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A parabola is a plane curve whose points are equally distant from a focus and a directrix. In suitable coordinates it has equation $y=a{x}^2$, with $a\ne 0$. Its quadratic geometry can also describe a contour deformation through a saddle point.

For example, take a focus of a conic at $(1,0)$ and a directrix of a conic $x=-1$. Equality of squared Euclidean distances gives $(x-1{)}^2+y^2=(x+1{)}^2$, hence $y^2=4x$. The point $(1,2)$ has distance $2$ to both the focus and the directrix. Its eccentricity of a conic is one. Unlike an ellipse, a parabola is unbounded and has only one matched focus-directrix description.

A parabola parametrized by $(t,t^2)$ is in bijection with $\mathbb{R}$ and is an uncountable set. Every affine line meets it in at most two points. A countable family of affine lines could therefore cover only countably many points of this parabola, and cannot cover the plane. More generally, an uncountable curve with finite intersection with every permitted member obstructs a countable covering by those members.

A quadric surface obtained by extending a parabola along a fixed direction outside its plane. For example, $x=c-y^2/(4c)$ with arbitrary $z$ is a parabolic cylinder parallel to the $z$ axis. Each cross-section of fixed $z$ has vertex $(c,0)$ and focus $(0,0)$ in its $xy$ plane. It occurs as a coordinate surface in parabolic cylindrical coordinates.

These orthogonal coordinates extend two confocal families of parabolas parallel to an axis. In the displayed scaling their scale factors of orthogonal coordinates are $h_u=h_v=2\sqrt{u^2+v^2}$ and $h_w=1$, so the Laplacian is $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/[4(u^2+v^2)]+{\varphi }_{ww}$ away from $u=v=0$. With $u\ge 0$, signed $v$ distinguishes the upper and lower half-planes; the two signs at $u=0$ describe the same negative $x$ half-axis. Fixed positive $u^2$ or $v^2$ describes a parabolic cylinder. This differs from the rotational three-dimensional parabolic coordinates, whose coordinate surfaces are elliptic paraboloids.

For speed $c(y)=c_0/\sqrt{1-ky}$, with $1-ky>0$, the elapsed-time integrand is $\sqrt{(1-ky)(1+(y^{\prime }{)}^2)}/c_0$. The Beltrami identity gives $(1-ky)/(1+(y^{\prime }{)}^2)=1-k{y}_0$, where $y_0$ is the turning height. Integration yields the displayed parabola. If the endpoints are $(\pm x_0,0)$, symmetry gives $x_c=0$ and $(k{x}_0{)}^2=4k{y}_0(1-k{y}_0)$. Hence $k{x}_0\le 1$, and the two algebraic turning heights are $[1\pm \sqrt{1-k^2{x}_0^2}]/(2k)$. Only heights lying within the specified optical layer are admissible.

These plane orthogonal coordinates use two families of confocal parabolas. On the region $y\ge 0$, take $u,v\ge 0$ and defineThen $x=(u^2-v^2)/2$ and $y=uv$. The coordinate tangent vectors $(u,v)$ and $(-v,u)$ have zero dot product, and both have length $\sqrt{u^2+v^2}$. Thus the Jacobian determinant is $u^2+v^2$, the area element is $(u^2+v^2)dudv$, and these orthogonal coordinates degenerate only at the origin. Fixed $u=a>0$ gives $x=(a^2-y^2/a^2)/2$; fixed $v=b>0$ gives $x=(y^2/b^2-b^2)/2$. This normalization is a rescaling of the plane section of parabolic cylindrical coordinates.

Without a restricted parameter domain, simultaneous sign reversal $(u,v)\mapsto (-u,-v)$ preserves the image, so these planar parabolic coordinates are globally two-to-one. The inverse function theorem supplies local inverse branches away from the origin. The nonnegative quadrant is one useful branch chart for the upper half-plane.

Use the condensate wavefunction convention $\int |\psi {|}^2{d}^{3}r=N$, so $n(\mathbf{r})=|\psi (\mathbf{r}){|}^2$ is the number density and $\psi /\sqrt{N}$ is a normalized one-particle orbital. The contact interaction strength is $U_0=4\pi {\hslash }^{2}a/m$ for dilute atoms with s-wave scattering length $a$. The chemical potential $\mu =\partial E/\partial N$ fixes the stationary time dependence $e^{-i\mu t/\hslash }$ and enforces normalization when minimizing the energy. The Gross–Pitaevskii equation follows from an energy functional that contains the pair-counting factor $1/2$:Varying $E-\mu \int |\psi {|}^2{d}^{3}r$ gives the stationary Gross–Pitaevskii equation; the interaction contribution to that equation is $U_{0}n\psi$, without the factor $1/2$.

For the broadening discussed here assume repulsive interactions, $a>0$. The Thomas–Fermi approximation for a condensate neglects the gradient kinetic energy compared with the trapping and interaction energies. At fixed spatial scale, increasing $N{U}_0$ increases the mean-field potential. Expansion then reduces the kinetic energy further: for a cloud radius $R$, its scale per atom is ${\hslash }^2/(m{R}^2)$, whereas the trap energy scale is $m{\omega }^2{R}^2$. Their ratio decreases as $(a_0/R{)}^4$. In the weak-interaction limit the kinetic energy instead balances confinement, giving the Gaussian wave packet ground state of the quantum harmonic oscillator.

The spherical Thomas–Fermi condensate hasIntegrating this density gives $N=8\pi \mu R^3/(15U_0)$. ThusAt fixed $N$, stronger repulsion produces a wider, lower wavefunction: the Gaussian wave packet crosses over towards the square root of an inverted parabola, with an almost flat central region and a steeper outer drop. The density is the inverted parabola; the wavefunction is its square root. The exact wavefunction remains smooth and has a tail, since kinetic energy cannot be neglected near the Thomas–Fermi condensate edge rounding region. The healing length must be small compared with the cloud size in the bulk. Attractive interactions would not follow this large-$N$ expansion argument.

The original plot compares the noninteracting endpoint with two normalized repulsive spherical Thomas–Fermi condensate profiles. The latter are asymptotic shapes, rather than exact intermediate solutions of the Gross–Pitaevskii equation. Each has the same integrated atom number; $\chi =Na/a_0$ labels the illustrative interaction strengths.

Choose $A$ real and positive. The Gaussian integral givesThe three zero-point energies add for each atom. The ideal-gas chemical potential is a one-atom energy; the total energy includes all $N$ atoms.

For weak interactions, the first-order correction in perturbation theory is the energy functional of the contact interaction evaluated on the unperturbed Gaussian wave packet. The integral needed isConsequently the Gaussian interaction energy of a trapped condensate isHere $\hslash \omega ={\hslash }^2/(m{a}_0^2)$. These are the mean-field expressions consistent with the supplied equation and $\int |\psi {|}^2=N$. Exact first-order perturbation theory for precisely $N$ atoms in one orbital replaces $N^2$ by $N(N-1)$ through the number of distinct pairs; this distinction is negligible in the large-$N$ regime of the crossover estimate.

To judge the ideal-gas approximation, compare the interaction energy with the total ideal energy, or compare the interaction energy per atom with $\hslash \omega$:Thus the crossover from Gaussian to Thomas–Fermi condensate is governed by $\chi =Na/a_0$: perturbation theory is reliable for $\chi \ll 1$, while the regime of the Thomas–Fermi approximation for a condensate requires $\chi \gg 1$. The characteristic crossover number is $N_c\sim a_0/a$; there is no sharp critical atom number here. If one specifically defines crossover by $E_I=E_0$ using the weak-interaction expression, it gives $N_c=(3\sqrt{2\pi }/2)a_0/a$. That prefactor is a convention-dependent estimate extrapolated to the edge of perturbation theory, not a phase transition. Comparing $E_I$ itself to a single $\hslash \omega$ would incorrectly omit a factor of $N$.

For the numerical estimate use $m\simeq 87\times 1.66054\times 10^{-27}\mathrm{kg}$ and $\hslash \simeq 1.05457\times 10^{-34}\mathrm{J}\mathrm{s}$. ThenThe particular equality $E_I=E_0$ would instead give about $1.15\times 10^3$ atoms. These quantify different choices within a smooth crossover; a well-developed spherical Thomas–Fermi condensate profile requires going substantially beyond the characteristic scale.

The method of characteristics replaces the first-order partial differential equation by the ordinary differential equationsAn integrating factor for the second equation gives $d(e^{-2s}y)/ds=\beta C{e}^{-s}$, so their general solution of a differential equation isHere $C,D,E$ are real constants. For $C\ne 0$, eliminating $s$ gives the characteristic projectionseach restricted to either $x>0$ or $x<0$. These are arcs of parabolas, with the straight-line exception $K=0$. Along an oriented characteristic curve, $|x|$ increases with $s$.

For $C=0$, the characteristic projections lie on $x=0$: $y=D{e}^{2s}$ describes either vertical half-line when $D\ne 0$. For $C=D=0$ the projection is the single point $(0,0)$, although $u=2s+E$ still changes. This last lifted characteristic curve cannot be contained in the function graph of a classical solution: at $(0,0)$ the original partial differential equation would require $0=2$.

Every point with $x>0$ has exactly one backward characteristic curve meeting $x=1$, at the point $(1,q(x,y))$. It meets the specified data precisely when $-1\le q\le 1$. Thus the whole region determined by these data, including the two bounding characteristic curves, isIts interior is the corresponding open domain for the partial differential equation. Uniqueness here means uniqueness in the class of continuous functions satisfying the characteristic equations for a transport equation, including the piecewise smooth function defining the solution of a differential equation above. The endpoints of the initial segment determine the two boundary values by continuity.

This region is maximal for determination by the given data: on $x>0$, arbitrary extensions of $g$ outside $[-1,1]$ give different values on the characteristic curves with $q\notin [-1,1]$. No characteristic curve from $x=1$ reaches $x\le 0$.

For $\beta =1$,The lower boundary is a straight line, the upper boundary is a parabola, and the characteristic carrying the corner is $y=x^2-x$. All three approach the excluded origin as $x\downarrow 0$.

Use the compatible open semicircles and set $r=\sqrt{x^2+y^2}$. The characteristic curve through $(\cos \theta ,\sin \theta ,-1)$ hasThe condition $s<1$ is essential: at $s=1$ there is finite-time blowup, so the opposite branch cannot be reached from the data. From $r=(1-s\sin \theta {)}^{-1}$ and $y=r\sin \theta$,Eliminating $s$ yieldsThe ratio inside the cube root is positive precisely on the two data-determined regionsOn $D_{+}$ each point traces to the upper semicircle, and on $D_{-}$ to the lower semicircle, with $s<1$. To check the parameter ranges explicitly, for $\eta =\sin \theta >0$ they are $-\infty <s<1$, whereas for $\eta <0$ they are $1/\eta <s<1$. At the latter lower endpoint the planar projection escapes to infinity. The inverse function theorem and uniqueness for the characteristic ordinary differential equations give a unique smooth solution on each region. Equivalently, each point has the unique foot $(x/r,y/r)$ and parameter $(r-1)/y$.

A direct check is particularly short. For $s=(r-1)/y$, let $\mathcal{D}=xy{\partial }_x+y^2{\partial }_y$. Then $\mathcal{D}r=yr$, $\mathcal{D}y=y^2$, andSince $dz/ds=-z^4/3$, the displayed formula solves the partial differential equation, and $r=1$ gives $z=-1$ on both open semicircles. The finite boundary where $r=1+y$, away from the axis, is the parabola $y=(x^2-1)/2$ and corresponds to blowup. Points on $y=0$ cannot be added to a classical continuation of these regions: where they are approached from a data-determined region, the height behaves like a nonzero multiple of $|y{|}^{1/3}$, or has incompatible limits at the endpoints. The formula is therefore the complete classical answer for the compatible arcs, while the full-circle problem as printed has no classical solution.

The figure illustrates Pierre de Fermat's use of adequality to construct a tangent to a parabola. Reading the local exam image is decisive: $D$ is the vertex of a parabola, $DC$ lies on its axis of symmetry, $B$ lies on the parabola, and the proposed tangent $BE$ meets the extended axis at $E$. The horizontal through the auxiliary point $O$ meets the axis at $I$; $O$ lies on $BE$, outside the parabola. In particular, $O$ must not be treated as a second point of the parabola. The target is the subtangent $CE$. The diagram and this construction occur in the tangent discussion associated with Varia opera mathematica by Pierre de Fermat. The 1679 book was posthumous: its date does not date the invention of the method.^[ref][ref]

Write $d=CD>0$, $a=CE>0$, and $e=CI$, with $I$ between $C$ and $D$. These are lengths, and the increment $e$ is distinct from the labelled point $E$. Then $DI=d-e$ and $EI=a-e$. The parabola's characteristic relation makes the square of an ordinate proportional to its distance from the vertex of a parabola. At height $I$, the actual ordinate of the parabola is shorter than $OI$. Hencewhere the equality follows from similar triangles $BCE$ and $OIE$.

The method of adequality replaces the comparison by a relation used to extract the condition of contact:Expanding and cancelling $d{a}^2$ gives $-2dae+d{e}^2\sim -a^{2}e$. Divide by the nonzero increment $e$, then suppress the remaining term containing $e$. The result is $a^2=2da$, so, since $a>0$,Thus $DE=DC$: the intersection with the axis lies as far above the vertex of a parabola as $C$ lies below it. This is a concrete construction, not just a rule for computing a numerical slope.

A modern check clarifies why the calculation succeeds without pretending that Pierre de Fermat used a modern limit. Put $D=(0,0)$, let $y$ increase down the axis, and write the parabola as $x^2=ky$, $k>0$. For $B=(b,d)$, with $b^2=kd$, the proposed tangent through $E=(0,-d)$ is $y=2bx/k-d$. Substitution givesIt meets the parabola with a double root at $B$, and for every other point its $y$ coordinate is less than that of the parabola. This also verifies the strict comparison used for $O$.

The significance is a systematic algebraic condition for geometric contact. Subtracting neighbouring expressions, dividing by an increment, and discarding its remaining terms anticipates differentiation. But the historical question is precisely what licenses the last step. The symbol $\sim$ above marks adequality, not ordinary equality at a finite displacement. Treating the construction as an already stated epsilon-delta definition of a limit would conceal the methodological innovation and its unresolved explanatory burden.

Standards of mathematical proof changed between the seventeenth and nineteenth centuries because both the objects studied and the operations performed on them changed. A finite geometrical construction, an infinite expansion, and an abstract group raise different questions about justification. There was no universal seventeenth-century indifference to rigour followed by its sudden nineteenth-century invention. The more defensible thesis is that successful methods repeatedly exposed unstated assumptions; later work made some of those assumptions into explicit definitions, hypotheses and existence arguments.

Pierre de Fermat's adequality illustrates the distinction between an effective method and its justification. For $F(a)=a(L-a)$, comparing neighbouring expressions givesDividing by $e$ and suppressing the remaining term in $e$ produces $a=L/2$. There is an independent finite verification:Thus the result can be secure even when the general procedure raises a question: why is the increment first nonzero for division and then discarded? In the parabola construction of question 2, the same issue appears in replacing a strict geometrical comparison by adequality and extracting the subtangent. A historical explanation must describe this operation on its own terms rather than silently equate it with a modern derivative.^[ref]

Isaac Newton and Gottfried Wilhelm Leibniz supplied different languages for the new calculus. Isaac Newton's Introduction to the Quadrature of Curves by Isaac Newton, first published with the 1704 Opticks, presents quantities generated by motion and considers their rates of change. Its mature language of first and last ratios seeks exact relations as increments vanish; it should not be confused with simply asserting that a small error is harmless. Gottfried Wilhelm Leibniz's 1684 Nova methodus pro maximis et minimis by Gottfried Wilhelm Leibniz presents a calculus of differentials and operational rules. Their common problem was to explain why procedures using vanishing increments establish exact results, while their chosen objects and modes of explanation differed.^[ref][ref]

George Berkeley's The Analyst by George Berkeley (1734) exposes the logical pressure point. For the cube,The finite identity does not become a legal division at $h=0$. The modern repair is to keep $h\ne 0$ throughout the identity and prove that its right-hand side tends to $3x^2$. That limit is different from a quotient evaluated at zero. George Berkeley's criticism therefore concerns the grounds of inference, not whether the established rules often produce correct answers. His theological polemic also reminds us that standards of mathematical proof could be contested outside specialist calculation.^[ref]

The infinite introduced another burden. Joseph Fourier's 1822 use of trigonometric series to represent arbitrary functions greatly extended what calculations in mathematical physics could express. But deriving Fourier coefficients by orthogonality only shows what those coefficients must be if the proposed operations are legitimate. It does not prove convergence, equality with the initial data, or interchange of an integral and an infinite sum. The example of a constant expanded in a Fourier sine series makes a hidden qualification visible: the terms all vanish at the endpoints, even when the assigned data do not. The correct theorem must specify the domain and mode of representation.^[ref]

Peter Gustav Lejeune Dirichlet's 1829 memoir supplies a particularly precise example of changing proof obligations. It criticises a convergence argument of Augustin-Louis Cauchy's which used complex substitution into an arbitrary function and an insufficient comparison between signed series. Peter Gustav Lejeune Dirichlet supplies the instructive pairThe first series converges by the alternating series test; the second diverges because their difference is the harmonic series. Nevertheless $b_n/a_n=1+(-1{)}^n/\sqrt{n}\to 1$. Thus asymptotic equality of terms does not transfer convergence between conditionally convergent signed series. A comparison principle valid for positive terms needs additional justification when cancellation is involved. The example refutes a mode of inference rather than merely correcting a numerical answer.^[ref]

Augustin-Louis Cauchy's own foundations programme is part of this change. The 1821 Cours d'analyse by Augustin-Louis Cauchy makes limits, continuity and series convergence central to elementary reasoning. Yet its Sum theorem of Cauchy also shows why announcing rigour does not make every proof adequate. In modern terminology, continuous functions can have a discontinuous pointwise limit. An explicit series version isEach term is a polynomial and the partial sum is $x-x^{N+1}$. The convergence does not hold in uniform convergence: with $x_N=1-1/N$, the error $x_N^{N+1}$ tends to $e^{-1}$ rather than zero. The modern uniform limit theorem repairs the inference by requiring one bound on the error to work throughout the domain. This example diagnoses the missing control under a pointwise reading; it does not settle every historical interpretation of Augustin-Louis Cauchy's infinitesimal language.^[ref]

The issue is the order of dependence in quantifiers. Pointwise convergence permits the threshold $N$ to depend on both the error tolerance and the point. Uniform convergence requires a threshold depending only on the tolerance. If $f_N$ is continuous near $x_0$ and the convergence is uniform, choose $N$ so that $|f-f_N|<\epsilon /3$ nearby, then use the continuity of this fixed $f_N$ to make $|f_N(x)-f_N(x_0)|<\epsilon /3$. The triangle inequality proves $|f(x)-f(x_0)|<\epsilon$. Writing out this proof reveals what a vague statement that the tail is small leaves unproved.

Another nineteenth-century development concerns the foundations of the real numbers themselves. Richard Dedekind's Continuity and Irrational Numbers by Richard Dedekind (1872) asks for an arithmetical account of the continuity often borrowed from a drawn line. A Dedekind cut separates the rational numbers into lower and upper classes. For instance,defines a gap in the rational numbers corresponding to $\sqrt{2}$. The cut is determined without assuming that a rational number fills it. The extended number system then supports existence theorems that a geometrical picture alone had suggested.^[ref]

This matters for apparently elementary proofs. Suppose $f$ is continuous on $[a,b]$ with $f(a)<0<f(b)$. Repeated bisection constructs nested intervals whose endpoint values have opposite signs and whose lengths tend to zero. To conclude that there is a common limiting point, one needs the completeness of the real numbers. Continuity then forces its value to be zero. Over the rational numbers, $f(q)=q^2-2$ changes sign between $1$ and $2$ but has no rational zero. Thus even a proof that avoids pictures must explain the existence of its limiting object. Richard Dedekind's account makes that requirement an explicit part of the mathematical setting, rather than an intuition about a continuous line.^[ref]

There is a parallel change in algebra. Heinrich Weber's 1882 definition in question 6 describes arbitrary elements through laws of composition. To assess such a definition one proves implications between axioms. The passage's finite set hypothesis implies that cancellative multiplication has an identity element and inverse elements; without that hypothesis, the positive integers under addition supply a counterexample. This is a new kind of proof obligation: one must establish that a proposed abstract definition has the intended consequences, rather than rely on their familiarity in particular numerical or permutation examples.^[ref]

The principal change was towards explicit control of operations, hypotheses and existence. Effective symbolic algorithms persisted, and geometric reasoning remained valuable. What became increasingly demanding was the passage from success in examples to a general claim: an infinitesimal step needed justification, an infinite expansion needed a convergence theorem, a limiting point needed a complete number system, and an abstract definition needed its consequences proved. The examples also prevent a simple progress story: nineteenth-century mathematicians disagreed about admissible reasoning, and a programme devoted to rigour could itself become the object of a successful critique.

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.

Set $r=\sqrt{x^2+y^2}$. Wherever $r>0$, the chain rule and the given implicit differential equation giveThus $r=x+C$ and $C=r-x\ge 0$. Squaring without losing the square-root sign restriction giveswith either sign of $y$ on its interval of definition. These parabolas satisfy $x+C>0$. Their vertex $(-C/2,0)$ is a regular point as a parametrized curve but cannot belong to a differentiable graph $y(x)$ satisfying the equation: there $y{y}^{\prime }=0$ while $-x+r=C>0$. The degenerate case $C=0$ gives the separate solution $\boxed{y=0\text{ for }x\ge 0}$ in the differential-form interpretation printed in the paper. It would be lost by dividing the equation by $y$.

For $z=x+iy$, the equation becomes $\sqrt{(x-2{)}^2+y^2}=x$. Its right side must be nonnegative. Squaring then gives $y^2=4(x-1)$, whose points already have $x\ge 1$, so no extraneous branch has been introduced:This is a parabola opening along the positive real axis, with vertex $(1,0)$, focus $(2,0)$ and directrix $x=0$. The original modulus equation equates distance from the focus with distance from that directrix.

Put $z=x+iy$ and $z^{\ast }=x-iy$. The first two squared terms are complex conjugates, so their sum is $2[(x-2{)}^2-(y+1{)}^2]$. Also $z{z}^{\ast }=x^2+y^2$. The equation therefore reduces toThus the locus is the upward-opening parabolawith vertex $1+2i$ and axis $x=1$ in the Argand diagram.

The boundaries intersect where $x^2=2x+8$, giving $x=-2$ and $x=4$. Between these values, the straight line lies above the parabola. ThereforeEvaluation givesA useful independent check uses the region's symmetry about $x=1$: its vertical width is $9-(x-1{)}^2$. Its area is ${\int }_{-3}^3(9-u^2)du=36$, and its centroid has $x$ coordinate one, so its first moment $\iint xdA$ is again $36$.

For $-1\le y\le 0$, a horizontal slice runs from $x=-1$ to the line $x=y$. For $0\le y\le 1$, it runs from $x=-1$ to the negative branch $x=-\sqrt{y}$ of the parabola. Thus the reversed iterated integral isThe positive branch $x=\sqrt{y}$ is outside the original integration region.

The integration region lies between the line $y=x$ and the parabola $y=x^2$, with $-1\le x\le 0$. It has a portion below the $x$-axis bounded above by $y=0$, and a portion above it bounded below by $y=0$. Horizontal slices change form at $y=0$; keeping that split is essential when reversing the order of integration.

Write $z=x+iy$. Dividing by two givesThe left side is nonnegative, so squaring is reversible only when $x+a\ge 0$. Squaring yields $y^2=4ax$.

If $a>0$, this equation implies $x\ge 0$, so the sign condition holds automatically. The locus is the parabolawith vertex at the origin, focus of a conic $(a,0)$ and directrix of a conic $x=-a$. Indeed the original equation equates distance from the focus of a conic with distance from the directrix of a conic on its right side.

If $a<0$, the squared equation forces $x\le 0$, making $x+a<0$. This violates the sign condition. Thus the locus is empty for $a<0$; squaring without checking that condition would introduce a spurious parabola.