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Classical mechanics describes the motion of bodies by assigning positions and velocities and specifying how their interactions change those quantities. A point mass is the simplest model: its size and internal structure are ignored, leaving a mass and a moving position. A rigid body retains orientation and rotational kinetic energy. Choosing a model means deciding which features affect the question being asked; a sliding block may be treated as a point mass, whereas predicting its rolling without slipping requires its moment of inertia.

Three complementary questions organize the subject. Newton's second law asks how the net force determines the instantaneous acceleration. The work-energy theorem asks how that force, acting through a displacement, changes kinetic energy. Conservation of mechanical energy asks which combinations of position and velocity are accessible when the interactions admit a time-independent potential energy and no other force supplies net power. These descriptions are compatible, but answer different questions: an energy calculation may find a maximum speed without first solving for position at every time.

The harmonic oscillator below makes this unity visible. A restoring force produces a repeating position trace; the corresponding phase plane has a closed ellipse; kinetic energy and potential energy exchange while their sum stays fixed. Lagrangian mechanics and Hamiltonian mechanics provide further ways to organize the same dynamics, especially when coordinates or constraints make individual forces awkward to track.

The scope of classical mechanics also matters. Newtonian mechanics approximates motion at speeds small compared with the speed of light; special relativity changes that description when this approximation fails. Quantum mechanics is needed for phenomena for which a classical assignment of simultaneous position and momentum is inadequate. These limitations do not prevent classical mechanics from being a useful predictive model within its regime. ^[ref]

For Newtonian mechanics with forces depending only on relative positions, a common Galilean transformation leaves relative positions and accelerations unchanged. Internal mechanical observations therefore cannot discriminate rival choices of a wholly uncoupled privileged uniform rest state. The qualification concerns uniform motion; it does not imply that rotation or acceleration is unobservable, or that special relativity makes identical predictions.

Free fall is motion under gravity alone: there is no supporting normal force, engine thrust or appreciable drag force. In the Newtonian mechanics model, a test particle of mass $m$ obeys $m\ddot{\mathbf{r}}=m\mathbf{g}(\mathbf{r})$, hence $\ddot{\mathbf{r}}=\mathbf{g}(\mathbf{r})$. The mass cancels. Bodies released at the same position with the same velocity follow the same path in this model, even if their gravitational weights differ. ^[ref]

A dropped stone is one example; a coasting satellite is another. Free fall does not require motion straight down or decreasing distance to the attracting body. An ideal circular orbit has constant radius while its velocity continually turns under centripetal acceleration. The circular orbit as continuous free fall connects the familiar drop with orbital motion.

Near a surface, treating the gravitational acceleration as constant gives a useful first prediction. Over larger heights, the inverse-square law changes the acceleration during the fall. Radial free fall from rest uses conservation of mechanical energy to account for that change exactly, within the specified Newtonian gravitational field. The contrast matters: the simplest formula is an approximation to a physical model, rather than a law with the same numerical $g$ everywhere.

History also makes the idealization tangible. Galileo's Two New Sciences (1638), Third Day, Theorem II, derives distance proportional to the square of time for uniformly accelerated motion from rest; its first corollary gives successive distances in the ratios $1,3,5,7,\dots$ over equal time intervals. This is a kinematic statement, predating the gravitational inverse-square law. ^[ref]

On Apollo 15's final lunar surface excursion, David Scott released a hammer and a feather together. The near-vacuum removed the large difference in drag force familiar on Earth. NASA's report describes their arrival together within the accuracy of the release; this demonstration illustrates the free fall prediction without establishing infinite experimental precision. ^[ref]

A geometric background for Newtonian mechanics consisting of a four-dimensional affine space, an absolute time covector $t_a$, a rank-three positive spatial inverse metric tensor $h^{ab}$ with $t_a{h}^{ab}=0$, and a flat torsion-free affine connection preserving both. A trajectory parametrized by absolute time has tangent $u$ with $t_a{u}^a=1$. Its absolute acceleration is ${\nabla }_{u}u$; an unforced particle obeys ${\nabla }_{u}u=0$. The structure distinguishes inertial motion without selecting an absolute state of uniform rest. Additional force laws, including a Newtonian gravitational force when wanted, specify a dynamical theory on this background.

In Newtonian mechanics, inertial mass is the coefficient in $\mathbf{F}=m_{\mathrm{i}}\mathbf{a}$, measuring a body's response to a net force. The weak equivalence principle makes its ratio to passive gravitational mass universal; choosing compatible units then sets the two equal.

An internal force is a force between members inside the chosen system boundary. Changing that boundary can change which forces are internal. Pairwise Newtonian mechanics interactions cancel through third-law interaction pairs in the system's total momentum balance.

In Cartesian spatial coordinates and absolute time $t$, the gravitational potential $\Phi (t,x)$ determines an affine connection by the displayed coefficients, all others zero. The geodesic equation with affine time is $d^2{x}^i/d{t}^2=-{\partial }_i\Phi$. It preserves $dt$ and the spatial inverse metric $h^{ij}={\delta }^{ij}$, reproducing freely falling Newtonian mechanics in a curved spacetime connection.

Newton–Cartan theory describes Newtonian mechanics geometrically using absolute time, a degenerate spatial inverse metric, and a compatible torsion-free affine connection. It is not a theory with a nondegenerate Lorentzian metric. Gravitationally accelerated particles are geodesics of that connection, whose curvature measures tidal gravity.

In the pair-interaction model of Newtonian mechanics, two bodies exert equal and opposite forces on each other:Both vectors are expressed in the same inertial frame at the same time. The subscripts name the agent and the recipient: $A$ acts on $B$, and $B$ acts on $A$. These are forces on different bodies, so they cannot cancel in the net force on either body. Newton's second law still uses all the forces on the particular body whose acceleration is sought. Identifying this third-law interaction pair does not require the bodies to be isolated: additional external forces can act at the same time.

The distinction makes Newton's third law useful both for drawing a free-body diagram and for deriving momentum conservation. A body-by-body calculation retains one member of each third-law interaction pair; a calculation for a collection containing both bodies cancels that pair in the total momentum balance. Equal interaction forces consequently coexist with nonzero, generally unequal accelerations.

Isaac Newton stated the equality and opposition of mutual actions in Law III of his 1687 Philosophiae Naturalis Principia Mathematica. The explanation distinguishes equal opposite changes of momentum from changes of velocity, which are inversely proportional to the bodies' masses when other influences are absent. Corollary III then explains why mutual interactions preserve the signed total momentum. The equations here are modern notation for that distinction. ^[ref]

The instantaneous body-pair equation has a scope: use it for the mechanical interactions assumed by this Newtonian mechanics model. In electromagnetism, bodies can exchange momentum with an electromagnetic field; their instantaneous mutual forces need not sum to zero. Momentum conservation then accounts for electromagnetic field momentum as well as matter, and for momentum crossing the chosen boundary. This complication occurs in classical electromagnetism, without invoking quantum mechanics.^[ref]

The time-translation Killing vector field of the Schwarzschild metric gives a conserved Killing energy per unit rest mass,For unit rest mass, this is the total energy relative to infinity, including rest energy. At release $dr/d\tau =0$, so the four-velocity normalization gives $dt/d\tau =f(r_0{)}^{-1/2}$ andSubstitution into the radial four-velocity normalization yieldsChoosing the inward square root and evaluating at the ground provesThis identity is exact within the assumed Schwarzschild metric, despite its familiar Newtonian mechanics form. The quantity $dr/d\tau$ is not exactly the locally measured speed. A Schwarzschild static observer measures $v_{\mathrm{loc}}=(dr/dt)/f=(dr/d\tau )/\sqrt{f(r_0)}$. Thus the distinction disappears in the stated weak-field approximation, giving $v_{\mathrm{loc}}^2\simeq 2gh$ for $h\ll R_E$.

Steven Shapin's challenge is persuasive as a criticism of a single, uniform founding event; it is much less persuasive if read as denying substantial changes in knowledge. His opening provocation introduces an argument about the diversity of early modern practices, purposes and standards. It should be assessed as a historical interpretation, rather than treated as a literal assertion that nothing happened. The useful question is whether the Scientific Revolution names an explanatory pattern or imposes an artificial unity on unlike developments.^[ref]

There is a strong case for retaining the category in a qualified form. Compare Aristotle's programme of explaining natural motion through the nature of bodies and different kinds of cause with Isaac Newton's organization of classical mechanics around definitions, Newton's laws of motion and mathematical consequences of specified forces. Newton's law of universal gravitation brought terrestrial falling bodies and planetary motion into one quantitative framework. That changed both the content of explanations and the problems they could solve: the same relation could constrain orbits, lunar motion and terrestrial gravitational acceleration. This is a substantial transformation even though Newtonian mechanics was later revised.^[ref][ref]

Instrumentation likewise altered what could count as evidence. Robert Boyle's air pump attached a transparent receiver to a mechanical means of removing air. Its value was not merely that an observer saw an unusual event. Successive operations could vary the condition of the enclosed gas, while observers compared such effects as the expansion of a partly inflated bladder and changes in combustion or animal breathing. The apparatus made contrasts that ordinary unaided observation could not easily produce. Yet its results depended on the quality of seals, the operator's skill and decisions about whether a trial had worked. Experiment was an organized achievement rather than nature speaking without mediation.^[ref]

The difficulty for an unqualified revolution narrative is that these changes did not have one method. Francis Bacon emphasized the disciplined ordering of instances and cautioned against premature generalization. Galileo Galilei deliberately reasoned about idealized motion, then connected its mathematical relations to particular trials. Isaac Newton built geometrical arguments concerning forces and used astronomical phenomena to select their application. These are related challenges to inherited authority, but not interchangeable versions of a universal method introduced on one date. Nor does a successful mathematical treatment of planetary motion establish an equally sudden transformation of all biology or medical practice.^{[ref][ref][ref]}

Continuities also matter. Euclid's Elements furnished a demonstrative tradition that early modern authors could adapt rather than invent. Existing astronomy supplied observations and problems without which a new celestial mechanics would have had little to explain. The Philosophiae Naturalis Principia Mathematica by Isaac Newton itself is predominantly geometrical in presentation: identifying modernity solely with replacement of geometry by symbolic calculus would misdescribe a central work of the supposed revolution. Equally, an investigator's religious commitments need not have disappeared for mathematical or experimental work to develop. The Philosophiae Naturalis Principia Mathematica by Isaac Newton's General Scholium combines claims about the limits of physical explanation with natural theology.^[ref][ref]

These qualifications expose a geographical and social selection problem. If the historian defines the event by a few successful European texts in astronomy and classical mechanics, then concludes that Europe suddenly acquired the scientific mentality, the selection has already determined much of the result. Robert Boyle's apparatus depended on skilled construction and operation; printed demonstrations depended on readers able to understand and assess them. A history of disembodied ideas misses the work that made claims credible. Conversely, locating that work in a social setting does not invalidate the measured relationships or make every contemporary theory equally successful.

The Scientific Revolution is defensible as a retrospective name for interconnected, uneven transformations, especially in astronomy, classical mechanics and experimental practice. It is misleading as a single moment when a universal scientific essence replaced ignorance. Steven Shapin's provocation improves the category by forcing its boundaries and assumptions into view. Abandoning the capitalized label is optional; abandoning the expectation of one origin, one method and one uniformly affected society is the more important historical correction.

Take $0\le \gamma \le 1$ for the coefficient of restitution. Conservation of energy in the initial vacuum fall givesThe fixed plate has zero velocity, so the restitution rule makes the first upward rebound speed $u_1=\gamma u_0$. The ratio of rebound to impact kinetic energy is ${\gamma }^2$; henceGravity preserves the speed magnitude between departure from and return to the same plate height. Therefore the upward speed after the $n$th bounce is ${\gamma }^n{u}_0$. The geometric bounce sequence hasThe total travelled distance for $\gamma <1$ is the first fall plus ascent and descent for every rebound. Summing the geometric series givesThe idealized infinitely many impacts accumulate in finite time, after which the limiting state is rest on the plate; $\gamma =0$ gives rest at the first impact.

For the atmospheric fall, use upward signed velocity $u$ and height $z$. Newtonian mechanics with quadratic drag givesBefore impact $u\le 0$, so $\dot{u}=-g+(\alpha /m)u^2$. Set $v=-u\ge 0$, $v_T=\sqrt{mg/\alpha }$, and $\beta =\sqrt{\alpha g/m}=g/v_T$. Separation gives $\mathrm{artanh}(v/v_T)=\beta t$, establishing the descent from rest under quadratic drag:The speed approaches the terminal velocity $v_T$, although the formula is used only until the first impact. This occurs when $\cosh (\beta t_{\mathrm{hit}})=e^{\alpha h_0/m}$. In particular, the atmospheric impact speed is $v_T\sqrt{1-e^{-2\alpha h_0/m}}$, and both speed and height formulas recover the vacuum fall as $\alpha \to 0$.

Outside a spherical planet, Newtonian mechanics gives the inverse-square central force equationThe satellite's mass cancels. The angular momentum is conserved, so the orbit lies in a plane. In that plane the general equations in polar coordinates areA synchronous circular orbit fixed above a surface point must be equatorial, prograde and have angular velocity $\omega =2\pi /T$. Substitution of constant $r$ into the radial equation givesThe radius is measured from the planet's centre; an external orbit requires it to exceed the planet's radius.

Now let the release speed be $v=(1+\eta )v_s$, with $0<\eta \ll 1$, in the same tangential direction. At release $\dot{r}=0$, butThus the satellite initially moves outward. Its specific total energy is $v^2/2-GM/r_s<0$ for this small increase, so the Kepler orbit is an ellipse, with the release point its pericentre. To make this explicit, put $b=v^2/v_s^2\in (1,2)$. The radial turning-point equation from energy and angular momentum conservation has rootsIts semi-major axis is $a=r_s/(2-b)>r_s$. The speed follows from the same conserved energy:It is greatest at pericentre and least at apocentre. In particular, the tangential speed at the two turning points satisfies $r_{\min }{v}_{\max }=r_{\max }{v}_{\min }$. The orbital period becomes $2\pi \sqrt{a^3/(GM)}>T$, so the orbit is no longer fixed over the same surface point.

On the horizontal road the total normal reaction is $Mg$. The kinetic friction force has magnitude $\mu Mg$, so Newtonian mechanics gives $\dot{u}=-\mu g$ until rest. ThusFor the wet high-speed stage, add the four tyre drags: the total linear drag is $\lambda u$, not $\lambda u/4$. Hence $M\dot{u}=-\lambda u$ and $u(t)=U{e}^{-\lambda t/M}$. At the switching speed $u_{\ast }=U/4$, the elapsed time and distance areThe remaining stage has the same constant kinetic friction deceleration as on a dry road, but starts at $U/4$. It therefore contributes time $U/(4\mu g)$ and distance $U^2/(32\mu g)$. Adding both stages of the two-stage stopping under linear drag and dry friction givesThe exponential first stage reaches its positive switching speed in finite time; it need not reach zero under linear drag.

The strong equivalence principle asserts universality of free fall even for bodies with gravitational self-binding, together with independence of local experimental outcomes, including gravitational experiments, from the location and velocity of a freely falling laboratory. “Local” means that external tidal variation across the apparatus can be neglected. It does not mean that all curvature, or a laboratory's own gravitational field, can be removed. The weak principle concerns freely falling test bodies without appreciable self-gravity; laboratory composition tests alone therefore do not establish the strong version.

Lunar laser ranging supplies evidence specifically sensitive to the strong version. The Earth and Moon have different fractions of gravitational self-energy. If those fractions affected gravitational and inertial mass differently, their accelerations toward the Sun would differ and the lunar orbit would acquire a periodically varying polarization, the Nordtvedt effect. Combining lunar ranging with laboratory bounds on composition-dependent differential acceleration separates this self-energy effect. A determination available before this examination found the self-energy contribution to the Earth–Moon difference in $m_G/m_I$ to be $(-2.0\pm 2.0)\times 10^{-13}$, consistent with zero; see the 2004 experimental analysis. Universality-of-free-fall experiments, gravitational redshift measurements and tests of local Lorentz invariance support the weaker constituent principles, but should not be confused with a direct self-gravity test.

In general relativity, all matter couples to one metric tensor, and freely falling test particles follow its geodesics. At any event, Riemann normal coordinates give $g_{ab}={\eta }_{ab}$ and ${\Gamma }^a{}_{bc}=0$. Nongravitational local equations reduce to their special-relativistic forms in that frame. Gravitational binding is governed by the same metric field equations rather than a separate composition-dependent force, and the local gravitational coupling is universal. These features implement the strong equivalence principle. Curvature still governs geodesic deviation, so this construction removes uniform acceleration, not tidal gravity.

To geometrize Newtonian mechanics, use a four-dimensional manifold with coordinates $(t,x^1,x^2,x^3)$, absolute time ${\tau }_a=(dt{)}_a$ and a degenerate spatial inverse metric $h^{ij}={\delta }^{ij}$, $h^{0a}=0$. For a gravitational potential $\Phi (t,x)$ define the torsion-free Newton–Cartan gravitational connection byIt satisfies ${\nabla }_a{\tau }_b=0$ and ${\nabla }_a{h}^{bc}=0$. An affine geodesic has $d^{2}t/d{\lambda }^2=0$; choosing $\lambda =t$ makes its spatial equationsThis is Newton's free-fall law. The geometry is Newton–Cartan theory, with an affine connection and degenerate metric data, rather than a nondegenerate four-dimensional Lorentzian metric. The spatial slices themselves remain Euclidean.

Use the curvature conventionIn these coordinates the quadratic connection terms vanish, and the only nonzero curvature components, apart from antisymmetry in $c,d$, areIn particular $R_{00}={\nabla }^2\Phi$ and all other components of the Ricci tensor vanish. This Newtonian tidal curvature gives relative acceleration $\delta {\ddot{x}}^i=-{\partial }_i{\partial }_j\Phi \delta x^j$. Spatially constant gravitational acceleration can be removed by an accelerating frame because its Hessian vanishes. The Poisson equation for Newtonian gravity becomes $R_{ab}=4\pi G\rho {\tau }_a{\tau }_b$. Fluid mechanics is compatible with the same connection: the covariant continuity equation is the usual mass-conservation equation, and the spatial Euler equation is $u^b{\nabla }_b{u}^i=-{\rho }^{-1}{\partial }_{i}p$, which includes the gravitational force through the connection.

Now replace this absolute-time geometry by the Lorentzian metric of general relativity, with signature $(-,+,+,+)$. A perfect fluid haswhere $\epsilon$ is rest-frame energy density. In units $c=1$, $\epsilon \simeq \rho$ for a nonrelativistic fluid. The Newtonian equation suggests a covariant curvature equation sourced by $T_{ab}$. However $R_{ab}=\kappa T_{ab}$ alone conflicts with generic stress-energy conservation, because the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Among expressions linear in Ricci curvature, conservation therefore selectsA constant multiple of the metric is also divergence-free, giving the possible cosmological constant term. This is a heuristic selection using locality, universal metric coupling and second-order metric equations; Newtonian matching by itself does not prove uniqueness among every possible gravity theory.

To fix the coefficient, first set the cosmological constant to zero. The trace-reversed equation would read $R_{ab}=\kappa (T_{ab}-\frac{1}{2}T{g}_{ab})$. In the fluid rest frame, $T=-\epsilon +3p$, so $R_{00}=\frac{\kappa }{2}(\epsilon +3p)$. In the weak static Newtonian limit, $g_{00}=-(1+2\Phi )$ and $R_{00}\simeq {\nabla }^2\Phi$. For negligible pressure the Poisson equation for Newtonian gravity then requires $\kappa /2=4\pi G$. Restoring $c$ uses $x^0=ct$, $R_{00}\simeq {\nabla }^2\Phi /c^2$ and $T_{00}\simeq \rho c^2$, giving $\kappa =8\pi G/c^4$. ThusPressure contributes to relativistic active gravity, while the low-pressure weak-field limit recovers Newton's Poisson equation. The comparison fixes the Newtonian normalization but does not fix $\Lambda$, which is an independent parameter.

For a comoving fluid volume, $V\propto a^3$ and $E=\rho c^{2}V$. The first-law relation $dE=-PdV$ gives $c^2(Vd\rho +\rho dV)=-PdV$. Since $\dot{V}/V=3\dot{a}/a$, the cosmological perfect-fluid continuity equation isFor the Newtonian energy argument, assume negligible pressure, spherical homogeneity, Newtonian gravity and constant enclosed mass. A test particle at physical radius $r=a\chi$ has constant specific energyDivide by $a^2{\chi }^2/2$ and set $k=-2ϵ/(c^2{\chi }^2)$ to obtain the Friedmann equationHomogeneity makes $k$ independent of the chosen comoving shell. In the relativistic interpretation it specifies the spatial-curvature sign: positive, zero or negative $k$ gives closed spherical, flat or open hyperbolic spatial geometry, with a conventional rescaling allowing $k=1,0,-1$. Newtonian mechanics itself derives the energy constant, not the spatial metric. When pressure is appreciable, the general relativistic equations still have the displayed Friedmann form, but the constant-mass Newtonian test-particle derivation cannot be applied without that qualification.

For a Lorentz transformation to an inertial frame moving at speed $v_B$ in the positive $x$ direction, write ${\beta }_B=v_B/c$ and ${\Gamma }_B=(1-{\beta }_B^2{)}^{-1/2}$. Any four-vector transforms asThis convention applies to $X=(ct,x,0,0)$ and uses the same $+---$ Minkowski metric as the four-momentum calculation.

The definition of proper time gives $d\tau =dt\sqrt{1-u^2/c^2}=dt/\gamma$ for $|u|<c$. Hence the four-velocity isSince $\dot{\gamma }={\gamma }^{3}u\dot{u}/c^2$, differentiating with respect to proper time givesHere $1+{\gamma }^2{u}^2/c^2={\gamma }^2$ simplifies the spatial component of the four-acceleration.

At each event, boost with $v_B=u$ to the instantaneous rest inertial frame. Then $A^{\prime 0}=0$ and $A^{\prime 1}=\gamma {\gamma }^4\dot{u}(1-u^2/c^2)={\gamma }^3\dot{u}$. Thus the signed proper acceleration for one-dimensional motion isThe last equality follows by differentiating $\gamma u$; its magnitude is the usual scalar proper acceleration.

If this signed proper acceleration is constant and the particle starts from rest, integration gives $\gamma u=at$, soIntegrating once more and imposing $x(0)=0$ yields the hyperbolic motion worldlineFor $a=0$ the continuous limiting solution is $x=0$. Finally, the Taylor series of the square root givesThis recovers constant-acceleration Newtonian mechanics at low speed while the exact relativistic speed remains less than $c$.

A molecule is an electrically neutral bound assembly of more than one atom. It can contain identical atoms, as in ${\mathrm{O}}_2$, or different atoms, as in ${\mathrm{H}}_{2}O$. Discrete molecules are not a universal description of matter: an extended ionic crystal does not consist of separate neutral molecules corresponding to each empirical formula unit. The IUPAC definition also specifies binding sufficient to support a vibrational state.^[ref]

Elasticity is a body's tendency to recover its original configuration when the deforming force is removed. Within a linear regime, Hooke's law makes stress proportional to strain, but elasticity does not require exact linearity. The elastic limit matters: permanent plastic deformation is a different response.

Inertia is the property expressed by Newton's first law: without a resultant external force, a body's velocity remains constant in an inertial frame of reference. Mass measures translational inertia. It is not a force acting against motion, and a moving body does not require a continuing forward force merely to keep moving.

A force describes an interaction that changes momentum. In Newtonian mechanics,A force has magnitude and direction; its point of application also matters for the torque and hence the body's rotational response.

The volume $V$ is the three-dimensional space occupied by the body. Its mass $m$ measures its inertia: in Newtonian mechanics the resultant force is $m\mathbf{a}$. Its gravitational weight is the force exerted on it by gravity,For a uniform body, $m=\rho V$, where $\rho$ is its mass density; more generally $m=\int \rho dV$. Volume can change with temperature or pressure while the body's mass remains fixed in the classical closed-system model. Equal volumes of different materials need not have equal masses.

The same mass need not have the same gravitational weight. The local gravitational acceleration changes between planets and with position on Earth. Outside a spherical Earth, $g(r)=G{M}_{\oplus }/r^2$, so increasing altitude reduces it. Earth's rotation and shape also affect terrestrial weighing: a scale at rest on the rotating Earth responds to the effective acceleration including the centrifugal contribution, rather than simply to a spherical gravitational field.

Distinguish gravitational weight from apparent weight. A supporting scale measures a contact force. In a vertically accelerating cabin, upward acceleration $b$ gives $N=m(g+b)$ while contact is maintained. During common free fall, $N=0$ although $mg$ remains nonzero. Immersion in a fluid can similarly reduce the support needed through buoyancy, without removing the gravitational attraction. These distinctions explain why a change in a balance reading is not automatically a change of mass.

The transition from phlogiston theory to an oxygen theory of combustion, and the transition from Newtonian mechanics to special relativity, support a restricted account of incommensurability. They show changes in conceptual organization and standards of explanation. They do not establish that rival scientists cannot communicate, compare results or give reasons for preferring a new account.

In the chemical case, a common phlogistic treatment makes a metal a calx combined with phlogiston, so calcination removes the principle responsible for its metallic character. Antoine Lavoisier's treatment reverses the decomposition: a metal combines with part of the air to form the calx. In a closed vessel, the calx's gain in mass must be understood together with what happens to the remaining gas. The mercury-and-air investigation and the recovery of the gas from the calx organize these operations into a different account of composition. This is a change in the relations between elements, compounds and chemical processes, not merely a new name for an unchanged substance.^[ref]

For Thomas Kuhn, a translation which replaces every occurrence of phlogiston with one modern substance fails to preserve those interconnected roles. Yet historians can learn the earlier vocabulary and reconstruct the relevant experiments. His later account explicitly distinguishes interpretation and language-learning from a word-for-word translation into a fixed successor vocabulary. Local untranslatability does not imply that every term or observation is inaccessible to the other community.^[ref]

The mechanical example concerns the reorganization of space, time and motion. In special relativity, distant simultaneity depends on an inertial frame and a specified clock-synchronization procedure. Newtonian mechanics instead uses a common time independent of inertial motion. This affects more than numerical corrections to one law: it changes which events are simultaneous and how velocity and momentum are related.^[ref]

Nevertheless, quantitative comparison is possible. For an invariant rest mass $m$,Here $c$ is the speed of light and $\gamma$ the Lorentz factor. At $v=0.6c$, the respective momenta are $0.75mc$ and $0.6mc$, a directly comparable difference. For $v/c\ll 1$, the leading agreement explains an important domain of earlier success. It would be inaccurate to claim that modern special relativity invariably makes mass velocity-dependent: invariant rest mass remains invariant in this convention.

These examples therefore support conceptual discontinuity without total evidential isolation. Some explanatory aims and classifications change; many preparations, measurements and quantitative relations remain comparable. No unique algorithm follows from shared values such as accuracy and scope, but the absence of such an algorithm is weaker than the absence of rational assessment. The most defensible Kuhnian conclusion is local and historical: comparison can require learning a conceptual system, while still permitting criticism of both systems and empirically warranted preference between them.