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The two endpoints of a nonzero line segment form a Euclidean Ramsey set. The finite configuration consists of those two points, rather than every point of the continuous line segment.

For a finite colouring with $q$ colours and a desired distance $d>0$, take the $q+1$ vertices of a regular simplex of side length $d$ in ${\mathbb{R}}^q$. The pigeonhole principle gives two vertices of the same colour. Every pair of distinct vertices has distance $d$, so this pair is an isometric copy of the prescribed endpoint pair. This also gives an explicit sufficient ambient dimension for every number of colours.

For an infinitely long thin strip of width $w$ carrying uniform longitudinal current $I$, the magnetic field a perpendicular distance $z>0$ above its centre has magnitude $B={\mu }_{0}I\arctan (w/(2z))/(\pi w)$. Its near-sheet limit is ${\mu }_{0}I/(2w)$ and its distant line-current limit is ${\mu }_{0}I/(2\pi z)$.

Matching constructs comparisons between units receiving different treatments but having similar pretreatment covariates. Similarity can be measured directly on selected covariates or through a propensity score. The design specifies the distance, allowed discrepancies, number of controls per treated unit and whether a control can be reused.

With one distinct control matched to each of $n_T$ treated units, a simple estimator isThis averages differences in the represented treated population. Dropping unmatched treated units changes that population, and reusing controls induces dependence between pair differences. Caliper matching explicitly limits allowed discrepancies.

Matching seeks covariate balance; it does not create a randomized controlled trial. Identifying a causal effect still requires conditional exchangeability given the matched pretreatment covariates, adequate positivity, and consistency in causal inference. Unmeasured determinants of assignment and outcome can remain different even when measured covariates are well balanced. Conditioning on a causal mediator or inducing collider bias is not repaired by calling the procedure matching.

For a freely moving unstable particle of fixed speed $v=\beta c$ and proper mean lifetime ${\tau }_0$, the mean travelled distance before decay is ${\ell }_d=v\gamma {\tau }_0$. Survival beyond distance $L$ has probability $e^{-L/{\ell }_d}$, since elapsed proper time is $L/(\gamma v)$. For rest mass $m$ and three-momentum magnitude $p$, ${\ell }_d=p{\tau }_0/m$.

The instantaneous distance from an attracting centre to an orbiting body. It is constant on a circular orbit but varies along an elliptic orbit. For Kepler's third law the relevant size is the semi-major axis, which equals the orbital radius only for a circular orbit.

The outer parallel body of a convex body $K\subset {\mathbb{R}}^n$ at distance $r\ge 0$ is $K_r=K+rB$, the Minkowski sum with the radius-$r$ closed ball. Equivalently it is the set of points at distance at most $r$ from $K$. A planar rectangle of sides $a,b$ expands into a rounded rectangle with area $ab+2r(a+b)+\pi r^2$: side strips give the linear term and four quarter-disks the quadratic term.

A normed vector space over $\mathbb{R}$ is a real vector space $X$ with a function $\Vert \cdot \Vert :X\to [0,\infty )$ satisfyingIts distance is $d(x,y)=\Vert x-y\Vert$. It is a Banach space if this metric is complete: every Cauchy sequence converges to an element of $X$.

Put $d=\mathrm{dist}(x,Z)$. If $d=0$, the zero functional works. If $d>0$, then $x\notin \overline{Z}$, and on $M=Z+\mathrm{span}\{x\}$ defineThe representation is unique, so this is linear. For $a\ne 0$,and the bound also holds for $a=0$. Thus $\Vert f_0\Vert \le 1$. The Hahn-Banach theorem extends it to $f\in X^{\prime }$ with the same bound, provingNo closedness assumption on $Z$ was required; distance is unchanged on taking its closure.

Let $Z=\overline{\mathrm{span}}\{x_n:n\ge 1\}$. If $Z\ne X$, part (a) provides a nonzero bounded functional annihilating $Z$; normalize it to have norm one. Density of the sequence in the dual unit ball gives an index $n$ with $\Vert f_n-f\Vert <1/4$. Since $f(x_n)=0$ and $\Vert x_n\Vert \le 1$,But the hypothesis and reverse triangle inequality givea contradiction. Therefore the span is dense.

If $X^{\prime }$ is separable, its unit ball has a countable dense set. For each nonzero $f_n$, the definition of its norm permits choosing $x_n$ in the unit ball with $|f_n(x_n)|\ge \Vert f_n\Vert /2$; choose zero for a zero functional. The preceding result gives a dense countable span over $\mathbb{Q}$, or over $\mathbb{Q}+i\mathbb{Q}$ in the complex case. Thus $X$ is separable.

The converse fails: ${\ell }^1$ is separable, since finite rational-coordinate sequences are dense, but $({\ell }^1{)}^{\prime }\cong {\ell }^{\infty }$ is not separable. The dual identification sends a bounded sequence $a$ to $x\mapsto {\sum }_n{a}_n{x}_n$, isometrically; conversely a bounded functional is determined by its values on the coordinate vectors and hence has this form by truncation. The uncountable family of zero-one sequences in ${\ell }^{\infty }$ has pairwise distance one, which no countable dense set can approximate within a radius smaller than $1/2$.

Let $d={\inf }_{y\in Y}\Vert x-y\Vert$ and choose $y_n\in Y$ with $\Vert x-y_n\Vert \to d$. The parallelogram law givesThe right side tends to zero. Thus $(y_n)$ is Cauchy. The closed subspace $Y$ of a Hilbert space is complete, so $y_n\to y_0\in Y$. Continuity of the norm gives $\Vert x-y_0\Vert =d$, proving the distance is attained.

Choose a spanning tree of the connected graph and root it at $x$. List vertices in nonincreasing order of their distance from $x$ in this tree, breaking ties arbitrarily. The root has distance zero and is last, so $v_n=x$.

Every other vertex has a parent one step closer to the root. That parent is adjacent to it and appears later in the ordering. HenceFor a graph of one vertex the condition is vacuous and the same ordering works.

Choose a spanning tree of the connected graph and root it at $x$. List vertices in nonincreasing order of their distance from $x$ in this tree, breaking ties arbitrarily. The root has distance zero and is last, so $v_n=x$.

Every other vertex has a parent one step closer to the root. That parent is adjacent to it and appears later in the ordering. HenceFor a graph of one vertex the condition is vacuous and the same ordering works.

A mathematician in Galileo Galilei's world was a practitioner situated within a particular division of knowledge and work. The word could embrace demonstrative geometry and arithmetic, but also astronomy, optics, mechanics, surveying, calendar calculation and the design or use of scientific instruments. Such mixed mathematics treated features of natural or practical objects through mathematical relations. It was neither identical to a present-day university department nor confined to calculation without claims about nature.

The distinction from a natural philosopher was important. Within a traditional university hierarchy, natural philosophy investigated the causes and principles of physical change. Mathematics could establish a relation between abstract quantities with great certainty, while opponents doubted whether it captured the constitution of a real body. Showing what follows for an ideal inclined plane, for example, leaves a further question about whether a particular apparatus realizes the assumptions closely enough. That question is not an objection which can be answered merely by producing a longer geometrical proof. The relation of abstraction, measurement and physical explanation was itself contested.^[ref]

There were also several routes to employment. A university teacher, an scientific instrument maker and someone serving a court or military patron might all perform mathematical work while possessing different income, status and audiences. Practical expertise in fortification, surveying or optics could bring rewards without establishing that its holder had authority over every question of natural philosophy. Conversely, a learned author could rely on the skills of people whose work was not credited equally in print. The important historical questions are what tasks were recognized, who paid for them, and whose testimony carried weight. A modern boundary between theoretical research and technical service would obscure these connections.

Galileo Galilei's career makes the contested boundary visible. The Sidereus Nuncius by Galileo Galilei communicated new observations through a manufactured telescope, successive drawings and an account intended to make unfamiliar objects credible. The work also addressed a courtly audience: naming the bodies accompanying Jupiter was part of presenting an achievement to a patron, not an incidental ornament added after a purely private discovery. The title and dedication situate mathematical work within relationships of employment and reputation.^[ref]

His mathematical treatment of motion likewise pressed beyond the role of calculating celestial appearances. In Two New Sciences by Galileo Galilei, a definition of uniform acceleration leads to a geometrical result in which travelled distance varies as the square of elapsed time. An inclined plane stretches the time of descent, making a comparison more manageable. In modern notation the relationship is $s=a{t}^2/2$ for release from rest. The notation is ours; the historically significant combination is an idealized proof, a proposed physical regularity and a reported experimental test.^[ref]

The label mathematician therefore cannot be read as a declaration that Galileo Galilei made no physical claims. Nor does his insistence on mathematical explanation prove that other occupations or disciplinary standards were irrelevant. His 1610 transition into Tuscan service, with the combined style of mathematician and philosopher, is evidence of the status being sought, rather than proof that the existing hierarchy had already disappeared. The case shows an investigator using scientific instruments, patronage and mathematical demonstration to enlarge the jurisdiction of his expertise. It also explains why a seemingly terminological distinction could become consequential in disputes over heliocentrism.^[ref]

Isaac Newton's achievement was not to introduce mathematics into a previously nonmathematical study of nature. Astronomy, optics and mechanics already had substantial mathematical traditions; Two New Sciences by Galileo Galilei, for example, connects geometrical reasoning with accelerated motion. The distinctive achievement of the Philosophiae Naturalis Principia Mathematica by Isaac Newton was to connect mathematical results about motion and forces with an ambitious, quantitatively constrained account of terrestrial and celestial phenomena.

The method separates a conditional mathematical proposition from its application. Given a specified force law and initial conditions, determine a trajectory. Conversely, given a suitably idealized trajectory and time law, infer what force would produce it. This makes the explanation more demanding than fitting a geometrical curve to positions: the timing and dynamical assumptions matter as well.

A modern calculation illustrates the connection. For a planar orbit under a central force, let $h=r^2\dot{\theta }$ be the conserved specific angular momentum and put $u=1/r$. An ellipse with the attracting point at a focus has $u=(1+e\cos \theta )/p$. Thus $u^{\prime \prime }+u=1/p$, and Binet's equation gives the radial accelerationThe orbit and equal-area timing therefore imply an inverse-square acceleration under the stated assumptions. The modern notation is explanatory; it is not an assertion that Newton wrote this particular differential-equation derivation. The geometrical treatment of central forces in Book I and its astronomical application in Book III supply the historical connection.^[ref]

The programme also joined apparently different domains. Newton's law of universal gravitation connects orbital motion with terrestrial falling motion through the same inverse-square dependence, with distance and time scales accounted for. That unification produced further tests rather than merely attaching one name to two phenomena. Departures from a perfect two-body ellipse, including disturbances by other gravitating bodies, became quantitative problems for the theory instead of automatic refutations of it.

Measurement did not disappear into deduction. Astronomical observations provide imperfectly known inputs; assumptions about mass distribution and neglected interactions restrict an ideal model. A discrepancy can expose a mistaken approximation, an observational error or a deficiency in the dynamical account. The mathematical proof alone does not decide which. The laws and initial data therefore create a disciplined relationship between measurement and explanation, not a proof of all nature from definitions.

Newton also did not claim that successfully describing a force mathematically supplied a complete mechanical account of its ultimate cause. Distinguishing a quantitative law from a speculative mechanism allowed a powerful programme of prediction while leaving explanatory questions open. The achievement was a new integration of mathematical demonstration, empirical inference and common dynamical principles. Its strength lies in the constraints this integration places on physical explanations, rather than in a simple opposition between mathematics and experiment.

There is no convincing single birthplace of science unless the definition is narrowed in advance. Systematic astronomy and mathematics preceded classical Greece; Greek natural philosophy developed distinctive explanatory and demonstrative ambitions; early modern Europe transformed the relationship between mathematics, experiment and institutions. These are different historical achievements. Treating any one of them as the creation of everything now called science confuses a particular form of inquiry with inquiry's entire history.

A good test begins with surviving practices rather than a list of supposed founding geniuses. The Babylonian astronomical diary VAT 4956 records positions of the Moon and planets relative to named stars, together with other observations, for 568–567 BCE. Such a record requires trained observers, a conventional vocabulary and a means of accumulating observations over time. Its intellectual value is not annulled because astronomy was entangled with divination and royal administration. However, this particular diary should not be credited with everything achieved by later Babylonian predictive astronomy: an observational record is evidence of systematic recording, not by itself evidence of a particular predictive algorithm.^[ref]

Egyptian mathematics provides a different kind of evidence. Problem 50 of the Rhind mathematical papyrus gives a procedure for the area of a circular field: remove one ninth of its diameter and square the remainder. Written in modern algebraic notation, the procedure isComparing this with the modern circle formula $A=\pi d^2/4$ gives the effective approximation $\pi \approx 256/81$. This is our reconstruction of the rule's accuracy, not a claim that its author used our symbol $\pi$ or formulated an abstract theory of that constant. A worked numerical procedure is mathematical knowledge even when it is not presented as a Greek-style proof.^[ref]

The case for Greece therefore needs a more specific claim. In Physics by Aristotle, nature is investigated through principles of motion and change, and explanation requires identifying different kinds of cause. His distinctions between material, formal, moving and final causes constitute an explicit programme of explanation, not merely a collection of useful recipes. Greek mathematics, exemplified by Euclid's Elements, made the organization of results into chains of proof unusually prominent. These traditions help explain why later European scholars gave Greek texts such authority. They do not establish that earlier or non-Greek practitioners lacked reasoning, nor that all Greek explanations were correct.^[ref][ref]

The European sixteenth and seventeenth centuries nevertheless deserve attention as more than the recovery of ancient truths. Galileo Galilei's Two New Sciences by Galileo Galilei joins precise definitions of motion to geometrical demonstrations and reported tests. An inclined plane lengthens a descent sufficiently to compare its distance and time; a mathematical dependence becomes something an apparatus can discriminate. Francis Bacon's Novum Organum by Francis Bacon instead urges a disciplined reconstruction of inquiry through ordered experience and the correction of intellectual prejudices. Their programmes differ, but both show participants deliberately contesting how knowledge should be made. Neither text authorizes the claim that observation or reasoning had only just been invented.^[ref][ref]

Institutional criteria also change the answer. If science means a modern profession with specialized laboratories, credentials and journals, even sixteenth-century Europe is too early for much of it. If it means any successful craft technique, the category becomes so broad that the first surviving written text cannot identify its beginning. If it means public, revisable claims about nature supported by disciplined observation and reasoning, several ancient traditions qualify, with different strengths and social settings. Each definition selects different evidence; none should be smuggled into a geographical verdict.

The strongest conclusion is consequently plural but discriminating. Greece was a major centre of explanatory natural philosophy and demonstrative mathematics; early modern Europe was a major site of their reorganization; neither was the unique beginning of systematic knowledge of nature. This conclusion recognizes real differences in standards of explanation without using proximity to present-day science as the sole measure of past achievement. It also leaves open the historical work of tracing transmission and transformation rather than imagining isolated civilizations independently possessing, or wholly lacking, a timeless scientific essence.

Isaac Newton made the greater contribution if the criterion is the scope and explanatory integration of mathematical natural philosophy; Galileo Galilei has the stronger claim if the criterion is the formative transformation of how terrestrial motion could be investigated. The first criterion gives a defensible overall verdict for Isaac Newton, but a serious comparison must show what each changed rather than award credit according to who worked later.

Galileo Galilei's contribution was not simply to attach numbers to familiar events. In the Third Day of Two New Sciences by Galileo Galilei, he defines uniformly accelerated motion by equal additions of speed in equal intervals of time, then derives consequences. Starting from rest, the modern notation is $v=at$; the mean speed over the interval is $at/2$, soHence distances in successive equal intervals have the ratios $1:3:5:7:\dots$. The novelty lies in making a precisely specified kind of motion the subject of mathematical demonstration, with observed falls assessed against its consequences. This is not the same as deriving the underlying physical cause of gravity.^[ref]

The reported apparatus shows how such an idealization could be investigated. A smooth bronze ball descended in a straight grooved wooden channel, raised at one end. Water discharged through a small opening into a vessel during the descent; its collected mass served as a measure of elapsed time. Repeating runs over different portions of the channel compared the predicted squared dependence of time on distance. The inclined plane slowed the event, while repetition and weighing made a temporal comparison possible without a modern stopwatch. The published report is evidence of Galileo Galilei's experimental presentation, not a surviving raw-data archive from which every reported precision can now be reconstructed.^[ref]

A necessary physical qualification is often missed. A ball in rolling without slipping is not a point particle sliding without friction. For a solid sphere on an ideal inclined plane, modern rigid body dynamics gives $a=5g\sin \alpha /7$, rather than $g\sin \alpha$. Both are constant accelerations, so both give $s\propto t^2$. The trial can therefore test the temporal form without directly establishing the free-fall coefficient. This distinction makes the experiment more intelligible: the evidence supports a specified relation under approximations, rather than every detail of a modern idealization.

The Fourth Day extends this approach to projectile motion. Combine uniform horizontal velocity $u$ with constant downward acceleration $g$, taking downward displacement positive. Then $x=ut$ and $y=g{t}^2/2$. Eliminating time givesa parabola. Actual projectiles experience air resistance, but the ideal curve organizes the effects that need separate treatment. Galileo Galilei thus gave a powerful example of how mathematical abstraction and physical observation could cooperate.^[ref]

Isaac Newton went further by connecting kinds of motion through a general account of force. Newton's second law links the change of momentum to the impressed force; for constant mass, its modern differential expression is $\mathbf{F}=m\mathbf{a}$. Newton's third law makes interaction reciprocal, enabling a consistent analysis of interacting bodies. The Philosophiae Naturalis Principia Mathematica by Isaac Newton develops consequences of central forces and then asks which laws fit astronomical phenomena. Mathematical deduction now connects measurable orbits to an underlying force law, rather than just describing each trajectory separately.^[ref]

A simple modern reconstruction shows the increase in explanatory reach. For a circular orbit of radius $r$ and orbital period $T$ about a much more massive central body, centripetal acceleration is $4{\pi }^{2}r/T^2$. Kepler's third law, $T^2\propto r^3$, therefore implies an acceleration proportional to $r^{-2}$. This circular argument is illustrative; Isaac Newton's treatment of elliptical orbits is more general. With Newton's law of universal gravitation,the two-body orbital period is tied to the semimajor axis $a$ and both masses. The same theory connects the Moon's orbital acceleration with terrestrial gravity, and supports further investigations of tides, comets and departures from a simple two-body orbit.^[ref]

This achievement depended on more than two individuals. Johannes Kepler's relations supplied crucial astronomical constraints, and both authors inherited geometry, observational records and technical practices. Nor should the Philosophiae Naturalis Principia Mathematica by Isaac Newton be described simply as the first application of symbolic calculus: its published arguments are largely geometrical. What distinguishes Isaac Newton is the demonstrative network uniting terrestrial and celestial problems, together with explicit idealizations whose failures generated further calculable questions.

On breadth, quantitative unification and the inference from motion to force, Isaac Newton's contribution was greater. Galileo Galilei's work supplied an exemplary way to mathematize terrestrial motion and test its idealized relations. Giving the former the overall verdict should therefore acknowledge the latter's distinct foundational achievement, not portray the Philosophiae Naturalis Principia Mathematica by Isaac Newton as a creation without predecessors.

Scientific knowledge is deeply shaped by historical context in the questions asked, evidence available and standards of credibility, but context does not make results arbitrary or determine their truth by itself. The strongest argument identifies mechanisms through which context enters inquiry and then examines what experimental or mathematical constraints can resist it. This avoids both the claim that facts emerge untouched by human activity and the opposite claim that a social explanation is a substitute for testing them.

First, inherited conceptual resources determine what an investigator initially thinks an observation might be. Joseph Priestley's experiments on oxygen illustrate this directly. Heating mercury(II) oxide produced an unusually combustion-supporting gas, but the experiment did not arrive labelled with a modern chemical formula. Phlogiston supplied an available vocabulary: the gas could be interpreted as exceptionally dephlogisticated air rather than a newly isolated chemical element. His own account acknowledges the difficulty of revising expectations even when striking observations appeared. This is specific evidence that interpretation has a history, rather than a general accusation that past investigators were irrational.^[ref]

Antoine Lavoisier's material balances reorganized the same field by treating the air consumed during calcination as a constituent incorporated into the product. The retort, enclosed air supply and quantitative examination of products made some interpretations more difficult to maintain. Yet his identification of oxygen as the universal acidifying principle was itself wrong: hydrogen chloride gives an chemical acid without containing oxygen. The example establishes two distinct conclusions. An interpretive framework helps make observations intelligible; further observations and successful alternative explanations can expose its limits. Historical situatedness is compatible with correction.^[ref]

Second, instruments and techniques shape what can become an observable object. In nineteenth-century embryology, an optical microscope did not simply magnify a transparent miniature adult. Investigators needed suitable organisms, controlled access to eggs and sperm, ways of preparing thin or fixed material, and conventions for ordering developmental stages. Transparent sea-urchin eggs and external fertilization made processes accessible that were much harder to observe in opaque mammalian material. Such a choice was productive but selective: knowledge gained from one organism needed further work before it could be generalized to others.^[ref]

The Wilhelm Roux–Hans Driesch comparison shows why the details of an intervention matter. Killing an attached frog blastomere and separating a living sea-urchin blastomere differ both in material and operation. Partial development in one setting and a complete small larva in another cannot honestly be flattened into an experiment with one obvious answer. Nevertheless, the second outcome constrains the universality of an exclusively fixed mosaic theory of development. A capacity for regulation is empirically established in the tested conditions even if disagreement remains about its explanation. Technical and theoretical contexts shape the result's significance without allowing the observer simply to choose its outcome.^[ref]

Third, social interests can supply a problem or analogy without fully fixing the resulting theory. Charles Darwin's adaptation of Thomas Robert Malthus's population argument is a clear case. A discussion of human subsistence helped him formulate a mechanism for changes among organisms. However, his broad struggle for existence includes dependence and physical conditions, not only direct competition. The resulting natural selection explanation must be evaluated through genetic variation, heredity and differential reproduction, not accepted or rejected merely because a stimulus came from political economy. The route by which a hypothesis is formed is a historical question; the evidence sustaining its application is a connected but distinct question.^[ref][ref]

Fourth, conventions of demonstration affect what a community can reliably criticize. Galileo Galilei's definitions of accelerated motion let readers derive a squared time relation and compare it with repeated descents on an inclined plane. A reader can dispute whether the apparatus sufficiently approximates the model, distinguish a rolling ball from a freely falling body, or check whether the mathematical conclusion follows. The idealization is historically chosen, but its consequences are not whatever the author wishes. For a constant acceleration from rest, the distance ratio for elapsed times in the ratio $1:2$ must be $1:4$; changing that conclusion requires changing the assumptions or the mathematics.^[ref]

These cases support degrees rather than a single all-purpose answer. Historical context strongly affects research priorities, available specimens, instruments, vocabularies and access to credibility. It also affects which uncertainties become urgent and which applications receive resources. Its influence on a particular measured relation or mathematical entailment must be demonstrated rather than presumed. The same social setting can sustain competing interpretations, and a mathematical framework can travel beyond the setting in which it was first developed. Conversely, the ability to reproduce a result elsewhere does not erase the historical work needed to make that reproduction possible.

Scientific knowledge is historically produced and materially constrained. Context helps explain why a claim was thinkable, testable and persuasive at a particular time; it does not alone tell us whether the claim survives comparison, intervention and criticism. A convincing history therefore traces both the circumstances that make knowledge possible and the resistant evidence through which those circumstances can themselves be challenged.

Normalize the normal vector $(2,4,3)$, whose Euclidean norm is $\sqrt{29}$. The plane has the formThe scalar $d$ is the signed perpendicular distance from the origin in the chosen unit normal direction; the unsigned distance is $|d|$. The nearest point is $d\mathbf{n}$.

The scalar product $\mathbf{r}\cdot \hat{\mathbf{n}}=d$ fixes the component of the position vector along the given unit normal vector. Thus the locus is a plane normal to $\hat{\mathbf{n}}$, through $d\hat{\mathbf{n}}$. Its signed distance from the origin along that normal is $d$, and its perpendicular distance from the origin is $|d|$. Any displacement lying within the plane is orthogonal to $\hat{\mathbf{n}}$.

The three vectors are coplanar precisely when their scalar triple product vanishes:This includes linearly dependent or zero vectors. It expresses that the determinant with those vectors as columns is zero.

For the two moving particles, distinguish closest approach at a common time from the geometric distance between their supporting lines. For fixed reference times, putTheir relative position is ${\mathbf{d}}_0+t\mathbf{v}$. If $\mathbf{v}\ne 0$, the closest approach of two uniformly moving particles occurs atIf $\mathbf{v}=0$, the distance is constant. If only $t\ge 0$ is permitted, replace $t_{\ast }$ by $\max (0,t_{\ast })$ before evaluating the distance.

Now suppose the velocities are nonparallel, and allow the relative starting-time offset to be chosen so that the particles reach the closest points of the two lines simultaneously. Write $\Delta ={\mathbf{a}}_2-{\mathbf{a}}_1$, $A=|{\mathbf{b}}_1{|}^2$, $B={\mathbf{b}}_1\cdot {\mathbf{b}}_2$, $C=|{\mathbf{b}}_2{|}^2$ and $D=AC-B^2>0$. The independent line parameters $s_1,s_2$ at the closest points satisfyThese equations make the connecting vector perpendicular to both velocities. Solving the two-by-two linear system givesThe corresponding distance between skew lines is $|\Delta \cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)|/|{\mathbf{b}}_1\times {\mathbf{b}}_2|$. Simultaneous arrival requires $t_1+s_1=t_2+s_2$, so $t_1-t_2=s_2-s_1$. Since the vector triple product givesthis condition is preciselyThus the printed formula has a valid interpretation as a synchronization condition for attaining the line-distance lower bound. It is not an identity that must hold for arbitrary fixed reference times at the time of closest approach.

For a counterexample to that stronger reading, take ${\mathbf{a}}_1=0$, ${\mathbf{a}}_2=(1,0,1)$, ${\mathbf{b}}_1=(1,0,0)$, ${\mathbf{b}}_2=(0,1,0)$ and $t_1=t_2=0$. The actual squared distance is $2(t-1/2{)}^2+3/2$, minimized at $t=1/2$. The printed right-hand side is $-1$, whereas the fixed difference $t_1-t_2$ is zero. The line-distance bound is one, but it is not attained by that synchronized motion. Parallel velocities also require separate treatment because the displayed denominator then vanishes.