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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.<Elements of Chemistry by Antoine Lavoisier>[Part I, chapters III–V; the mercury-air investigation and the division and naming of atmospheric constituents]{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.<Commensurability, Comparability, Communicability by Thomas Kuhn>[Sections 3–4, pp. 673–679: chemical vocabulary, translation and interpretation]{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.<On the Electrodynamics of Moving Bodies by Albert Einstein>[Sections 1–3 on simultaneity, clock synchronization and coordinate transformation]{ref}
Nevertheless, quantitative comparison is possible. For an invariant rest <mass> $m$,
$$
p_{\mathrm{rel}}=\gamma mv,\qquad p_{\mathrm N}=mv,\qquad
\gamma=\frac{1}{\sqrt{1-v^2/c^2}}=1+\frac{v^2}{2c^2}+O(v^4/c^4).
$$
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 $v/c\ll1$, 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. \b[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.
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