The Laws of Motion - PhilSci-Archive



The Mechanical Philosophy

1660 – 1675

MechPhil2white, 2 August 2007

E B Davies[1]

Abstract We study the state of mechanics and astronomy between 1660 and 1675 in order to understand the extent of the commitment to the mechanical philosophy of Kepler prior to the writing of Principia.

Introduction

This article studies the state of mechanics and astronomy between 1660 and 1675. This period was not dominated by any one person, but saw many momentous developments. Among these were the foundation of the Royal Society in 1660 and the Academie des Sciences in 1666. The first issue of the Journal des Sçavans appeared early in 1665, a few months before the start of the Philosophical Transactions (of the Royal Society). These heralded a key change in scientific philosophy – that even small developments in scientific understanding should be documented quickly rather than being treated as personal property, in some cases for decades.

Henry Oldenburg was the first editor of the Philosophical Transactions, as well as being the founding Secretary of the Royal Society. His correspondence is of great value because many scientific communications were made via him.[2] This correspondence was the origin of the modern practice of scientific refereeing. Birch[3] provides a detailed record of the meetings of the Royal Society from its inception, in strict temporal order.

The period covered by this article excludes the controversies involving Hooke and Newton in the period leading up to the publication of Principia in 1687. We pay particular attention to some selected publications of Wallis because less has been written about him than about many of the other major contributors to our subject in the seventeenth century. We should, however, mention an ongoing project to record all of his correspondence[4] and a brief general biography by Scott.[5] Wallis was one of the outstanding mathematicians of the seventeenth century, but his reputation suffered from the fact that his contributions were overshadowed by those of Newton in Principia.[6] This is a pity, because they were of substantial importance in their own right and provide fascinating evidence of the level of scientific understanding in Europe during the period. In 1671 Wallis published his magisterial treatise on mechanics, ‘Mechanica: sive De Motu, Tractatus Geometricus’, which, more than almost anyone else, provided the mathematical underpinning of mechanics that was later so important to Newton. Huygens was as great a scientist as Wallis, but he did not publish his fundamental treatise “De motu corporum ex percussione” of 1656, which therefore had less influence that it should have.

In the body of the paper we try to avoid references to Principia, but we discuss the influence of Wallis and others on Newton in the final section.

Kepler

Johannes Kepler is important not only for his eponymous Laws, but because of his role in promoting the world view often known as the mechanical philosophy. In 1609 (before Galileo’s dramatic discoveries using his telescope) he argued in Astronomia Nova that astronomical theory should not be about appearances, and that it should be based upon and integrated into natural philosophy (physics).[7] He wrote

“ Indeed all things are so interconnected, involved, and intertwined with one another that after trying many different approaches to the reform of astronomical calculations, some well trodden by the ancients and others constructed in emulation of them and by their example, none other could succeed than are founded upon the motions’ physical causes themselves, which I establish in this work.[8]”

Voelkel and others have emphasized that, as Kepler’s philosophical starting point, this was an essential ingredient in his later derivations of his Laws.[9] Although Kepler’s controversial proposal was of great importance, he could not implement it fully, because the understanding of mechanics was so defective at that time.

Kepler's use of mathematics to describe the physically caused orbits of the planets must be contrasted with geometrical descriptions of the appearances of the orbits. He consistently used the word ‘orbita’ only when referring to the former.[10] However, his first Law was not trusted, even by Newton, who pointedly did not derive the inverse square Law of Gravitation from the ellipticity of planetary orbits in Volume 3 of Principia.[11] On the other hand Newton declared that Kepler’s third Law, T2 ~ R3, was accepted by everyone.[12] This was also the view of Thomas Streete, the author of the extensive astronomical tables in “Astronomia Carolina”, published in 1661.

As an example of his philosophy, Kepler used the notion of centre of gravity, taken from mechanics, in his discussion of the effects of gravity on the Moon and Earth.

“If the moon and the earth were not each held back in its own circuit by an animate force or something else equivalent to it, the earth would ascend towards the moon by one fifty-fourth part of the interval, and the moon would descend towards the earth about fifty three parts of the interval, and there they would be joined together; provided, that is, that the substance of each is of the same density.[13]”

It appears from these passages that he considered that an explanation of the orbits of the primary and secondary planets was needed, but he did not identify gravity as providing this. He supposed that the Sun emitted an immaterial species (somewhat like light[14]) that was dispersed as it got further away from the Sun. This species rotated around the Sun as a result of the Sun’s own rotation, and the planets were impelled by it to move in orbits around the Sun.[15] He also argued that the species might well be magnetic in nature, but kept this conjecture separate from the remainder of Chapter 34.[16] His theory, partly inherited from Gilbert, differed from the later, but equally incorrect, vortex theory of Descartes. Both survived as long as they did in spite of there being no observational support for them because no better proposals were forthcoming.

The Vacuum

It is well known that René Descartes played an important role in the eventual defeat of the scholastic philosophy. Anticipating opposition, and mindful of the fate of Galileo, he delayed publishing his views about physics, but his “Principles of Philosophy” eventually appeared in 1644. His agreement with the Aristotelians about the non-existence of the vacuum therefore became public after Evangelista Torricelli’s construction of the first barometer in 1643, creating a vacuum at the top of a 35 foot column of water. In 1644 Torricelli wrote that “We live submerged at the bottom of an ocean of the element air, that by unquestioned experiments is known to have weight.[17]” Torricelli’s work created considerable controversy, but in 1647, with the help of Perier, Pascal confirmed his ideas by showing that air pressure decreased as one goes up a mountain. These discoveries provided further support for the mechanical philosophy and implied that at a sufficient height the atmosphere would come to an end.

In 1660 Robert Boyle published his “New Experiments Physio-Mechanicall, Touching the Spring of the Air and its Effects”, describing a series of experiments performed in a 38cm glass chamber that had been evacuated using an air pump designed by his assistant Robert Hooke.[18] His discovery of the relationship between pressure and volume (Boyle’s Law) reinforced the idea that air was a material substance that could be understood within the framework of the mechanical philosophy. Huygens was shown Boyle’s apparatus in 1661 when he visited London, and was stimulated to design his own apparatus and carry out further experiments with it.

A 36 page article of John Wallis on gravitation was presented to the Royal Society on 12 November 1674,[19] and shows that Wallis had fully absorbed the lesson of Torricelli and others about the existence of the vacuum. He rejected the arguments of Descartes, Linus and others as follows:

“And had the Ancients been aware of what we find; That the Air hath a positive Gravity; and, consequently, though it be but small in proportion to that of other bodies, yet a great height of Air may countervail a lesser height of a Heavier Liquor; (like as we see that a greater height of Water may countervail a lesser height of Quick-silver:) They would not, I presume, have troubled themselves with a Fuga vacui.[20]”

Wallis on the Earth-Moon system

We next discuss a paper of Wallis on the tides[21], dated 6 August 1666, which was copied from a letter written to Boyle on 25 April 1666. Wallis was by no means an astronomer, but, like most others by this time, he was fully committed to the explanation of astronomical phenomena within the mechanical philosophy, about which he knew as much as anyone.

“For, since that Galilaeo, and (after him) Torricellio, and others, have applied Mechanick Principles to the salving of Philosophical Difficulties; Natural Philosophy is well known to have been rendered more intelligible, and to have made a much greater progress in less than an hundred years, than before for many ages.[22]”

He further wrote

“We shall first take for granted, what is now adayes pretty commonly entertained by those, who treat of such matters; That a Body in motion is apt to continue its motion, and that in the same degree of celerity, unless hindered by some contrary Impediment; (like as a Body at rest, to continue so, unless by some sufficient mover, put into motion:)[23]”

He applied this rule to the centre of gravity of the Earth and Moon regarded as an aggregate body; in other words he applied the law of conservation of momentum, derived by studying collisions of small bodies on Earth, to a pair of astronomical bodies that were associated or tied by unknown means. This idea was based upon a leap of imagination rather than evidence. He went beyond Kepler’s counterfactual statement about what would happen to the earth and moon in the absence of his ‘animate force’, by making a positive (and correct) statement about how the Earth and Moon actually moved; see below. He made many references to the orbital motions of the Earth, Moon and other planets, but did not suggest that these were controlled by gravitational forces. Indeed his conjecture that the motions were in some way the result of the rotation of the bodies involved, was taken directly from Kepler’s Astronomia Nova.[24]

Wallis stated that he did not know the cause of the constant association of the Earth and Moon.[25] However, since they were so associated, the laws of statics suggested that they should be considered as one aggregate body with a common centre of gravity.

“Now supposing the Earth and Moon, jointly as one Body, carried about by the Sun in the great Orb of the Annual motion; this motion is to be estimated, (according to the Laws of Staticks, in other cases,) by the motion of the common Center of Gravity of both Bodies. For we use in Staticks, to estimate a Body, or Aggregate of Bodies, to be moved upwards, downwards, or otherwise, so much as its Common Center of Gravity is so moved, howsoever the parts may change places amongst themselves.[26]”

His calculations suggested that the centre of gravity of the Earth and Moon (or more properly what we would now call their centre of mass), which necessarily lies on the straight line joining them, is about 4/3 of the Earth’s radius from the centre of the Earth towards the Moon, i.e. slightly outside the Earth’s surface.[27]

Wallis emphasized that his explanation of the tides was conjectural, and that he only presented it so that others might examine the ideas there. He rightly criticized Galileo’s theory for not explaining the daily progression of the tides.[28] To resolve this weakness, he (wrongly) suggested that the circular motion of the Earth around the centre of gravity of the Earth-Moon system (now called the Earth-Moon wobble[29]) would induced a monthly component to the tides. Although Wallis discussed the effects of both Sun and Moon on the tides, he was in no position to consider the Earth-Sun-Moon three-body dynamics, whose mathematical complexities later defeated even Newton. Wallis also suggested that the wobble might well be responsible for inexplicable perturbations of the orbit of the Moon, but once again this was unfounded.[30] Following Newton’s lead, Clairaut, Laplace and others eventually proved that the various anomalies in the Earth’s orbit and of its axis of rotation depended on solving the three-body problem and/or using the oblateness of the Earth. Newton himself showed that the only effect of the mutual interaction of the Earth and Moon (assumed to be spherical) on their relative motion was to alter their effective masses slightly, but since the mass of the Moon was unknown this could not have been detected.

There is an interesting Appendix to his article,[31] in which Wallis replied to some objections to his ideas. The first of these was that there was no explanation of how two bodies that had no tie could have a common centre of gravity, or could act in the same manner as if they were connected. Wallis’s reply to this was characteristic of his general attitude, and was copied by Newton. His first response was by analogy with the (magnetic) attraction that ties together Loadstones and needles that are not in contact. His second response was that he might not be able to say how the Earth and Moon were connected, but nobody could doubt that they were connected: the Moon and Earth “keep a respective position to one another” (i.e. keep at a constant distance). For Wallis, as for Newton later, providing a mathematical account of a phenomenon was itself a major achievement, whether or not a physical explanation was also available.

Parallax Measurements

Observational evidence for the Earth-Moon wobble would most plausibly be obtained by finding a monthly variation in the position of Mars, i.e. by observing its parallax when the Earth is at opposite sides of its orbit around the Earth-Moon centre of gravity. Wallis raises this possibility in the Appendix,[32] but admits that the parallax of Mars had at that time not been observed. Parallax due to the wobble does indeed exist, but observing it was beyond the technology of that period, for the following reasons.

Seventeenth century astronomers often claimed greater accuracy for their observations than could be justified by the instruments that they were using. A detailed discussion of this may be found in an article of Chapman.[33] He examined the accuracy of the graduated arcs of six instruments, but was careful not to make any statements about the overall performance of the complete instruments in their original state. Although the larger arcs tend to be more accurate, the abilities of the craftsmen who made them were very important. Obtaining an accuracy of better than 40 seconds over the entire range of an arc was clearly extremely hard, but the mural arc built by Sharp for Flamsteed in 1688, which had a radius of 7 feet and is now lost, might have had an accuracy of about 15 seconds. One also needs to distinguish between precise measurements of the angle between two widely separated points in the sky, and between two points that are very close to each other. In the second case, which is relevant to some parallax observations, one might expect to measure the angle involved more accurately.

The first observations of the parallax of Mars were made in 1672. In that year Richer went to Cayenne to measure the position of Mars. At the same time Cassini measured it in Paris. From the observed parallax they obtained the first measurement of the scale of the Solar System, or of the mean distance from the Earth to the Sun, called the Astronomical Unit. Their observed parallax of 24 seconds of arc indicated that the AU was about value of 140 × 106 km, and should be compared with the modern value of 149.6 × 106 km. On 6 October 1672, when Mars was in opposition, Flamsteed measured the parallax between two observations of Mars separated by six hours, and deduced a value of the AU of about 131 × 106 km. All of the above measurements were so close to the limits of what was possible with the instruments available, that it was not surprising that the values obtained were not accurate. Distances recorded in multiples of AUs were much more accurate and were the basis of almost all calculations for many more decades.

Measuring the position of Mars to find a monthly variation would have been harder than either of the above measurements. Using modern data we can calculate that the parallax of Mars at opposition (i.e. when it is closest to the Earth) when measured from opposite sides of the Earth-Moon orbit is very roughly 25 seconds. In order to obtain the largest possible parallax one would need to wait two weeks (a half of the Moon’s orbital period), by which time the position of Mars in the sky would have changed substantially. One would either have to calculate and then compare the expected change in position of Mars in the absence and in the presence of the Earth-Moon wobble, or look for statistical evidence of a small monthly oscillation over a long time period. The theoretical understanding of planetary orbits was not sufficient to make the first possible, while statistical analysis was not a technique that had been developed.

It is worth mentioning that in 1669 Hooke made great efforts to detect the parallax of the star Gamma Draconis.[34] Although he had good reasons for choosing this star, its distance from the Sun is about 45 parsecs. This corresponds to an angular variation of about 0.04 seconds of arc as the Earth orbits the Sun, and was far beyond the resolving power of Hooke's telescope. He thought that he had observed the parallax, but actually the effect that he had seen had a different explanation.

1668/9 The Laws of Motion

This section describes in some detail the contents of and background to three papers by Wallis, Wren and Huygens.[35] Oldenburg invited them to submit articles about their Laws of Motion, and duly published these in the Philosophical transactions in 1669. The article of Wallace is the only one of the three to use mathematical equations in a substantial way.[36] It has the title:

“A Summary Account given by Dr John Wallis, of the General Laws of Motion, by way of Letter written by him to the Publisher, and communicated to the R. Society, Novemb. 26 1668”.

Wallis’s article is excessively brief, but he was presumably too busy writing his magnum opus on the subject to want to make it longer. One has to infer his knowledge of the Laws from calculations involving a variety of examples. He wrote to Oldenburg several times during December 1668 to further explain his ideas. His treatment is novel in considering inelastic collisions, in which two bodies combine during the collision and continue afterwards as one. The first few items of Wallis’s article provided a formula for calculating the ‘Vis’ (momentum in our language) of a body.

Item 10 states (in modern language) that if a body of mass (Pondus) P and speed (Celeritas) C impinges on a body of mass mP and speed nC, then the resulting body has speed

C' = (1 + mn)C/(1 + m).

The calculation involves adding the momentum V = PC of the first body to the momentum mnV = mnPC of the second to obtain the total momentum (1 + mn)V of the combined body. In other words Wallace uses the law of conservation of momentum, without stating it in words.

In Item 11 Wallis considers two bodies initially travelling in opposite directions and then colliding to form a single body whose speed is

C' = (1 – mn)C/(1 + m) .

He then adds that this is towards the right or the left depending on whether 1 or mn is larger. This shows that celeritas should be translated as velocity rather than speed in this context, because its direction depends on its sign. Failure to take this issue into account properly had led Descartes into serious error.[37] Indeed several of Descartes’ rules governing collisions are so far from the truth that he clearly made no attempt to test them experimentally.[38]

The article of Wallis is followed immediately by that of Wren.[39] Its title is

“Dr Christopher Wrens Theory concerning the Same Subject; imparted to the R. Society Decemb. 17 last, though entertain’d by the Author diverse years ago, and verifyd by many Experiments, Made by Himself and that other excellent Mathematician M. Rook before the said Society, as is attested by many Worthy Members of that Illustrious Body”.

Lawrence Rooke, not to be confused with Robert Hooke, was a founder member of the Royal Society, and died in 1662. Hall translates this article in its entirety, and comments that several of the key notions are not defined, but that it was agreed by all concerned that it agreed in content with the other two papers. We will not discuss it further.

Much of Huygens’ work on the collisions of bodies was carried out in the period between 1652 and 1656, and he wrote a connected account of the subject in the treatise “De motu corporum ex percussione”. Unfortunately he did not publish it, possibly because of discouragement by Van Schooten in 1654.[40] A detailed account of the treatise, which was printed posthumously in 1702, and of Huygens’ correspondence may be found in his 22 volume Oeuvres Complètes.

The publication of Huygens’s article was preceded by several exchanges between him and the editor, Oldenburg. Huygens knew that he had scientific priority over Wren and Wallace, and was angry that his article was not published alongside the other two in the Philosophical Transactions. As a result Huygens decided to submit a French version of his article to the Journal des Scavans.

The article of Huygens later appeared in the Philosophical Transactions, where it has the title

“A Summary Account of the Laws of Motion, communicated by Mr. Christian Hugens in a Letter to the R. Society, and since printed in French in the Iournal des Scavans of March 18, 1669. st. n.[41]”

The controversy mentioned above is alluded to in an opening statement by the editor, Oldenburg, namely

“Before these Rules of Motion be here deliver’d, ‘tis necessary to preface something, whereby the worthy Author of them may receive what is unquestionably due to him, yet without derogating from others, with whom in substance he agreeth.”

After a preamble describing the history of the Laws, Huygens stated his Laws of Motion. His procedure (Law 4) for determining the outcome of a collision between two hard bodies was graphical in nature. Law 5 concerns the conservation of momentum in collisions. It states

“The quantity of motion of two bodies may be increased or decreased by their collision; but the same quantity always remains in any direction, after subtracting the quantity of contrary motion.”

The first part may have been contrasting his law with the incorrect formulation of Descartes. In the second part the emphasis on directions and the need to subtract contrary motions was crucial; it was copied very closely by Newton in Principia.

Huygen’s Law 6, which we would now call the law of conservation of kinetic energy, is only applicable to bodies that are perfectly elastic. It states

“The sum of the products obtained from the size of each hard body, multiplied by the square of its speed, is always the same before and after their collision.”

1671 Wallis’s De Motu

The monumental treatise of Wallis entitled ‘Mechanica: sive De Motu, Tractatus Geometricus’ occupies 493 pages of volume 1 of his collected works.[42]

It was published in three parts, dated 1669, 1670 and 1671. Scott[43] has provided a brief discussion of the contents of the treatise, but it can only be appreciated properly by being read. Part 2, namely Chapters 4 and 5, about centres of gravity, makes up more than a half of the entire work, and is highly mathematical. Chapters 6 to 9 describe applications of his mechanics to a variety of machines and pulley systems, many beautifully illustrated. The following list of chapter headings gives some idea of its range, which does not, however, include astronomical applications of mechanics.

Dedication

Chap 1 De Motu Generalia;

Chap 2 De Gravium Descensu & Motuum Declivitate;

Chap 3 De Libra;

Chap 4 De Centro Gravitatas;

Chap 5 De Calculo Centri Gravitatas;

Chap 6 De Vecte;

Chap 7 De Axe in Peritrochia & Machinis cognatis;

Chap 8 De Trochlea, seu Polyspasto;

Chap 9 De Cochlea;

Chap 10 De Motibus Compositis, Acceleratis, Retardatis & Projectorum;

Chap 11 De Percussione;

Chap 12 De Cuneo;

Chap 13 De Elatere & Resilitione seu Reflexione;

Chap 14 De Hydrostaticus;

Chap 15 Epilogus, ex Miscellaneis

It is not easy to penetrate Wallis’s use of the words vis, momentum and impetus in the treatise, but fortunately it is not necessary. His mathematical formulae are crystal clear and demonstrate his complete understanding of the subject. In particular his frequent use of the formula V = PC where P is the weight and C is the speed of the body, makes clear that V is what we now call momentum; he refers to Prop. 27, Chap. 1, p. 592 for this formula.

Prop. 3, Chap. 11, p. 1004 – 5 is a slight modification of item 10 of Wallis’s 1669 paper on inelastic collisions, which we have already discussed, but uses the term ‘momentum’ in place of ‘vis’. Chapter 13 derives the analogous laws for a variety of collisions involving perfectly elastic bodies. Prop. 8, Chap. 13, p. 1028 states in modern language that if an elastic body A with weight mP and speed rC collides with a stationary body B with weight nP, then the speed of A after the collision is

(m-n)rC/(m+n) while the speed of B is 2mrC/(m+n). In order to derive these (correct) formulae Wallis used the conservation of total momentum and a formula for the relative speeds before and after the collision.

The impact of the treatise can be gauged by the fact that it continued to be praised for many decades. It is one of the main reasons why Newton did not attempt to claim originality for the Laws of Motion as written in the ‘Axioms’ section preceding Book 1 of Principia. Principia Book 1 is indeed very impressive, but because of the mathematical theorems that he is able to deduce from the Laws, not because of the Laws themselves.

Horologium Oscillatorium

Huygens’ book ‘Horologium Oscillatorium sive de motu pendulorum’ appeared in 1673.[44] It starts with a detailed description of the pendulum clock that he had invented and that was already transforming many aspects of science. Its five parts have the titles

1. Pendulum clocks;

2. Concerning the descent of moving bodies, and the motion of these on a Cycloid;

3. Concerning the evolutes and lengths of curved lines;

4. Concerning the centre of Oscillation;

5. In which the construction of another kind of clock is considered from the motion of a circular pendulum; and Theorems on Centrifugal Force.

Part 5 contains 13 unproved theorems, each of which makes a specific and substantial statement about the motions of particles in circles. For example Theorem 6 (correctly) states

“On the hollow surface of a paraboloid of a cone, which has a vertical axis, all the circular orbits of small objects, travelling around circumferences parallel to the horizontal, however small or large they may be, are carried out in equal times: which individual times are equal to two oscillations of the pendulum, the length of which is equal to half the latus rectum of the parabola.[45]”

A modern proof of this involves determining the horizontal and vertical components of the normal contact force, using the formula a = v2/r for the radial acceleration and then writing down a first order differential equation giving the dependence of the orbital period of the small object moving horizontally on the surface of revolution. The periods of the small object and of the pendulum are eventually found to be

T = 2π (2a/g )1/2

where a is the distance between the focus and vertex of the parabola and g is the gravitational constant. Doing all this by purely geometric methods without introducing g explicitly would be a significant task.

Newton’s discussion of the simple pendulum used the notion of centripetal force,[46] and one of his rare references to centrifugal force in Principia came in his comparison of his method with that in Horologium Oscillatorium, which he described as an “excellent treatise”.[47] He also rederived Huygen’s results on the cycloidal pendulum, referring to both Huygens and Wren.[48]

Onwards to Newton

Robert Hooke was the only person before Newton to conjecture that the motions of the planets could be explained by a gravitating force directed towards the Sun, or towards the relevant planet in the case of its satellites. He gave two lectures, in 1666 and 1670, in which he conjectured that the planets moved in their orbits as a result of an attractive force which was directed towards the Sun and decreased with distance from it.[49] Although he undoubtedly started Newton on the route that led to Principia, the two lectures were very brief and no detailed consequences or predictions were made; his Cutler lecture of 1670 only devoted one page to this radically new idea.[50] Wallis accepted without hesitation that the Laws of Motion applied to the Moon and planets, but did not know what influence controlled their orbits. Huygens was astonished at Newton’s success in obtaining Kepler’s laws from the inverse square attraction due to gravity, but he was never convinced that this was a physically correct explanation.

Wallis’s understanding of the Laws of Motion was far better than that of his contemporaries, apart possibly from Huygens. Newton acknowledged his debt to Wallis, Wren, Huygens and Mariotte for these Laws in Principia, pp. 424, 425. Newton’s extension of the Laws to arbitrary actions between bodies rather than just collisions is made explicit in Principia Law 3, p. 417 and Corollary 3, p. 420. Newton had no observational evidence to support this extension of the Laws apart from a brief mention of experiments with a lodestone and iron on p. 428. His discussion of Rule 3 on p. 796 and of the planets on p. 806 follows the mechanical philosophy advocated by Kepler. Although writing about gravitation in both places, he takes it for granted in both places that the planets are simply material bodies and subject to the third Law of Motion.

“For no one doubts that Venus, Mercury, and the rest [of the planets, primary and secondary,] are bodies of the same kind as Jupiter and Saturn. And since by the third law of motion, every attraction is mutual, …[51]”

The same can be said of his comments about the centre of gravity of the Earth-Moon system on p. 804 and p. 818, which follow those of Wallis discussed above. Although he said that the Earth-Moon wobble was ‘sensible’, the fact that he made no attempt to demonstrate it suggests that he knew that it was too small to be observable.

Newton’s statement in the General Scholium on p. 943 that the Laws of Motion had been found by using his method of deduction from the phenomena and made general by induction was not correct.[52] They emerged after many decades of trial and error, but their application to the planets depended on the gradually strengthening belief, and eventually the conviction, that the celestial bodies were ordinary material objects to which the mechanical philosophy could be applied, rather than entities of a wholly different type from those on the Earth. The evidence for this came from telescopic observations by Galileo and later by others of the planets and their moons, rather than from mathematical calculations. Newton provided the key that produced the detailed theoretical and mathematical support for the belief, but it took almost a hundred years for everyone to be convinced that his key would unlock all of the doors. Newton was correct in stating that, after taking the Laws of Motion as axioms, his Universal Law of Gravitation had been derived by a process of deduction from the phenomena. From a Popperian perspective one might say that he was simultaneously testing the applicability of the Laws of Motion to the planets and the Universal Law of Gravitation, but Newton would not have agreed. The application of the Laws of Motion to the planets was an intrinsic part of the world view that characterized the Scientific Revolution. The status of Newton’s Universal Law of Gravitation was quite different. It was the creation of a single person, and turned out to provide the extra ingredient needed to combine terrestrial and celestial motion into a single subject.

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[1] Department of Mathematics, King’s College London, Strand, London WC2R 2LS, England

[2]See also The Correspondence of Henry Oldenburg, ed. and transl. by A R Hall and M B Hall, in several volumes, Madison, Univ. of Wisconsin Press, 1965 – .

[3] Thomas Birch: The History of the Royal Society of London, vol. 1-4, 1756-57.

[4] Correspondence of John Wallis, eds. Philip Beeley and Christoph Scriba, Oxford Univ. Press, Oxford, Vol. 1 (1641 - 1659), 2003; Vol. 2 (1660 - September 1668), 2005.

[5] J F Scott: The Reverend John Wallis, F.R.S. (1616-1703). Notes and Records of the Royal Society 15 (1960) 57-67.

[6] We will always refer to the English translation of the third edition of Principia by I B Cohen and A Whitman, published by the University of California Press at Berkeley in 1999.

[7] This is discussed more fully in R Martens: Kepler’s Philosophy and the New Astronomy. Princeton Univ. Press, Princeton, 2000.

[8] See p. 48 of Johannes Kepler, New Astronomy, transl. by W H Donahue, Camb. Univ. Press 1992.

[9] J R Voelkel: Commentary on Ernan McMullin, “The Impact of Newton’s Principia on the Philosophy of Science”, Phil. Sci. 68 (2001) 319 – 326.

[10] B R Goldstein and G Hon: Kepler’s move from Orbs to Orbits: Documenting a Revolutionary Scientific Concept. Perspectives on Science 13.1 (2005) 74 – 111.

[11] See Principia, Prop. 2, p. 802.

[12] See Principia, Phen. 4, p. 800.

[13] Kepler, loc. cit. p. 55.

[14] Kepler, loc. cit. p. 386.

[15] Kepler, loc. cit. pp. 386, 387.

[16] Kepler, loc. cit. p. 390, 391.

[17] Letter of Torricelli to Michelangelo Ricci, [1644]. English translation of relevant pages in ‘High Altitude Physiology’, edited by West J B, Stroudsburg, PA, Hutchinson Ross, 1981.

[18] The Works of Robert Boyle, 14 vols., Eds. Hunter M and Davis E B., London: Pickering and Chatto, 1999; and

J B West: Robert Boyle’s landmark book of 1660 with the first experiments on rarified air. J Appl Physiol 98 (2005) 31–39, 2005.

[19] J Wallis, A discourse of gravity and gravitation, grounded on experimental observations, London, 1675, listed as Royal Society Tracts C1.

[20] Wallis, 1675, p. 34.

[21] An essay of Dr John Wallis, exhibiting his hypothesis about the flux and reflux of the sea, Phil.Trans. 6 (1666) 263 – 281.

[22] Wallis, 1666, p. 264.

[23] Wallis, 1666, p, 268.

[24] Wallis, 1666, pp. 270 – 271.

[25] Wallis, 1666, p. 271.

[26] Wallis, 1666, p. 272.

[27] The correct value is about 3/4, not 4/3, so the centre of mass is within the body of the Earth.

[28] Wallis, 1666, pp. 269 – 270.

[29] This wobble is quite different from various other 'wobbles', particularly of the axis of rotation of the Earth.

[30] Wallis, 1666, p. 280.

[31] Phil.Trans. 6 (1666) 281 – 289.

[32] Loc. Cit. p. 289.

[33] A Chapman: The design and accuracy of some observatory instruments of the seventeenth century, Ann. Sci. 40 (1983) 457 – 471.

[34] R. Hooke: An attempt to prove the motion of the Earth from observations, London, 1674.

[35] Translations of parts of these may be found in A R Hall: Mechanics and the Royal Society 1668 – 1670, Brit. J. Hist. Sci. 3 (1966) 24 – 38.

[36] J. Wallis, Phil. Trans. 3 (1669) 864-866.

[37] See pp. 313 – 318 of The Cambridge Companion to Descartes, ed. J Cottingham, Camb. Univ. Press, 1992.

[38] Cottingham, loc. cit. pp. 322 – 324.

[39] C Wren, Phil. Trans. 3 (1669) pp. 867, 868.

[40] See p. 113 of C D Andriesse: Huygens, the Man behind the principle. Camb. Univ. Press, 2005.

[41] Phil. Trans. 4 (1669) 925 – 928.

[42] See pp. 571 – 1063 of Johannes Wallis, Opera Mathematica, Vol. 1, Oxford, 1695.

[43] See pp. 106 – 126 of J F Scott: The mathematical work of John Wallis, Taylor and Francis, London, 1938.

[44] ‘Christiaan Huygens' the Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula As Applied to Clocks’ transl. by Richard J. Blackwell, Iowa State Press 1986. See also the translation by Ian Bruce at

[45] See Bruce, loc. cit. Part 5, p. 3.

[46] See Principia Prop. 4, p. 449 and Cor. 9, p. 451.

[47] See Principia p. 452.

[48] See Principia p. 555.

[49] See Birch pp. 91-92 for the record of a lecture of Hooke to the Royal Society on 23 May 1666, and R. Hooke: An attempt to prove the motion of the Earth from observations, London, 1674.

[50] A detailed discussion of Hooke’s insights and of the experimental basis for them may be found in M Nauenberg: Robert Hooke’s seminal contribution to orbital dynamics. Phys. Persp. 7 (2005) 4 – 34.

[51] Principia, p. 806. The text in square brackets is an interpolation by the translator.

[52] E B Davies: Some reflections on Newton’s ‘Principia’. Brit. J. Hist. Sci. to appear.

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