Mit seinem Hauptwerk "Philosophiae Naturalis Principia Mathematica" begründete Isaac Newton im Jahr die klassische Physik. Kein Ladenhüter: Als der große Gelehrte Isaac Newton im Jahr sein Hauptwerk "Principia" veröffentlichte, legte er den Grundstein für die. Quis jam talis haberi poterit, nisi ratio pura ejusque principia practica, cum ratio sit praestantissimum Dei donum mortalibus concessum, quo homo imaginem.
Isaac Newtons „Principia“ war ein BestsellerEx his autem jam patet, Philosophos hos & inter fe, & ab ipfo Thalece suo antelignano quoad rerum principia dissensille. §. XI. Jam verò, si reli & ta fe & ta. Philosophiae Naturalis Principia Mathematica, oft auch Principia Mathematica oder einfach Principia genannt, ist das Hauptwerk von Isaac Newton. Der lateinische Titel bedeutet übersetzt Die mathematischen Grundlagen der Naturphilosophie. Principia bezeichnet: Principia (Stabsgebäude), das Stabsgebäude in römischen Militärlagern; The System of the World, Buch von Neal Stephenson, siehe.
Principia Search form VideoNewton's Principia Explained Part I
In and , Newton dealt only with orbital dynamics; he had not yet arrived at the concept of universal gravitation.
Nearly five years later, in August , Newton was visited by the British astronomer Edmond Halley , who was also troubled by the problem of orbital dynamics.
Already Newton was at work improving and expanding it. Significantly, De Motu did not state the law of universal gravitation. For that matter, even though it was a treatise on planetary dynamics , it did not contain any of the three Newtonian laws of motion.
Only when revising De Motu did Newton embrace the principle of inertia the first law and arrive at the second law of motion.
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If an internal link led you here, you may wish to change the link to point directly to the intended article. With this, the body can be viewed as driven from one instantaneous circle to the next by the component of force tangential to the motion, a component that disappears in the case of uniform circular motion.
Newton illustrates the value of Proposition 6 with a series of examples, the two most important of which involve motion in an ellipse.
But if the force center is at the center C of the ellipse, the force turns out to vary as PC, that is, linearly with r.
This contrast raises an interesting question. What conclusion can be drawn in the case of motion in an ellipse for which the foci are very near the center, and the center of force is not known to be exactly at the focus?
Newton clearly noticed this question and supplied the means for answering it in the Scholium that ends Section 2. Section 10 includes a philosophically important result that has gone largely unnoticed in the literature on the Principia.
Newton's argument that terrestrial gravity extends to the Moon depends crucially on Huygens's precise measurement of the strength of surface gravity.
This theory-mediated measurement was based on the isochronism [ 36 ] of the cycloidal pendulum under uniform gravity directed in parallel lines toward a flat Earth.
But gravity is directed toward the center of the nearly spherical Earth along lines that are not parallel to one another, and according to Newton's theory it is not uniform.
So, does Huygens's measurement cease to be valid in the context of the Principia? Newton recognized this concern and addressed it in Propositions 48 through 52 by extending Huygens's theory of the cycloidal pendulum to cover the hypocycloidal pendulum — that is, a cycloidal path produced when the generating circle rolls along the inside of a sphere instead of along a flat surface.
Proposition 52 then shows that such a pendulum, although not isochronous under inverse-square centripetal forces, is isochronous under centripetal forces that vary linearly with the distance to the center.
Insofar as gravity varies thus linearly below the surface in a uniformly dense sphere, the hypocycloidal pendulum is isochronous up to the surface, and hence it can in principle be used to measure the strength of gravity.
A corollary to this proposition goes further by pointing out that, as the radius of the sphere is increased indefinitely, its surface approaches a plane surface and the law of the hypocycloidal asymptotically approaches Huygens's law of the cycloidal pendulum.
This not only validates Huygens's measurement of surface gravity, but also provides a formula that can be used to determine the error associated with using Huygens's theory rather than the theory of the hypocycloidal pendulum.
Thus, what Newton has taken the trouble to do in Section 10 is to show that Huygens's theory of pendulums under uniform parallel gravity is a limit-case of Newton's theory of pendulums under universal gravity.
At the end of Section 2 he points out in passing that this limit strategy also captures Galileo's theory of projectile motion.
In other words, Newton took the trouble to show that the Galilean-Huygensian theory of local motion under their uniform gravity is a particular limit-case of his theory of universal gravity, just as Einstein took the trouble to show that Newtonian gravity is a limit-case of the theory of gravity of general relativity.
Newton's main reason for doing this appears to have been the need to validate a measurement pivotal to the evidential reasoning for universal gravity in Book 3.
From a philosophic standpoint, however, what is striking is not merely his recognizing this need, but more so the trouble he went to to fulfill it.
Section 10 may thus illustrate best of all that Newton had a clear reason for including everything he chose to include in the Principia.
Section 9 includes another often overlooked result that is pivotal to the evidential reasoning for universal gravity in Book 3. Proposition 45 applies the result on precessing orbits mentioned earlier to the special case of nearly circular orbits, that is, orbits like those of the then known planets and their satellites.
This proposition establishes that such orbits, under purely centripetal forces, are stationary — that is, do not precess — if and only if the centripetal force governing them is exactly inverse-square.
This result is striking in three ways. Third, even when an orbit does precess, once such a fractional departure of the exponent from -2 is shown to result from the perturbing effect of outside bodies, then one can still conclude that the force toward the central body is exactly This is precisely the strategy Newton follows in concluding that the centripetal force on the Moon, once a correction is made for the perturbing effects of the Sun, is inverse-square.
Propositions 1 and 2 establish that a motion is governed purely by centripetal forces if and only if equal areas are swept out in equal times.
The second and third corollaries of Proposition 3 then yield the conclusion that a motion is quam proxime governed purely by centripetal forces if and only if equal areas are quam proxime swept out in equal times.
These propositions— which Newton has taken the trouble to show still hold in a quam proxime form — are the very ones he invokes in Book 3 to conclude that the forces retaining bodies in their orbits in our planetary system are all centripetal and inverse-square.
A failure to notice these quam proxime forms in Book 1 blinds one to the subtlety of the approximative reasoning Newton employs in Book 3.
The purpose of Book 2 is to provide a conclusive refutation of the Cartesian idea, adopted as well by Leibniz, that the planets are carried around their orbits by fluid vortices.
Newton's main argument, which extends from the beginning of Section 1 until the end of Section 7, occupies 80 percent of the Book. Section 9, which ends the Book, offers a further, parting argument.
We best dispense with this second argument before turning to the first. The argument has two shortcomings, both of them recognized by Newton's opponents at the time.
Second, his analysis of the vortex generated around a rotating cylinder or sphere involves fundamentally wrong physics: it defines steady state in terms of a balance of forces instead of torques across each shell element comprising the vortex.
The argument that carried much more weight at the time — it convinced Huygens, for example — is the one that extends across the first seven sections of the Book.
The thrust of this argument is clear from its conclusion, as stated more forcefully in the second and third editions than in the first:.
To reach this conclusion Newton had to show that 1 the inertia of the fluid does indeed produce a resistance force proportional to its density, a force that 2 is independent of the tenacity that is, surface friction and the friction of the parts that is, viscosity of the fluid.
The theory in Book 1 is generic in that it examines centripetal forces that vary as different functions of the distance from the force center. The theory in Book 2 is generic in that it examines motion under resistance forces that vary as the velocity, the velocity squared, the sum of these two, and ultimately even the sum of two or three independent contributions, each of which is allowed to vary as any power of velocity whatever.
Because Newton's goal was to reach a conclusion about the contribution to the total resistance made by the inertia of the fluid, and he recognized that surface friction and viscosity can contribute to the resistance as well, his empirical problem became one of disaggregating the inertial contribution from the total resistance, that is, the contribution that alone varies with the density of the fluid.
Fortunately, because gravitational forces so totally dominate celestial motions, this need to disaggregate different sorts of forces did not arise in Book 3.
Some preliminary pendulum-decay experiments showed promise for doing this, leading him in the first edition to rely solely on this phenomenon.
The idea was to start a pendulum from several different heights in order to cover a range of velocities and then to use simultaneous algebraic equations to fit a two or three term polynomial to two or three lost-arc data-points, changing the exponents until the polynomial achieved good agreement with the other lost-arc data points.
The theoretical solutions for pendulum motion under resistance forces in Section 6 would then allow him to infer the forces from the rate of decay of the pendulum.
These theoretical solutions covered resistance forces that vary not only as velocity to the powers 0, 1, and 2, but also as any power at all of velocity.
In principle, therefore, he saw himself in a position to infer laws for resistance forces on spheres from the phenomenon of pendulum decay in full parallel with his deduction of the law of universal gravity from the phenomena of orbital motion in Book 3.
And he could then conclude from the total absence of signs of resistance forces acting on the planets and, most especially, comets that the density of any fluid in the celestial regions must be exactly or very nearly zero.
Unfortunately, pendulum-decay turned out not to be as well behaved a phenomenon as Newton anticipated it was going to be while he was working on the first edition.
The General Scholium following Section 6 [ 39 ] reports detailed decay-rate data for an impressive range of experiments, including different size bobs in air and bobs moving as well in water and mercury.
The reader is also shown in detail how to proceed from the data in each case to a polynomial as above defining the resistance force. Any reader who worked through the data discovered what Newton knew, but was less than candid about: no polynomial fit the data.
The experiments did clearly indicate that resistance forces involve no power of velocity greater than 2, and they provided good evidence that a velocity squared effect was dominant, even to the extent of masking any effect involving some other power.
Newton also managed to extract some highly qualified evidence that the velocity squared effect varies as the density of the fluid and the frontal area that is, the square of the diameter of spheres.
The approach to resistance in the first edition relied entirely on pendulum-decay experiments. The disappointing evidence they yielded led to a far weaker statement of the conclusion about the absence of fluid in the celestial regions in the first edition than the conclusion in the subsequent editions quoted above.
Not long after the first edition was published, Newton initiated some vertical-fall experiments in water that persuaded him that the phenomenon of vertical-fall in resisting media would yield much better behaved data.
In the second and third editions, therefore, even though the pendulum-decay experiments are still fully reported, the central argument in Book 2 relies on vertical-fall experiments including ones from the top of the dome of the newly completed St.
Paul's Cathedral to establish a resistance effect on spheres that is proportional to the density of the fluid, the square of the diameter, and the square of the velocity.
The data from these experiments were very good — indeed, even better than Newton realized, for small vagaries in them that he dismissed as experimental error were in fact not vagaries at all, but evidence that no polynomial of the sort he was seeking is adequate for resistance forces.
While the vertical-fall experiments put Newton in a position to make his concluding rejection of vortex theories more forceful, they also posed a methodological complication.
The vertical-fall experiments offered no way of disaggregating the contribution to resistance made by the inertia of the medium from the total resistance.
But the argument against vortices required him to show that, no matter how perfectly free of friction and viscosity the celestial fluid might be, its inertia would still give rise to resistance forces that would affect the motions of comets, if not planets as well.
From the resistance measured in the pendulum-decay experiments, Newton could conclude that the forces in air and water are dominated by a contribution that varies as the velocity squared.
In the vertical-fall experiments in air and water the measured forces varied to first approximation as the product of the density and the velocity squared, but only to a first approximation, leaving room to question whether a purely inertial contribution had been isolated.
Newton dealt with this problem by offering a rather ad hoc theoretical derivation for the purely inertial contribution, showing how closely it agreed with the vertical-fall results, and proposing that the differences between the theoretical and the measured resistances could be used to investigate other contributions.
Success of such a program in characterizing the contributions made by surface friction and the viscosity would have provided compelling support for Newton's theory of the inertial contribution.
Still, the approach left Newton with not so straightforward a derivation of the laws of resistance forces from phenomena as he had hoped for in the first edition.
In fact, there is a deep mistake in Newton's approach to resistance forces that came to be understood only at the beginning of the twentieth century.
Resistance forces do not arise from independent contributions made by such factors as the viscosity and inertia of the fluid. Consequently, no polynomial consisting of a few always positive terms in powers of velocity can ever be adequate for resistance forces.
The first indication of this came when d'Alembert, unhappy with Newton's ad hoc theory for the inertial contribution, analyzed the flow of what we now call a perfect fluid about spheres and bodies of other shapes, discovering in all cases that the net force of the fluid is exactly zero.
Consequently, contrary to Newton, there is no such thing as the contribution made to resistance purely by the inertia of the fluid.
Resistance forces always arise from a combination of viscous and inertial effects, however low the viscosity of the fluid may be. Newton's assumption that resistance forces can be represented as a sum, one term of which gives the contribution made purely by the inertia of the fluid, was wrong empirically, much as his assumptions about simultaneity and space being Euclidean turned out to be wrong.
Unlike the latter assumptions, however, the assumption about resistance amounted to a dead end. All Newton achieved in Book 2 with resistance forces was merely a curve-fit.
Nevertheless, the main body of it does consist of four clearly separate parts: 1 the derivation of the law of gravity Props. These parts will be discussed in sequence below.
Newton's first two rules of reasoning appeared in the first edition there labeled as hypotheses [ 41 ] , the third rule was added in the second edition, and the fourth rule, in the third edition.
These are rules intended to govern evidential reasoning in natural philosophy, akin to rules of deductive reasoning except for their very much not guaranteeing true conclusions from true premises.
The philosophic question why Newton's rules are appropriate is best addressed not by asking how they increase the probability of truth, but by asking whether there is some strategy in ongoing research for which these rules will both promote further discoveries and safeguard against dead-end garden paths that ultimately require all the supposed discoveries to be discarded.
The ellipse, by the way, is not one of the phenomena. In Phenomenon 3 Newton rules out the Ptolemaic system, just as Galileo had in his Dialogue Concerning the Two Chief World Systems , by appealing to the phases of Mercury and Venus and their absence in the case of Mars, Jupiter, and Saturn to conclude that these five orbits encircle the Sun.
But this Phenomenon and all the others are carefully formulated to remain neutral between the Copernican and Tychonic systems. In Phenomenon 4 Boulliau's calculated orbits are treated on a par with Kepler's, indicating that the phenomena do not rule out the possibility that Boulliau's alternative to the area rule is correct.
Phenomenon 6 explicitly grants that the area rule holds only approximately for the Moon, with a further remark indicating that none of the phenomena are being put forward as holding exactly.
This points the way to the most reasonable reading of all of the phenomena: they describe to reasonably high approximation, but not more than that, the observations of the planets and their satellites made by Tycho and others over a finite period of time — roughly from to the time of Newton's writing.
On this way of viewing the Phenomena, they are in no way contentious or problematic. They leave entirely open not only questions about whether any claims concerning the orbits made by Kepler and his contemporaries hold exactly, but also questions about whether any of these claims hold even remotely in other eras, past or future.
The Phenomena are thus not inconsistent with Descartes' insistence that the motions are constantly changing. Of course, this means that the deduction shows only that the conclusions, most notably the law of gravity, hold quam proxime over the restricted period of time for which the premises hold.
The Rules of Reasoning then license the conclusion to be taken exactly, without restriction of space or time. The conclusions, so taken, do indeed then show that the premises hold only quam proxime , and not exactly.
This conclusion in no way contradicts the premises. While, however, the orbits of Venus, Jupiter, and Saturn were considered to be very nearly circular, the motion in them had been known from before Ptolemy not to be uniform.
The phenomena really are sufficient to reach the conclusion in this weak form. So, the complaint has bite only when the law of gravity is taken to be exact.
That this be the case amounts to a requirement on the deduction from phenomena: the leap to taking the law of gravity as exact is justified only if it yields circumstances in which the phenomena from which it was inferred would hold exactly.
But the magnitude for the action of the Sun that he gives in Proposition 3 [ 49 ] is twice the value he later in Book 3 indicates is the correct value.
This lacuna was not resolved by Alexis-Claude Clairaut until two decades after Newton died. Second, when Newton invokes the third law of motion in the corollaries to Proposition 5, he is tacitly assuming that, for example, Jupiter and the Sun are, in effect, directly interacting.
In other words, he is ignoring the alternative favored by Huygens that some unseen medium is effecting the centripetal force on Jupiter, a medium that can in principle absorb the linear momentum which Newton is assuming is being transferred to the Sun.
Huygens may well have perceived this lacuna, to which Cotes explicitly called Newton's attention while he was preparing the second edition. The group of propositions following the deduction of universal gravity gives indications of the evidential strategy that lies behind the leap to taking this law to be exact.
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