= Solution
\b[There is no unconditional logical dependence of <Newton's first law> on <Newton's third law>.] There is, however, a dependence if one applies the first to an extended, isolated composite and interprets its <motion> as the <motion> of its <centre of mass>. The difference is between the behaviour of an elementary body with zero total <force> and a theorem about cancellation of internal <forces>. <Isaac Newton>'s own first-law discussion includes projectiles and spinning bodies, so the composite-body issue is physically significant.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Axioms, Laws I–III and Corollary III; distinguish impressed forces and the common centre of gravity]{ref}
For a constant positive <mass> in an <inertial frame>, <Newton's second law> gives $m\ddot{\mathbf x}=\mathbf F$. Setting the total <force> to zero yields uniform rectilinear <motion>; it uses no action–reaction assumption. The first law additionally identifies a common class of <inertial frames> in which such a statement applies. One cannot establish that class by arbitrarily straightening a single <trajectory>, and one cannot test the claim by defining every observed <acceleration> to be an unspecified <force>.
A concrete countermodel separates these issues. Suppose two bodies obey the second-law equations with internal <forces> $\mathbf F_{12}=\mathbf f$ and $\mathbf F_{21}=-2\mathbf f$, with nonzero constant $\mathbf f$. Every elementary body with zero total <force> still has constant <velocity>, but the two-body <momentum> satisfies
$$
\frac{d}{dt}(\mathbf p_1+\mathbf p_2)=-\mathbf f.
$$
Thus the elementary-body first law is compatible with failure of the third. Calling the pair isolated does not make its internal total <force> zero: that is precisely what has not been assumed. This countermodel establishes logical independence, not a proposal for a successful fundamental interaction.
Now strengthen the first-law claim: every isolated composite has an inertially moving <centre of mass> whenever the total <external force> vanishes. With $M=\sum_i m_i$, <Newton's second law> gives
$$
M\ddot{\mathbf R}=\sum_i\mathbf F_i^{\rm ext}
+\sum_{i<j}(\mathbf F_{ij}+\mathbf F_{ji}).
$$
For an isolated pair, the strengthened claim requires $\mathbf F_{12}+\mathbf F_{21}=0$. This is the weak, equal-and-opposite form of <Newton's third law>. It does not by itself require collinearity with the bodies' separation; the central-force condition is stronger and bears on <conservation of angular momentum>. For many bodies, cancellation of the sum does not alone imply pairwise cancellation without assumptions about the decomposition into interactions.
The right conclusion is therefore conditional. A first law for individual force-free bodies does not presuppose the third law. A first law for all isolated material composites, combined with the usual <centre of mass> and <Newton's second law>, imports an internal-<momentum>-balance condition. In theories with dynamical <fields>, material bodies can exchange <momentum> with those <fields>; <conservation of momentum> then concerns the combined system and need not imply instantaneous pairwise action–reaction between particles.
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