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\b[<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 $at/2$, so
$$
s=\frac12at^2.
$$
Hence <distances> in successive equal intervals have the ratios $1:3:5:7:\ldots$. 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>.<Two New Sciences by Galileo Galilei>[Third Day: definition of naturally accelerated motion, Theorem I on mean speed and Theorem II with its corollaries]{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.<Two New Sciences by Galileo Galilei>[Third Day, discussion following Theorem II: grooved channel, bronze ball, water timing and repeated trials]{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 $y=gt^2/2$. Eliminating <time> gives
$$
\boxed{y=\frac{g}{2u^2}x^2,}
$$
a <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.<Two New Sciences by Galileo Galilei>[Fourth Day, Theorem I: the parabolic trajectory formed by uniform horizontal and naturally accelerated vertical motion]{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.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Definitions, Laws II–III; Book I, section II, propositions I–III; section III, propositions XI–XIII]{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 $4\pi^2r/T^2$. <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>,
$$
\boxed{F=\frac{GMm}{r^2},\qquad T^2=\frac{4\pi^2a^3}{G(M+m)},}
$$
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>.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Book I, proposition XV and section XI; Book III, proposition IV and the subsequent treatment of universal gravitation]{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.
\b[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.
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