n−3 and e₂=n−1; for r=2, e₁=3n−6 and e₂=2n−3; for r=3, e₁=4n−10 and e₂=3n−6; for r=4, e₁=5n−15 and e₂=4n−10; generally, e₁=(r+1)n − (r+1)(r+2)/2 and e₂=rn − r(r+1)/2. The e₁ column is used when conditions about the sense of distances are made; e₂ when only absolute magnitude is given. Between n points, n(n−1)/2 distances are thinkable, generally more than a space of any given dimensionality can accommodate, which shows how the spatial representation of molecular systems imposes restrictions that are not inherent in the phenomena themselves.
The General Theorem of Excluded Perpetual Motion
The passage opens by establishing a broad, abstract theorem: for any group of phenomena (x, y, z) to serve as a continuous source of work—that is, to cause continual variation in another group (a, β, γ)—the first group itself must be undergoing continual variation. This formulation is presented as the clearest and most unambiguous expression of the principle of excluded perpetual motion. Crucially, this theorem is not confined to mechanics; it applies universally to all phenomena, with the mechanical principle of excluded perpetual motion being merely one special case within this wider framework.
The text immediately cautions, however, that this remark cannot be inverted. It is entirely possible to conceive of systems in a state of continual variation that produce no corresponding change in another group of phenomena; such systems are described as being “shut up in themselves.” Only direct experience can teach us how such groups are divided—which phenomena depend on one another and in what manner. The abstract law of causality, by itself, offers no guidance on this matter. Without the support of positive experience, the theorem of excluded perpetual motion is as empty and formal as the law of sufficient reason or any other purely logical principle. History demonstrates that the theorem has found fruitful application in physics only as positive knowledge has advanced: first in mechanics, then in the theory of heat, and finally in the theory of electricity. Abstract theorems alone lead nowhere, and the text quotes Poinsot to reinforce this point: “Rien ne vous dispense d’étudier les choses en elles-mêmes, et de nous bien rendre compte des idées qui font l’objet de nos spéculations” (Nothing dispenses you from studying things in themselves, and from giving a clear account of the ideas that are the object of our speculations).
Several concrete examples illuminate the theorem. The vibrations of a tuning-fork are periodical variations; they can serve as a lasting source of work only if they undergo a lasting variation themselves, such as the gradual diminution of their amplitude. Indeed, we hear a tuning-fork only because its vibrations are decreasing. Similarly, a rotating top can perform work only if its angular velocity is decreasing. The mere side-by-side placement of a copper and a zinc plate generates no electric current, for there is no continual variation in the system; but if a continual chemical change occurs, the production of an electric current becomes plausible.
The text then provides an example illustrating why the inversion of the theorem is impermissible. A top protected from all resistance can rotate uniformly without becoming a source of work. Its angular velocity remains constant, even though its angle of rotation varies continually. This does not contradict the principle, because the varying angle of rotation is not a source of work. Experience alone teaches us that only variations of velocity, not variations of position, can become sources of other variations. However, it would be a mistake to think that the top’s variation of position is connected to no other variation whatsoever—it is, in fact, connected to the increasing angle of rotation of the earth. This observation leads to a peculiar conception of the law of inertia, which the passage declines to discuss further at this point.
The Principle as a Form of the Law of Causality
The principle of excluded perpetual motion is fruitful for an experienced investigator navigating known territory, but it is useless in unexplored areas of experience. People place great value on the fact that the sum of the store of work and the vis viva, or energy, remains constant. While this “commercial or housekeeping” expression is convenient and well-suited to human nature, which is inherently economical, there is nothing essentially more profound in this law than in any other law of nature. The law of causality supposes a dependence between phenomena; it matters little how the equations representing this dependence are written. An equation can be expressed as f(a, β, γ) = 0, or as a = φ(β, γ), or as P(a, β, γ) = const., with no special wisdom inherent in the last form. The conservation of work differs from other laws only in this formal respect; any law can be cast in a similar form, such as Mariotte’s law written as log p + log v = const. The passage expresses no enthusiasm for the mysticism that some have attempted to attach to this theorem.
The passage concludes that the theorem of excluded perpetual motion is merely a special form of the law of causality, which itself results from the fundamental supposition of the dependence of phenomena—a supposition that precedes all scientific investigation. This theorem is unconnected with the mechanical view of nature but is consistent with any view that retains a strict rule by laws. The riches that investigators have accumulated over time include actual knowledge, superseded theories, and useless philosophemes; reviewing these treasures allows us to discard what is worthless. The true object of natural science is the connexion of phenomena; theories are like dry leaves that fall away when they cease to serve as the lungs of the tree of science.
Notes on the Law of Inertia
The passage then shifts to a series of notes. The first note discusses the law of inertia, quoting Newton’s formulation from the Principia. Since Newton’s time, the law has attained the dignity of a papal dictum, accepted without question. The author remarks that he noted a great indefiniteness in its formulation: it is not stated with respect to which body direction and velocity are determined. He first drew attention to this difficulty in his lectures in the summer of 1868. C. Neumann later discussed the same point and found identical difficulties. The author’s solution differs from Neumann’s, who considered all motion to be absolute by positing a hypothetical body. The author argues that it does not matter whether we say the earth turns or is at rest while the celestial bodies revolve around it; these are geometrically the same relative rotation. If the earth is at rest, there is no flattening at the poles and no Foucault’s pendulum experiment. Neumann preferred to speak of absolute motion; the author prefers to say that the law of inertia is wrongly expressed. It must be conceived in such a way that both suppositions give the same result, with due regard paid to the masses of the universe. In ordinary terrestrial cases, reckoning relative to a tower or the corner of a room suffices; in astronomical cases, the fixed stars suffice. But a system of coordinates has value only if it is determined by bodies. The author compares this to paper money: not needing a definite piece of money does not mean that the currency need not be funded at all. If the stars showed noticeable movements, choosing a reference star would cease to be immaterial; since none could be preferred, the influence of all would have to be considered. If the whole heavens moved in confusion, how could the law be applied? All bodies, each with its share, are important. A free body acted on by an instantaneous couple moves with its central ellipsoid rolling on a tangent-plane; this is motion in consequence of inertia. The distant heavenly bodies have their role in this conception.
On Immediate Evidence and the Law of Inertia
Mach criticizes the tendency to assert propositions as “immediately evident.” The law of inertia is often claimed to be such a proposition, requiring no experiential proof. Mach insists that it “can only have grown out of experience.” If masses imparted velocities (rather than accelerations) dependent on distance, there would be no law of inertia; only experience can determine which state of affairs actually holds. If we had only heat sensations, there would be merely “equalizing velocities” that vanish with temperature differences.
Regarding the language of cause and effect in the motion of masses, one may say “The effect of every cause persists” just as correctly as “Cessante causa cessat effectus”—it is merely a verbal distinction. If we call the resulting velocity the “effect,” the first proposition holds; if we call the acceleration the “effect,” the second holds. Similarly, attempts to deduce the parallelogram of forces a priori must tacitly assume that forces are independent of one another—which makes the derivation superfluous.
The Conception of Mass
Mach identifies two difficulties in the foundations of mechanics: first, the unsuitable arrangement of its first conceptions and theorems; second, the silent passing over of important presuppositions. The usual definitions of mass, such as m = P/g and P = mg, involve either a circle or require conceiving force as “pressure,” which is problematic when statics precedes dynamics. Newton’s principle “Actioni contrariam semper et aequalem esse reactionem” requires “actio” to mean pressure, or it is unintelligible without prior conceptions of force and mass. Pressure seems strange at the head of modern phoronomical mechanics, though it is avoidable.
If only one kind of matter existed, the law of sufficient reason would show that two completely similar bodies impart only equal and opposite accelerations to each other. Assuming the mutual independence of forces: body A (containing m bodies of type a) and body B (containing m’ bodies of type a) give accelerations φ and φ’; then φ:φ’ = m’:m. Mass means the number of times a body contains the unit body a; accelerations vary inversely as masses. Experimentally, m/m’ = φ’/φ (attending to sign). This definition can apply to different kinds of matter, but we cannot know a priori whether other comparison bodies or forces would yield different mass values. Chemical combination ratios (a:b, a:c) do not guarantee b:c beforehand—only experience teaches that bodies behaving as equal masses to a third also behave so to each other.
With gold versus lead, the law of sufficient reason fails; both might accelerate in the same direction, yielding negative masses. However, the transitivity of equal mass is likely, since its contrary would contradict the law of conservation of work. Mach illustrates this with three bodies A, B, C on a smooth fixed ring: if A and B, and A and C, behave as equal masses, then B and C must also; otherwise vis viva would increase with each revolution—contradicting known facts. With mass thus defined, force as the product of mass and acceleration remains valid; Newton’s law becomes a mere identity. Weight provides a convenient measure of mass, under the two suppositions mentioned. Mach proposes this arrangement: first, a theorem of experience—bodies communicate opposite accelerations along their line of junction (which includes the law of inertia); second, a definition—equal mass means equal and opposite accelerations; third, a theorem of experience—mass-values remain unaltered with other forces and comparison bodies; fourth, a theorem of experience—accelerations are mutually independent (which includes the parallelogram of forces); fifth, a definition—force is mass-value times acceleration. This note is dated Prague, November 15, 1867.
Generalization of Thermodynamics and Other Notes
A further note reproduces an 1871 note (drawn up in 1870) generalizing the second law of thermodynamics: replacing quantity of heat with “potential” and absolute temperature with “potential function,” the equation P/T + P'(1/T’ − 1/T) = 0 applies beyond heat phenomena. For the impact of inert masses, vis viva is the potential and velocity the potential-function.
In 1862, Mach wrote a physics compendium rigorously applying mechanical atomic theory, which revealed its insufficiency. Studying psychophysics and Herbart convinced him that spatial intuition is bound to sensory organization; we cannot ascribe spatial properties to things not perceived by the senses. He stated that atoms need not be thought of spatially. Herbart’s arbitrary limitation of dimensions struck him; relations like space with any number of dimensions are thinkable. Attempts to explain spectra mechanically strengthened this view, though he dared not speak candidly before orthodox physicists. These views were first communicated in lectures in the summer of 1864 and the winter of 1864–65, and published in 1865–66; Riemann’s paper on the subject (1867) was unknown to him at the time.
Mach has long held that science concerns convenience and the saving of thought; from the economist E. Herrmann he adopted the view that “Science has a problem of economy or thrift.” Quoting his own 1866 essay, he discusses the idea of imagining a “something” behind matter with states (e.g., pressure); physics expresses mutual action as a function of distance—but spatial relations are recognized only through forces. The fundamental law of force need not contain spatial relations but only dependence between states. For the universe, “all positions are functions of time” means all positions depend on one another; since positions are recognized only through states, all states depend on one another. “Physical space” containing time is nothing other than this web of interdependence.
The Classification of Sciences and Final Notes
The main text continues with a numbered classification of sciences. Physics is defined as the discovery of laws governing the connection of sensations (perceptions). Psychophysics is defined as the clear establishment of laws governing the connection of sensations and presentations. If these laws of connection are considered mathematical, establishing them presupposes that everything they embrace is measurable. The text acknowledges that much remains wanting in this regard. Fechner, in his Psychophysik, succeeded in measuring even single sensations, but doubt remains as to the meaning of this measure. A sensation of greater intensity is always also of another quality, making Fechner’s measure more physical than psychical. However, these difficulties are said to be not insurmountable.
The author’s notes to the second edition (1909) address various points. Confusion from using “force” in different significations appears in a communication by Faraday of 1857, in a paper “On the Conservation of Force”; many eminent investigators of that time committed the same fault. The law of inertia was treated at length in Mach’s Mechanics, where all the relevant literature is noticed; the last important work known to Mach is J. Petzoldt’s article “Die Gebiete der absoluten und relativen Bewegung.” Further developments appear in Mach’s Mechanics. Publications with analogous considerations include Mach’s Mechanics; Josef Popper’s Die physikalischen Grundsätze der elektrischen Kraftübertragung (1884); Helm’s Die Lehre von der Energie (1887); Wronsky’s Das Intensitätsgesetz (1888); and Mach’s own “Geschichte und Kritik des Carnot’schen Wärmegesetzes” (1892) and Wärmelehre. Such considerations were first mentioned after Carnot by Zeuner in 1866. Mach held the double resolution MV²/2, MV²V/2 possible, retaining “velocity” rather than “velocity squared” as Ostwald later did in 1892. Mach soon recognized that the potential-level is a scalar V²/2, not a vector V or V/2. Popper gave a sufficient exposition of the correspondence between masses and quantities, as did Friedrich Wolfgang Adler in 1905.
Spaces of many dimensions seem not essential for physics; Mach would uphold them only if atoms are indispensable and the freedom of working hypotheses is upheld. The economy-of-thought principle is developed in Mach’s later writings. Mach repeatedly expressed that physics’ foundation may be thermal or electric, in his Mechanics and Analysis of the Sensations. Space and time are not independent entities but forms of dependence of phenomena on one another; Mach subscribes to the principle of relativity, upheld in his Mechanics and Wärmelehre. The sensationalistic standpoint was attained through physiology-of-senses studies, developed in his Bewegungsempfindungen (1875), Analysis of the Sensations, and Erkenntnis und Irrtum. Nervous, subjectivistic apprehensions about the physics of Mars inhabitants are groundless.
The translator’s notes add context: Kant’s Prolegomena influenced Mach when he was fifteen. The investigators referred to are not Kirchhoff and Helmholtz, whose works appeared later; Kirchhoff is still regarded by many as a pioneer of descriptive physics. References to Stevinus, Galileo, Huygens, d’Alembert, vis viva, Torricelli, and virtual velocities are found in Mach’s Mechanics and other works. Gauss’s paper is reprinted in Ostwald’s Klassiker. From Wärmelehre, the translator cites Carnot’s principle, Mayer and Joule’s principle, and their uniting by W. Thomson and Clausius. Many-dimensional spaces serve as mathematical helps; H. Weber’s edition of Riemann used a particle in n-dimensional space to represent Hertz’s “position of a system.” On impact and gravitation theories, the translator refers to J. B. Stallo’s The Concepts and Theories of Modern Physics. Clerk Maxwell’s “General Maxim of Physical Science” resembles Fechner’s law of causality, and Bertrand Russell discusses causality in The Principles of Mathematics. The translator attempted to formulate Mach’s principle of causality mathematically, discussing analytic functions, Taylor’s theorem, and nature’s “uniformity” through mathematical models, with references to Poincaré, Kleinpeter, Boltzmann, Petzoldt, and Ward following.
Main Arguments
The central argument of the book is that the principle of the conservation of energy, understood as the impossibility of perpetual motion, is not a recent discovery but has been implicitly recognized and used by the greatest investigators since the early seventeenth century. The author demonstrates through detailed historical analysis that Stevinus, Galileo, Huygens, and their successors all appealed to the impossibility of perpetual motion as a foundational principle, even when they did not state it explicitly. This historical continuity, the author argues, reveals that the principle is not a product of the mechanical worldview but has a deeper logical root in the human mind.
A second major argument concerns the relationship between the conservation principle and the mechanical conception of nature. The author insists that the principle does not stand or fall with the attempt to reduce all phenomena to motions of molecules. On the contrary, the principle is consistent with any view of nature that maintains strict rule by laws. The mechanical worldview is presented as one possible framework among others, not as a necessary foundation for the conservation principle. The author supports this claim by showing how the principle has been applied successfully beyond mechanics, first to heat and then to electricity, and by arguing that these applications do not require any particular theory of the underlying nature of these phenomena.
The author also argues for the conventionality of scientific concepts and measurements. He develops this point through a detailed comparison of heat and electricity. The quantity of heat, he notes, is measured in such a way that it appears to be conserved when no work is done, but decreases when work is performed. The quantity of electricity, by contrast, is measured in a way that remains constant even when work is done. The author shows that this difference is not grounded in nature but in the historical accident of which measuring instrument was invented first. If Riess’s thermoelectrometer had preceded Coulomb’s torsion balance, our conception of electricity would be entirely different. This argument is generalized to all physical quantities: what we choose to call the “quantity” of a substance is a matter of convention, not of nature.
A further argument concerns the nature of scientific explanation itself. The author maintains that in investigating nature, we deal only with knowledge of the connection of appearances. What we represent behind appearances—atoms, forces, substances—exists only in our understanding and has the value of a memoria technica, a convenient formula whose form is arbitrary and varies with our culture. The mechanical conception of physics, which reduces all phenomena to the equilibrium and movement of molecules, is criticized for imposing the limitations of the visible and tangible on creations of thought. The author argues that this restriction is unnecessary and potentially misleading, and he suggests that other modes of representation, such as those based on auditory or thermal sensations, are equally justified.
Finally, the author argues that the conservation principle, in its most abstract form, is a special case of the law of causality. For a group of phenomena to be a source of work on another group, the first group must itself be undergoing continual variation. This theorem, the author insists, is not specific to mechanics but applies to all phenomena. However, he is careful to note that the theorem cannot be inverted: not every system of continual variation produces work on another system. Experience alone can teach us which phenomena depend on one another and how. The law of causality, he concludes, says nothing about the specific divisions of phenomena; it merely supposes that phenomena are dependent on one another, a supposition that precedes all scientific investigation.
Key Concepts
The author introduces and develops several key concepts that structure his analysis. The most important is the principle of excluded perpetual motion, which he identifies as the historical and logical root of the conservation of energy. This principle, which states that it is impossible to create work out of nothing, is shown to have been used by Stevinus, Galileo, Huygens, and others as a foundation for deriving laws of equilibrium and motion. The author prefers to call the conservation law the “law of conservation of work” rather than “conservation of force” or “conservation of energy,” because this name is understood by all and prevents the wrong ideas that arise from ambiguous terminology.
The concept of “work” itself is central to the author’s analysis. He traces the historical development of this concept, noting that different investigators used different terms for what we now call work: Euler called it “effort,” Poncelet “travail,” and Mayer “force.” The author insists on the importance of clear terminology, citing Faraday’s misunderstandings of the “law of conservation of force” as an example of the confusion that arises when terms are used loosely. He argues that the theorem of conservation of work can be expressed in two equivalent forms: one mathematical, involving the sum of masses times velocities squared, and one verbal, stating the impossibility of perpetual motion. The second form, he argues, is historically prior and logically more fundamental.
The author develops a distinction between the “quantity” of a substance and its “capacity for work.” This distinction is crucial for his argument about the conventionality of scientific measurement. He shows that what we call the quantity of heat is, historically, its work-value, and thus it decreases when work is performed. The quantity of electricity, by contrast, is measured by a method that does not track work-value, and thus appears constant. The author argues that these choices are arbitrary and historically conditioned, and he illustrates this with a thought experiment: if Riess’s thermoelectrometer had been invented before Coulomb’s torsion balance, we would naturally measure electricity by its heat-producing capacity, and electricity would appear to be a motion rather than a substance.
The concept of “mass” is subjected to critical scrutiny. The author identifies two difficulties with the usual treatment: the unsuitable arrangement of first conceptions and theorems, and the silent passing over of important presuppositions. He proposes a redefinition of mass in terms of the accelerations that bodies communicate to one another, arguing that equal masses are those that impart equal and opposite accelerations. This definition, he claims, avoids the circularity of the usual definitions and provides a firm foundation for the concept of force as the product of mass and acceleration. He also discusses the possibility of negative masses, though he notes that the transitivity of equal mass is likely to hold, since its contrary would contradict the conservation of work.
The author introduces the concept of “physical space” as distinct from the space of geometry. He argues that spatial relations are recognized only through forces, and that the fundamental law of force need not contain spatial relations but only dependence between states. This leads to a radical reconception of the role of space and time in physics: they are not independent entities but forms of dependence of phenomena on one another. The author connects this view to the principle of relativity, which he endorses, and to his broader sensationalist epistemology, according to which the laws of physics are laws governing the connection of sensations.
Finally, the author develops the concept of the “economy of thought” in science. He argues that science is concerned with convenience and the saving of thought, and he adopts from the economist E. Herrmann the idea that science has a problem of economy or thrift. This concept underlies his critique of the mechanical worldview: if a way of presenting phenomena becomes so inflexible that it hinders the following of phenomena, it should not be used. Theories, he suggests, are like dry leaves that fall away when they cease to be the lungs of the tree of science.
Themes
A recurring theme in the book is the relationship between history and scientific understanding. The author argues that knowledge of the historical development of scientific ideas is essential for genuine understanding, not merely for antiquarian interest. He suggests that the historical method reveals the contingency of scientific views, showing that what seems natural and inevitable to one generation was once novel and controversial. This historical consciousness is presented as a safeguard against dogmatism and a stimulus to further inquiry. The author’s own practice exemplifies this theme: he does not merely state the conservation principle but traces its development through the works of Stevinus, Galileo, Huygens, and others, showing how each investigator built on the insights of his predecessors.
Another important theme is the critique of the mechanical worldview. The author challenges the assumption that all physical phenomena must ultimately be explained in terms of the motions of molecules and atoms. He argues that this view imposes unnecessary restrictions on scientific thought, forcing us to think of all phenomena in spatial terms when other modes of representation are equally possible. He notes that we are as justified in thinking of phenomena in terms of sound or heat as in terms of space, and that the preference for spatial representation is a historical accident rather than a logical necessity. This critique is connected to his broader epistemological position that in investigating nature, we deal only with the connection of appearances, and that what we represent behind appearances is a matter of convention.
The conventionality of scientific concepts is a related theme that runs throughout the book. The author shows that our choice of what to measure, and how to measure it, is not dictated by nature but by historical circumstances. The distinction between “substances” and “forces,” the division of phenomena into conserved and non-conserved quantities, and even the choice of spatial representation for atoms are all presented as conventional decisions that could have been made differently. This theme is developed most fully in the author’s comparison of heat and electricity, where he shows that the different treatment of these phenomena is “purely historical, wholly conventional, and wholly indifferent.”
The theme of intellectual freedom is also prominent. The author repeatedly insists that scientists should not be bound by inherited ways of thinking if those ways prove unsuitable for higher knowledge. He criticizes the tendency to treat certain propositions as “immediately evident” and to elevate the law of inertia to the dignity of a papal dictum. He argues that ideas of matter made at a lower stage of culture must be given up if they are unsuitable for higher knowledge. This theme is connected to his advocacy of the historical method, which reveals that scientific views have changed in the past and therefore may change in the future.
A final theme is the unity of science and the continuity of its development. The author shows how the principle of excluded perpetual motion, first applied in mechanics, was extended to heat and then to electricity, and he argues that this extension was natural and inevitable given the logical character of the principle. He also emphasizes the continuity of scientific development, showing how each investigator built on the work of his predecessors, often without fully recognizing the implications of what he had done. This theme is balanced by an acknowledgment of the discontinuities and revolutions in scientific thought, as when the author notes that the theorem of conservation of work “does not stand or fall with the mechanical view of the world.”
The Author’s Conclusions
The author concludes that the principle of the conservation of energy, understood as the impossibility of perpetual motion, is not a recent discovery but has been implicitly recognized and used by the greatest investigators since the early seventeenth century. Through detailed historical analysis, he shows that Stevinus, Galileo, Huygens, and their successors all appealed to this principle as a foundation for their work, even when they did not state it explicitly. This historical continuity reveals that the principle has a deeper logical root than the mechanical worldview that later came to be associated with it.
The author argues that the principle of excluded perpetual motion is a special form of the law of causality, which results from the supposition of the dependence of phenomena—a supposition that precedes all scientific investigation. In its most abstract form, the principle states that for one group of phenomena to be a source of work on another group, the first group must itself be undergoing continual variation. This theorem is not specific to mechanics but applies to all phenomena, and it is consistent with any view of nature that maintains strict rule by laws. The author emphasizes, however, that the theorem cannot be inverted: not every system of continual variation produces work on another system. Experience alone can teach us which phenomena depend on one another and how.
The author concludes that the mechanical conception of physics, which reduces all phenomena to the equilibrium and movement of molecules and atoms, is not a necessary foundation for the conservation principle. He criticizes this worldview for imposing unnecessary restrictions on scientific thought and for treating as absolute what is merely conventional. He argues that what we represent behind appearances—atoms, forces, substances—exists only in our understanding and has the value of a convenient formula whose form is arbitrary and varies with our culture. The choice of what to measure and how to measure it is not dictated by nature but by historical circumstances, as shown by the different treatment of heat and electricity.
The author also concludes that the law of inertia, as usually stated, is imprecise because it does not specify with respect to which body direction and velocity are determined. He argues that the law must be conceived so that different suppositions about the state of motion of the earth give the same results, with regard paid to the masses of the universe. He compares the need for a frame of reference to paper money: not needing a definite piece of money does not mean it need not be funded at all. In ordinary cases, reckoning relative to a tower or a corner of a room suffices; in astronomical cases, the stars suffice. But a system of coordinates has value only if it is determined by bodies.
The author’s final reflections concern the nature of scientific knowledge itself. He maintains that the object of natural science is the connection of phenomena, and that theories are like dry leaves that fall away when they cease to be the lungs of the tree of science. He expresses no enthusiasm for the mysticism that some push forward with the theorem of conservation of work, noting that the “commercial or housekeeping” expression of the law is convenient and suited to human nature, which is economical, but that there is nothing essentially more in this law than in any other law of nature. The riches investigators have accumulated include actual knowledge, superseded theories, and useless philosophemes; reviewing these treasures allows us to discard the worthless and to see clearly the path by which our ideas were formed.
Significance
The book occupies an important place in the history and philosophy of science, representing a sustained attempt to understand a fundamental physical principle through historical investigation rather than purely formal analysis. The author’s method—tracing the development of the conservation principle through the works of Stevinus, Galileo, Huygens, and others—established a model for how the history of science could illuminate the logical status of scientific concepts. His demonstration that the principle of excluded perpetual motion was implicitly used by the founders of modern mechanics, long before it received formal statement, challenged the ahistorical presentation of physics that was common in textbooks and lectures.
The author’s critique of the mechanical worldview anticipated later developments in the philosophy of science, particularly the recognition that scientific theories are not unique descriptions of nature but conventional frameworks that could be replaced by alternatives. His argument that the choice of what to measure and how to measure it is historically conditioned, illustrated by the comparison of heat and electricity, was a pioneering contribution to the understanding of scientific measurement. His insistence that what we represent behind appearances has the value of a memoria technica, whose form is arbitrary and varies with our culture, pointed toward a pragmatic and instrumentalist view of scientific theories.
The author’s reflections on the law of inertia and the relativity of motion were also significant, contributing to the critical examination of Newtonian mechanics that would culminate in the theory of relativity. His argument that the law of inertia must be conceived so that different suppositions about the state of motion of the earth give the same results, with regard paid to the masses of the universe, anticipated the general principle of relativity. His proposal to redefine mass in terms of the accelerations that bodies communicate to one another, avoiding the circularity of the usual definitions, was a contribution to the foundations of mechanics that influenced later discussions of the concept of mass.
The book is also significant for its advocacy of the historical method in science education. The author’s argument that knowledge of the historical development of one’s science is a form of “classical education” for the scientist, essential for genuine understanding and a safeguard against dogmatism, has been influential in debates about the role of history in the teaching of science. His invocation of Heraclitus’s image of the river—one cannot step into the same stream twice—captures the dynamic, unfinished character of science that he sought to convey. The book remains a valuable contribution to the understanding of the conservation principle and a compelling demonstration of the power of historical analysis to illuminate the foundations of scientific knowledge.
