The Mechanical Philosophy
1660 – 1675
MechPhil2white, 2 August 2007
E B Davies1
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. Birch3
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 correspondence4 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
1
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, England
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.
2
1
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.
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.
2
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 light14) 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
13
Kepler, loc. cit. p. 55.
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
14
3
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 tides21, 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
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.
4
‘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 wobble29)
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.
Wallis, 1666, pp. 270 – 271.
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.
24
25
5
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
Phil.Trans. 6 (1666) 281 – 289.
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.
31
32
6
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 EarthMoon 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.
34
R. Hooke: An attempt to prove the motion of the Earth from observations, London, 1674.
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.
35
7
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.
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.
37
8
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. Scott43 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;
Phil. Trans. 4 (1669) 925 – 928.
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.
41
42
9
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;
‘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 http://www.17centurymaths.com/contents/huygenscontents.html
44
10
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
45
See Bruce, loc. cit. Part 5, p. 3.
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.
46
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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.
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.
50
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