LAWS NEWTON'S

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ti
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nd
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Ourfirstcase studyencompasses essentiallythewhole ofwhatcan be consideredthemodern
scientific process. Unlike the other case studies, it requires little mathematics but a great
deal interms ofintellectual imagination. Forme, itis aheroic tale ofscientists ofthe highest
genius lying the foundations ofmodern science. Everything is there - the roles ofbrilliant
experimental skill, of imagination in the interpretation of observational and experimental
dataandoftheremarkable leaps ofthe imaginationwhichwereto laythe foundations forthe
Newtonianpicture ofthe world. This achievementmaynot at first sight seem so remarkable
to the twenty-first-century reader, but closer inspection shows that in fact it is immense.
As expressed by Herbert Butterfield in his Origins ofModern Science,l the understanding
of Illotion was one of the rnost difficult steps that scientists have ever undertaken. In the
quotation by Douglas Gough in Chapter 1, he expresses eloquently the 'pain' experienced
on being forced to discard a cherished prejudice in the sciences. How much more difficult
must have been the process oflaying the foundations ofmodern science, when the concept
that the laws ofnature can be written in mathematical form had not yet been formulated.
How did our modern appreciation ofthe nature ofour physical Universe come about? I
make no apology for starting at the very beginning. In Chapter 2, the first ofthree chapters
that address Case Study I, we set the scene for the subsequent triumphs, and tragedies, of
two of the greatest minds of modern science - Galileo Galilei and Isaac Newton. Their
achievements were firmly grounded in the remarkable observational advances of Tycho
BraheandinGalileo's skill as anexperimentalphysicistandastronomer. Galileo andhistrial
by the Inquisition are considered in SOllle detail in Chapter 3, the emphasis being upon the
scientific aspects ofthis controversial episode in the history ofscience. The issues involved
can be considered as the touchstone for the modern view ofthe nature ofscientific enquiry.
Then, with the deep insights of Kepler and Galileo established, Newton's extraordinary
achievements are placed in their historical context in Chapter 4.
It may seelll somewhat strange to devote so much space at the beginning ofthis text to
what many will consider to be ancient history, a great deal ofwhich we now understand to
be wrong and misleading. Having been through this material, I feel very differently about
it. It is a gripping story and full of resonances about the way we practice science today.
There are, in addition, other aspects to this story which I believe are important. The great
13

2. 1 4 Case Study I . The origins of Newton's laws of motion and of gravity
Figure 1.1: Tycho Brahe with the instruments he constructed for accurate observations ofthepositions
ofthe stars and planets. He is seen seatedwithinthe 'great mural quadrant', which produced the most
accurate measurements of the positions of the stars and planets at that time. (After Astronomiae
lnstauratae Mechanica, 1602, p. 20, Niirnberg. From the Crawford Collection, Royal Observatory,
Edinburgh.)
scientists involved in this case study had complex personalities and, to provide a rounder
picture of their characters and achievements, it is helpful to understand their intellectual
perspectives as well as their contributions to fundamental physics.
1 . 1 Reference
1 Butterfield, H. (1950). The Origins ofModern Science. London: G. Bell, New York:
Macmillan (1951).

3. 2 Fro m Pto l e my 0 Kep l e r - the
Copern ica n revo l uti o n
2. 1 Ancient history
The first ofthe great astrononlers ofwhomwe haveknowledge isHipparchus,whowas born
in Nicaea in the second century BC. Perhaps his greatest achievement was his catalogue of
thepositions andbrightnesses of850 stars inthe northern sky. The cataloguewas completed
in 127 Be and represented a quite monumental achievement. A measure ofhis skill as an
astronomer is that he compared his positions with those ofTimochatis made in Alexandria
150 years earlier and discovered theprecession ofthe equinoxes, the very slow change in
direction ofthe Earth's axis ofrotation relative to the frame ofreference ofthe fixed stars.
Wenowknow thatthis precession is causedbytidaltorques due to the SunandMoon acting
upon the slightly non-spherical Earth. At that time, however, the Earth was assumed to be
stationary and so the precession ofthe equinoxes had to be attributed to a movenlent ofthe
'sphere offixed stars'.
The lllOst famous of the ancient astronomical texts is the Almagest of Claudius
Ptolomeaus, or Ptolemy, who lived in the second century AD. The word 'Almagest' is a
corruption of the Arabic translation of the title of his book, the Megele Syntaxis or Great
Composition, which in Arabic becomes al-majisti. It consisted of 13 volumes andprovided
a synthesis ofall the achievenlents ofthe Greekastronomers and, inparticular, leantheavily
upon the work ofHipparchus. Within the Almagest, Ptolemy set out what became known
as the Ptolemaic system ofthe world, which was to dominate astronomical thinking for the
next 1500 years.
How did the Ptolemaic system work? It is apparent to everyone that the Sun and Moon
appear to move in roughly circular paths about the Earth. Their trajectories are traced out
against the sphere ofthefixed stars, which also appears to rotate about the Earth once
per day. In addition, five planets are observable by the naked eye, Mercury, Venus, Mars,
Jupiter and Saturn. The Greek astronomers understood that the planets did not move in
simple circles about the Earth, but had somewhat more complex motions. Figure 2.1 shows
Ptolemy'S observations of the motion of Saturn in AD 133 against the background of the
fixed stars. Ratherthan move in a smooth path across the sky, the path ofthe planet doubles
back upon itself.
The challenge to the Greek astronomers was to work out mathematical schemes which
could describe these nlotions. As early as the third century BC, a few astronomers had
suggested that these phenomena could be explained if the Earth rotated on its axis and
15

4. 1 6 2 From Ptolemy to Kepler - the Copern ican revolution
1",4 -
1 ".3 -
1°. 2 -
1 °.1-
1'.0 -
0'.9 -
I
256'
I
255'
Figure 2. 1 : The motion of Saturn from 5 December AD 1 32 to 20 December AD 133 as observed
by Ptolemy against the background of the fixed stars. (From O. Pedersen and M. Pihl, 1974, Early
Physics andAstronomy, p. 71, London: McDonald and Co.)
even that the planets orbit the Sun. Herac1eides of Pontus described a geo-heliocentric
system whichwe will meet again inthe work ofTycho Brahe. Mostremarkably, Aristarchos
proposed that the planets move in circular orbits about the Sun. In The Sand Reckoner,
Archimedes wrote to King Gelon,
You are not unaware that by the universe most astronomers understand a sphere the centre ofwhich
is at the centre of the Earth . . . . However, Aristarchos of Samos has published certain writings on
the [astronomical] hypotheses. The presuppositions found in these writings implythatthe universe is
much greater than we mentioned above. Actually, he begins with the hypothesis that the fixed stars
and the Sun remain without motion. As for the Earth, it moves around the Sun on the circumference
ofa circle with centre in the Sun.1
These ideas became the inspiration for Copernicus roughly eighteen centuries later. They
were rejected at the time ofAristarchos for a number ofreasons. Probably the most serious
was the opposition ofthe upholders ofGreek religious beliefs. According to Pedersen and
Pihl (1974),1
Aristarchos had sinned against deep-rooted ideas about Hestia's fire, and the Earth as a Divine Being.
Such religious tenets could not be shaken by abstract astronOlnical theories incomprehensible to the
ordinary man.2
From our perspective, the physical arguments against the heliocentric hypothesis are of
equal interest. First, the idea that the Earth rotates about an axis was rejected. Ifthe Earth
rotated then when an object is thrown up in the air it would not come down again in the
same spot - the Earth would have moved, because ofits rotation, before the object landed.
No one had ever observed this to be the case and so the Earth could not be rotating. The

5. 2 . 1 Ancient h istory 1 7
Figure 2.2: The basic Ptolemaic system ofthe world showing the celestial bodies from the Earth in the
order, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn and the sphere offixed stars. (From Andreas
Cellarius, 1661, Harmonia JJ//acrocosmica Amsterdam. Courtesy of F. Bertola, from Imago Mundi,
1995, Biblios, Padova.)
second problem resulted from the observation that if objects are not supported they fall
under gravity. Therefore, ifthe Sun were the centre ofthe Universe rather than the Earth,
everything ought to be falling towards that centre. Now, if objects are dropped they fall
towards the centre of the Earth and not towards the Sun. It follows that the Earth must
be located at the centre ofthe Universe. Thus, religious beliefwas supported by scientific
rationale.
According to the Ptolemaic geocentric system of the world, the Earth is stationary at
the centre ofthe Universe and the principal orbits of the other celestial objects are circles
in the order Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn and finally the sphere of
the fixed stars (Fig. 2.2). The problem with the elementary Ptolemaic system was that
it could not account for the details of the motions of the planets, such as the retrograde
lTIotion shown in Fig. 2.1, and so the model had to become lTIOre complex. There was one
central concept from Greek mathematics which played a key role in refining the Ptolemaic
systen1. Part of the basic philosophy of the Greeks was that the only allowable motions
were uniform motion in a straight line and uniform circularmotion. PtolelTIY himselfstated
that uniform circularn10tion was the only kind ofmotion 'in agreement with the nature of

6. 1 8 2 From Ptolemy to Kepler - the Copern ican revolution
Figure 2.3: Illustrating circular epicyclic motion about a circular orbit according to the epicyclic
model of Appolonios. (From O. Pedersen and M. Pihl, 1974, Early Physics and Astronomy, p. 83,
London: McDonald and Co.)
Divine Beings'. Therefore, itwas supposedthat, in additionto their circular orbits about the
Earth, the planets, as well as the Sun and Moon, had circular motions about the principal
circular orbit (Fig. 2.3); the circles superimposed upon the main circular orbit were known
as epicycles. It can be readily understood how the type of orbit shown in Fig. 2.1 can be
reproduced by selecting suitable speeds for the motions ofthe planets in their epicycles.
One ofthe basic rules ofastrometry, meaning the accurate measurement ofthe positions
andmovements ofbodies onthe celestial sphere, isthatthe accuracywithwhichtheir orbits
are determined improves the longer the time span over which the observations are made.
As a result, the simple epicyclic picture hadto become more andmore complex, the longer
the time base of the observations. To improve the accuracy of the Ptolemaic model, the
centre of the circle of a planet's principal orbit was allowed to differ from the position of
the Earth, each circular component ofthe motion remaining uniform. As a consequence, it
was found necessary to assume thatthe centre ofthe circle about which the epicycles took
place also differed from the position ofthe Earth (Fig. 2.4). An extensive terminology was
usedto describe the details ofthe orbits, butthere is no needto enter into these complexities
here (see Pedersen andPihl (1974) for more details1). The keypoint is that, by considerable
geometrical ingenuity, Ptolemy and later generations of astronomers were able to give a
good account of the observed motions of the Sun, Moon and the planets, but the models
involved a considerable number ofmore or less arbitrary geometrical decisions. Although
complicated, the Ptolemaic model was used in the preparation of almanacs and in the
determination ofthe dates ofreligious festivals until after the Copernican revolution.
2.2 The Copernican revol ution
By the sixteenth century, the Ptolemaic system was becoming more and more complicated
as a tool for predicting the positions of celestial bodies. Nicolaus Copernicus revived the

7. 2.2 The Copernican revolution 1 9
Figure 2.4: Illustrating the cOlYlplexity of the Ptolemaic theory of the motion of the outer planets.
(From O. Pedersen and M. Pihl, 1974, Early Physics and Astronomy, p. 94, London: McDonald
and Co.)
idea ofAristarchus that a sinlpler model, in which the Sun is at the centre ofthe Universe,
ITIightprovide a simplerdescription ofthenlotions oftheplanets. Copernicus, born inTorun
in Poland in 1473, first attendedthe University ofKrak6w and then went to Bologna, where
his studies included astronomy, Greek, mathematics and the writings ofPlato. In the early
1500s, he spent fouryears at Paduawhere he also studied medicine. By the time he returned
to Poland, he had mastered all the astronomical and mathematical sciences. Copernicus
Inade some observations himselfand these were published between 1497 to 1529.
His great works were, however, his investigations of whether a heliocentric Universe
could provide a simpler account of the motions of the planets. When he worked out the
mathematics of this model, he found that it gave a remarkably good description. Again,
however, he restricted the ITIotions ofthe Moon and the planets to uniform circular orbits,
according to the precepts ofAristotelian physics. In 1514 he circulated his ideas privately
in a short manuscript called 'De hypothesibus motuum coelestium a se constitutis com­
mentariolus' (A commentary on the theory ofthe motion ofthe heavenly objectsfrom their
arrangements). The ideas were presented to Pope Clement VII in 1533, who approved
of them and who in 1536 nlade a formal request that the work be published. Copernicus
hesitated, but eventually wrote his great treatise summarising what is now known as the
Copernican model ofthe universe in DeRevolutionibus Orbium Coelestium (On theRevo­
lutions oftheHeavenly Spheres).3 The publication ofthe workwas delayed, but eventually
it was publishedby Osiander in 1543. It is said thatthe first copywas broughtto Copernicus
on his death-bed on 24 May 1543. Osiander had inserted his own foreword into the treatise

8. 20 2 From Ptolemy to Kepler - the Copern ica n revol ution
Figure2.5: TheCopernicanUniversefromCopernicus' treatiseDeRevolutionibus Orbium Celestium,
1 543, opposite p. 1 0, Nurnberg. (From the Crawford collection, Royal Observatory, Edinburgh.)
stating that the Copernican model was no more than a calculating device for sim­
plifying the predictions of planetary motions, but it is clear from the text itself that
Copernicus was in no doubt that the Sun really was the centre of the Universe, and not
the Earth.
Figure 2.5 shows the famous picture which appears opposite p. 10 ofCopernicus' trea­
tise, showing the planets intheir familiar orderwith the Moon orbiting the Earth andthe six
planets orbitingthe Sun. Beyondthese lies the sphere ofthefixedstars. The implications of
the Copernican picture were profound, not only for science but also for the understanding
of our place in the Universe. The scientific implications were twofold. First, the size of
the Universe was vastly increased as compared with the Ptolemaic model. Ifthe fixed stars
were relatively nearby then they ought to display parallaxes, apparent motions relative to
more distant background stars, because of the Earth's motion about the Sun. No such
stellar parallax had ever been observed, and so the fixed stars must be very distant
indeed.
In England, these ideas were enthusiastically adoptedby the most important astronomer
of the reign of Queen Elizabeth I, Thomas Digges, who was also the translator of large
sections of De Revolutionibus into English. In his version of the Copernican model, the
Universe is of infinite extent and the stars are scattered throughout space (Fig. 2.6). This
is a remarkably prescient picture and one whichNewton was to adopt, but it leads to some
tricky cosmological problems, as we will see.
The second fundamental implication ofthe Copernican picture was that something was
wrong with the Aristotelian concept that all objects fall towards the centre ofthe Universe,

9. 2.3 Tycho Brahe - the lord of U ra n iborg
Folio_f>
A perfitdefcription oftheCadefiiallOrbes1
�d#rtl11Ig t4 tiJ(RlIlt tllII/f.tmIlJ,{/Tl/U Ij'tbt
I'pilJ!/ITUIIS. <ft.
* '1f.
* * )(­
* * * 1'
'f' "" 'II- )(- *
'*'" -+ *' -1c- * '*
,.,.. � .fro
* * * .. 1f; * * * *
2 1
Figure 2.6: Thomas Digges' version ofthe Copernican picture ofthe world showing the solar system
embedded in an infinite distribution ofstars. (From Thomas Digges, 1 576, A PeTiit Description ofthe
Caelestiall Orbes, London.)
whichisnowoccupiedbythe Sun. ThisproblemwasonlyresolvedwithNewton's discovery
ofthe nature ofthe law ofgravity, namely, that it is an inverse- square law.
2.3 Tycho Brahe - the lord of U ra n iborg
Copies of Copernicus' De Revolutionibus Orbium Coelestium circulated remarkably
quickly thoughout Europe. One of the motivations behind Copernicus' researches was to
produce simpler mathematical procedures for working out the motions of the Sun, Moon
and the planets for the purposes of determining the exact date of the vernal equinox.
These were needed to establish the correct dates for religious festivals and this was
perhaps one reason for the favourable reception ofCopernicus' heliocentric model by Pope
Clelnent VII.
Until 1543, the predictions ofthe motions ofthe celestial bodies had been taken from
what were known as the Alphonsine Tables, which were derived frolnthe Ptolemaic system
as refined by the Arabic astronomers. These tables had been prepared by the Rabbi Isaac
ben Sid ofToledo andpublished in manuscript form in the LibrosdelSabredeAstronomica
in 1277 underthe patronage ofAlfonso X ofCastile, also known as Alfonso the Wise. The

10. 22 2 From ptolemy to Kepler - the Copernica n revolution
tables were copied in manuscript form and were quickly disseminated throughout Europe.
They were only published in the modern sense in 1483.
Modern scholarship has suggested that, in fact, predictions using the Copernican model
were often not much better than those using the Alphonsine Tables. Predictions using the
data in De Revolutionibus were made by Erasmus Reinhold, who produced what became
known as the Prutenic, or Prussian, Tables, giving the positions of the stars and planets.
These were published in 1551, no more than eight years after the first publication ofDe
Revolutionibus.
The next hero of our story is Tycho Brahe, who was born into a noble family in 1546
at the family home of Knudstrup at Skane, Denmark. He formed an early passion for
astronomy. Toprepare him forthe life ofa nobleman, however, he was sent to the University
of Leipzig in March 1562 to study law, but he kept up his interest in astronomy, making
observations secretly atnightwhile his tutor slept. He also spent all the moneyhe could save
onastronomicalbooks, tables andinstruments. Atthistimehe acquiredhis owncopiesofthe
Alphonsine andPrutenicTables. The inspirationforhis futureworkcame fromthepredicted
conjunction ofSaturn andJupiterin 1563. He foundthatthe predictions ofboth tables were
in error, by about a month ifhe used the Alphonsine Tables andby a few days ifhe usedthe
Prutenic Tables. The need to improve the accuracy with which the planetary motions were
known was one ofthe prime motivations for the monumental series ofobservations which
he began in the late 1570s.
Once established in Denmark, he gave lectures at the University ofCopenhagen where
.he discussed the Copernican theory.
Tycho spoke ofthe skill ofCopernicus, whose system, although not in accord withphysical principles,
was mathematically admirable and did not make the absurd assumptions ofthe ancients.4
Tychocontinuedhis internationalcollaborationwithLandgraveWilliamIV, whomhevisited
inKassel in 1575 and, during thatvisit, hefirstbecame aware ofthe importance ofthe effect
of refraction by the Earth's atmosphere on astronomical observations. Tycho was the first
astronomer to take account of refraction in working out accurate stellar and planetary
positions.
Tycho was determined to carry out a programme of measurement of the positions of
the stars and planets to the very highest accuracy achievable. Frederick II ofDenmark was
persuaded that Tycho was an outstanding scientist who would bring honour to Denmark,
and so, in 1576, to prevent Tycho 'brain-draining' to Germany made him an offer which he
could not refuse. In the words ofFrederick, he provided Tycho with
. . . our land ofHven with all our and the Crown's tenants and servants who thereon live, with all the
rent and duty which comes from that . . . as long as he lives and likes to continue and follow his studia
mathematica . . . 5
The islandofHvenlieshalfwaybetween Denmarkand SwedenandTycho consideredthis
peaceful islandretreat ideal for his astronomical ambitions. He was allowedto use the rents
for the upkeep andthe running ofthe observatories he was to build. In addition, he received
regularsupplementary funds fromFrederickIIto enablehimtobuildhis greatobservatories
and to construct the instruments needed for making astronomical observations. In Tycho's

11. 2.3 Tycho Bra he - the lord of U ra n i borg 23
own words, he believed that the total enterprise had cost Frederick 'more than a tun of
gold'.6 Victor Thoren estimates that Tycho received an annual income ofabout 1% ofthe
Crown's revenue throughout his years at Hven.7 This was the first example of 'big science'
in the modern era.
Thefirstpartoftheprojectwastobuildthemainobservatory,whichhenamedUraniborg,
orthe Heavenly Castle. Besides adequate space forall his astronomical instruments, he built
a1chelnical laboratories onthe groundfloorwhere he carriedout chemical experiments. The
building also included a papermill andprinting press, so thatthe results ofthe observations
could be published promptly.
Uraniborg was completed by about 1580 and Tycho then began the construction of a
second observatory, 'Stjerneborg', which was located at ground level. He now realised the
value of constructing very solid foundations for the astronomical instruments, and they
were clustered in a much more compact area in the new observatory. These astronomical
instruments were the real glory ofthe observatory. They predated the telescope; all obser­
vations were made with the naked eye and consisted ofmeasuring as accurately as possible
the relative positions ofthe stars and planets in a systematic way. Two ofthe instruments
seen in Figure 1.1 are worthy of special mention. The first is the Great Globe, which can
be seen on the ground floor ofUraniborg in Fig. 1.1 . On it, Tycho marked the positions of
all the stars he recorded, thus forming a permanent record ofthe positions of, in the end,
777 stars.
The secondwasthe greatmural quadrant, also seenin Fig. 1.1 beingpointed outbyTycho
himself. The mural quadrant is fixed in one position and had radius 6� feet. Observations
were made by observing stars through a hole inthe wall at the centre ofthe circle described
by the quadrant. Observation ofthe position ofa star consisted ofmeasuring the angle from
the horizon. Because ofthe large radius ofthe quadrant, very accurate positions could be
Ineasured. Tokeep an accurate track oftin1e, Tycho had four clocks so that he would notice
ifany one ofthem was not keeping proper time.
Tycho's technical achievements were quite remarkable. He was the first scientist known
to have understood the crucial importance of taking account of systematic errors in his
observations. There are two beautiful examples of these. We have already mentioned the
first ofthese, regarding atmospheric refraction. The second arises because large instruments
bendundergravity,producing asystematicerrorinthe sensethatiftheinstrumentispointing
vertically it does not bend whereas if it is horizontal the ends bend downwards and give
incorrect angles. Tycho understood the necessity of eliminating these types of systematic
error.
Anotherimportantadvance was his understanding ofthe needto estimate howprecisethe
observations were once the effects ofsystematic errors had been removed, in other words,
the Inagnitude ofthe random errors in the observations. Tycho was also probably the first
scientist to work out accurately the random errors in his observations. They turned out to
be about ten times smaller than those ofany earlier observations. This is a quite enormous
increase inprecision. A furtherkeyfeatureoftheobservationswasthattheyweresystematic,
in the sense that they were 111ade continuously over a period ofabout 20 years from 1576
to 1597 and systematically analysed by Tycho and his team ofassistants. Throughout that
period, hemeasuredpreciselythepositions ofthe Sun, Moon,planets andthe fixed stars. His

12. 24 2 From ptolemy to Kepler - the Copernican revolution
NOVA MVNDANI SYSTEMATIS HYPOTYPOSIS AB
AUTHORE NUFER ADINUENTA. QUA TUM VETUS ILLA
PTOLEMAICA REDUNDANTIA &. INCONCINNITAS;
TUM ETIAM RECENS COPERNIANA IN MOTU
TERRJE PHYSICA ABSURDITAS; EXCLU*
DUNTUR� OMNIAQUE APPAREN�
TUS C<ELESTIBUS APTISSIME
CORRESPONDENT.
Figure 2.7: The Tychonic system of the world. (From V.E. Thoren, 1 990, The Lord of Uraniborg,
p. 252, Cambridge: Cambridge University Press.)
final catalogue containedthe positions of777 stars measured with a precision ofabout I to
2 minutes of arc. The crucial importance of knowing this figure will become apparent in
the next section.
This was the summit ofTycho's achievement. He created the greatest observatory ofits
day, andDenmark andHvenbecame the centre ofastronomy in Europe. After Frederick II's
deathin 1588, however, the supportforpure sciencewanedunderhis successor, ChristianIV
Inaddition, Tycho wasmismanagingthe island. Matters came to ahead whenhe left forexile
in 1597, taking with him his observations, instrunlents and printing press. He eventually
settled outside Prague under the patronage of Emperor Rudolf II at the castle of Benatek,
where he began setting up again his magnificent set ofinstruments. During the remaining
years ofhis life, a priority was the analysis ofthe mass ofdata he had secured and he had
one stroke of great good fortune: one of his last acts, in 1600, was to employ Johannes
Kepler to reduce the data from his observations ofthe planet Mars.
Tycho developedhis owncosmology,whichheregardedas a synthesis ofthe Copernican
and Ptolemaic views ofthe Universe, in about 1583 (Fig. 2.7). Inthis model, the Earthis the
centre ofthe Universe andthe Moon and Sun orbitthe Earth, while allthe otherplanets orbit
the Sun. Tycho wasjustlyproud ofhis model, whichbears morethan apassing resemblance

13. 2.4 Johannes Kepler and heavenly ha rmonies 25
to the model ofHeracleides mentioned in Section 2.1 . Until one understands centrifugal
forces and the law ofgravity, there is not llluch wrong, but Tycho's own observations were
to lead to the demolition ofhis theory and the establishment ofthe law ofgravity.
Tycho'sachievementsrank amongthegreatestinobservationalandexperimental science.
We recognise inhis workall the very best features ofmodern experimental science, and yet
the rules had not been written in his day. The achievement is all the greater when we recall
thatthe ideaofquantitative scientific llleasurement scarcely existed atthattime. Experiment
andprecise measurement did not exist -astronomical observationwas by farthe most exact
of all the physical sciences. It should come as no surprise that Tycho's legacy played a
central role in the Newtonian revolution.
2.4 Johannes Kepler and heavenly harmonies
We now introduce a very different character, Johannes Kepler. He was born in December
1571 in the Swabian city ofWeil der Stadt. Kepler was a weak boy but a talented student,
who entered the University of Tiibingen in 1589. His first encounter with astronomy was
through Michael Maestlin, who taughthim mathematics, Euclid and trigonometry. In 1582,
Maestlin had publishedhis treatise EpitolneAstronomiae, which contained a description of
Copernicus's heliocentric Universe. Maestlin was, however, very cautious and, under the
influence ofthe church, relTI,ained a Ptolemaian. In contrast, Kepler had no doubts. In his
words:
Already in Tiibingen, when I followed attentively the instruction of the famous magister Michael
Maestlin, I perceived how clumsy inmanyrespects is the hitherto custon1ary notion ofthe structure of
the universe. Hence, I was so very delightedby Copernicus, whommyteacher very oftenmentioned in
his lectures, that I not only repeatedly advocated his views in disputations ofthe candidates [students],
but also Inade a careful disputation about the thesis that the first motion [the revolution ofthe heaven
offixed stars] results from the rotation ofthe Earth.8
Thus, from the very beginning, Kepler was a passionate and convinced Copernican. His
instructionattheuniversityincludedbothastronomyandastrologyandhebecameanexpert
at casting horoscopes, which was to stand him in very good stead.
Kepler's serious studies began in 1595 when he asked some basic questions about the
Copernican picture ofthe Solar System. Why are there only six planets in orbit about the
Sun? Why are their distances so arranged? Whydo theyn10ve more slowly iftheyare further
away from the Sun? Kepler's approach is summarised in two quotations from Max Casper.
First,
He was possessed and enchanted by the idea ofhannony.9
Second,
, Nothing in this world was created by God without a plan; this was Kepler's principal axiom. His
undertaking was no less than to discover this plan of creation, to think the thoughts of God over
. 10
agam . . .

14. 26 2 From Ptolemy to Kepler - the Copernica n revolution
Figure 2.8: Kepler's model of nested polyhedra which he developed to account for the number of
planets and their radial distances from the Sun. (From Dictionary ofScientific Biography, Vol. VII,
1 973, p. 292, Charles Scribner's Sons, © 1 970-80. Reprinted by permission ofthe Gale Group. After
Kepler's original drawing in his Mysterium Cosmographicum, 1 597.)
Itwas duringhisteachingofmathematics andgeometrythatthe germofanideawasplanted.
The moment ofrevelation is described in his book Mysterium Cosmographicum in his own
inimitable style.
Behold, reader, the invention and whole substance of this little book! In memory of the event, I
am writing down for you the sentence in the words from that moment of conception: the Earth's
orbit is the measure ofall things: circumscribe around it a dodecahedron and the circle containing it
will be Mars; circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter;
circumscribe about Jupiter a cube and the circle containing this will be Saturn. Now inscribe within
the Earth an icosahedron and the circle contained in it will be Venus; inscribe within Venus an
octahedron, and the circle contained in it will be Mercury. You now have the reason for the number of
planets.II
How does this work? It is a well-known fact ofsolid geometry that there are only five
regular solids, thefiveplatonic solids, in which all the edges are lines ofequal length and
in which the faces are all identical regular figures. Remarkably, by his choice of the order
ofthe platonic solids Kepler was able to account for the radii ofthe planetary orbits to an
accuracy of about 5%. In 1596, he explained his model of the solar system to the Duke of
Wiirtemburg and designs were made to construct areal model ofthe solar system (Fig. 2.8),
but it was never built. But Kepler went further. Copernicus had given no special physical
significance to the Sun as the centre ofthe Solar System, but Kepler argued that the Sun
was the origin ofthe forces which held the planets in their orbits.
Now many ofKepler's speculations are wrong and irrelevant to what follows. I seem to
have devotedrather a lot ofspace to them, buttherearetwo reasons for this. The first is that
Kepler was seekingphysical causes for the phenomena discovered by Copernicus. No-one

15. 2.4 Johannes Kepler and heavenly harmonies 27
had attempted to make this leap ofthe imagination before. The second is that this model is
the first example ofKepler's fascination with harmonic and geometric principles.
With unbounded confidence and enthusiasm, Kepler published these ideas in 1597 in
his Mysterium Cosmographicum (The Mystery ofthe Universe). He sent copies to many
distinguished scientists ofthe day, including Tycho Brahe and Galileo. Galileo simply ac­
knowledgedreceiptofthebook, whileTychoBrahewascautiouslypositive andencouraging
in his response: he invited Kepler to come and work with him in Benatek.
The Mysterium Cosmographicum made a considerable impact upon astronomical think­
ing. Looking back much later, Kepler wrote ofhis book
. . . nearly all astronomical books which I published since that time have been related to some one
of the main chapters in this little book, presenting themselves as its more detailed argument or
perfection . . . The success which my book has had in the following years loudly testifies that no one
ever produced a first book more deserving ofadmiration, more auspicious and, as far as its subject is
concerned, more worthy.1 2
Kepler realised that what he needed to test his theory was much more accurate data on
the orbits ofthe planets. The only person who had access to such data was Tycho Brahe.
After various toings and froings, he ended up in Tycho's employ in 1600. There was a very
great difference in outlook between Tycho and Kepler. When Kepler moved to Benatek,
Tycho vvas 53 years old and Kepler 28. Tycho was the greatest astronomer ofhis time and of
noble origin; Kepler was the greatest mathematician in Europe and ofhumble origin. Tycho
wanted Kepler to work on the 'Tychonic' theory ofthe Solar System, whereas Kepler was
already an ardent Copernican.
Just before Tycho died in 1601, he set Kepler to work on the problem of the orbit of
Mars. On his deathbed, Tycho urgedKeplerto complete anew set ofastronomical tables to
replace the Prutenic Tables. These were to be known as the 'Rudolphine Tables' in honour
ofthe emperor Rudolph II, who had provided Tycho with the castle ofBenatek as well as
an enonnous salary, 3000 gulden. Withintwo days ofTycho's death, Kepler was appointed
Imperial Mathematician and the greatest period ofhis life's work began.
Atfirst, Keplerassumedthatthe orbitofMarswas circular, as inthe standardCopernican
picture. His first discoverywasthatthe motion ofMars could not be describedbythis model
ifitwasreferredtothe centre ofthe Earth's orbit. Rather, themotionhadtobereferredtothe
true position ofthe Sun. This was an important advance. Kepler carried out an enormous
nurnber of calculations to try to fit the observed orbit of Mars to circular orbits, again
following implicitly the precept that only circular motions should be used to describe the
orbits ofthe planets. After a great deal oftrial and error, the best orbits he could find still
disagreed with the observations of Tycho Brahe by an error of 8 minutes of arc. This is
where the knowledge ofthe errors in Tycho's observations were critical. As Kepler stated:
DivineProvidencegrantedtous suchadiligentobserverinTycho Brahethathisobservations convicted
this Ptolemaic calculation ofan error of 8 minutes of arc; it is only right that we should accept God's
gift with a grateful mind . . . Because these 8 minutes of arc could not be ignored, they alone have led
to a total reformation of astronomy.l3
In other words, the random errors in Tycho's final determinations of the planetary orbits
amounted to only about 1 to 2 minutes of arc, whereas the minimum discrepancy which

16. 28 2 From Ptolemy to Kepler - the Copern ica n revol ution
Kepler could find was at least four times this observational error. Before the time ofTycho,
the random errors were about ten times greater and therefore Kepler would have had no
problem in fitling these earlier observations to models involving circular orbits.
To paraphrase Kepler's more eloquent words, this disagreement was unacceptable and
so he had to start again from the beginning. His next attempts to describe the solar system
were based upon the use of ovoids (egg-shaped figures), in conjunction with a magnetic
theory for the origin ofthe forces which hold the planets in their orbits. He found that it
was very complicated and tedious to work out the orbits according to this magnetic theory,
and so he adopted intuitively an alternative approach in which the motions of the planets
are such that they sweep out equal areas in equal times. Whatever the actual shape ofthe
orbit, the result is that the planet must move faster when it is closer to the Sun so that the
area swept out by the line from the Sun to the planet is the same in equal time intervals.
It turned out that this theory gives excellent predictions of the longitudes of the planets
about the Sun and also of the Earth's orbit about the Sun. This great discovery is what
we now know as Keplers second law ofplanetary motion. Formally the statement is as
follows:
Equal areas are swept out by the line from the Sun to a planet in equal times.
Keplerproceededwiththemammothtaskoffitting ovoids andthe areallawto the motion
ofMars, but he could not obtain exact agreement, the minimum discrepancy amounting to
about 4 minutes of arc, still outside the limits of Tycho's observational errors. In parallel
with these researches, he was writing his treatise A Commentary on the Motion ofMars
and he reached Chapter 5 1 before he realised that what he needed was a figure intermediate
between an ovoid and a circle, an ellipse. He soon arrived at the key result that the orbit
ofMars and indeed those ofthe other planets are ellipses with the Sun lying at one focus.
The treatise on Mars was renamedAstronomiaNova, or TheNewAstronomy, with a subtitle
Based on Causes, or Celestial Physics. It was published in 1609, four years after he had
madethe discovery ofthis law, whichwenowknowasKeplersfirstlawofplanetarymotion.
To state it formally,
The planetary orbits are ellipses with the Sun in one focus.
Notice the imaginative leap needed to place the Sun at the foci ofthe ellipses. Kepler
alreadyknewthatthe motion ofMars could not be referred to the centre ofthe Earth's orbit
and so the focus is the next most obvious place. Kepler had discovered a crucial fact about
the orbits of the planets, but he had no physical explanation for it. It turned out to be one
ofthe key discoveries for the proper understanding ofthe law ofgravity but it had to await
the genius ofIsaac Newton before its full significance was appreciated.
The next development was due to Galileo. We have to anticipate the great discoveries
which he made in 1609 with his astronomical telescope. These were published in his book,
the Sidereus Nuncius or the SiderealMessenger, in 1610. Galileo was aware ofthe publi­
cation of Kepler's Astronomia Nova in the previous year and sent a copy of the Sidereus
Nunciusto the TuscanAmbassadoratthe Imperial CourtinPrague, askingforawrittenopin­
ion from K.epler. Kepler replied in a long letter on 19 April 1610, which he then published

17. 2.4 Johannes Kepler and heaven ly harmonies 29
o
- 0
Figure 2.9: Two pages from Galileo's Sidereus Nuncius, showing his drawings ofthe movements of
the four Galilean satellites. (Courtesy ofthe Royal Observatory, Edinburgh.)
in May under the title Dissertatio cum Nuncio Siderio or Conversation with the Sidereal
Messenger. As might be imagined, Kepler was wildly enthusiastic and, while Galileo pre­
sented his observations with some cautious attempts at interpretation, Kepler gave full rein
to his imagination.
The most ilnportant discoveries for Kepler were the moons of Jupiter. These fitted in
beautifullywiththeCopernicanpicture. Herewasaminiature solarsystem: thefourbrightest
moons ofJupiter - 10, Europa, Ganymede and Callisto - were seen orbiting the planet. In
the Sidereus Nuncius, Galileo shows the motions of the four satellites - examples of the
diagranls appearing in the Sidereus Nuncius are shown in Fig. 2.9. Kepler was in full flow.
Here is the part ofhis letter explaining why Jupiter has moons:
The conclusion is quite clear. Our moon exists for us on Earth, not for the other globes. Those four
little moons exist for Jupiter, not for us. Each planet in turn, together with its occupants, is served by
its own satellites. From this line of reasoning we deduce with the highest degree of probability that
Jupiter is inhabited. 14
What can one say? It is intriguing that this fonn of extreme lateral thinking was part ofthe
personality ofthe greatest ll1athematician in Europe at the tilne.
Kepler's third law was deeply buried in his treatise, HarmonicesMundi or TheHarmony
ofthe World. According to Max Casper, this work, published in 1619, was his crown­
ing achievelnent, in which he attenlpted to synthesise all his ideas into one harmonious

18. 30 2 From Ptolemy to Kepler - the Copernican revolution
picture ofthe Universe. His harmonic theory was to encompass geometry, music, architec­
ture, metaphysics, psychology, astrology and astrononlY, as can be seen from the contents
page ofthe five books which make up the treatise (Fig. 2.10). In modern terms, this was
his 'grand unified theory', a concept which has haunted much of physics throughout the
centuries.
By 1619, Kepler was no longer satisfied with an accuracy of 5% in the comparison of
the radii ofthe planetary orbits with his harmonic theory. He now hadmuch more accurate
mean radii for the orbits ofthe planets and suddenly, at the time he had reached the writing
ofBook V, Chapter III, Eighth Division, ofthe Harmony ofthe World, he discovered what
is now known as Keplers third law ofplanetary motion:
The period of a planetary orbit is proportional to the three-halves power ofthe mean distance ofthe
planet from the Sun.
This was the crucial discovery which eventually led to Newton's law of gravity. Notice
that this law is somewhat different from the solution he had proposed in the Mys­
terium Cosmographicum. It is not in conflict with it, however, since there is nothing
in Kepler's third law to tell us why the planets have to lie at particular distances from
the Sun.
It is important to note that Kepler's discoveries of the law of equal areas and of the
ellipses which describe the paths of the planets, as well as the third law, were intuitive
leaps ofthe imagination rather than the result offollowing some prescribed set ofstandard
mathematical techniques.
Itmightbe thoughtthatthe support ofKepler, apassionate Copernican, would have been
invaluable to Galileo during his prosecution for advocating the Copernican picture ofthe
world. As we will see, matters were not quite as simple as this. Galileo was rather cautious
about Kepler's support and we can identify at least two aspects ofhis concerns. Two years
after Kepler's death on 15 November 1630, Galileo wrote:
I do not doubt but that the thoughts of Landsberg and some of Kepler's tend rather to the diminution
ofthe doctrine of Copernicus than to its establishment as it seems to me that these (in the common
phrase) have wished too much for it . . .
1 5
We have already quoted Kepler's purple prose, which indicates part of the worry.
It is difficult to be taken seriously as a research scientist if you spend time worrying
about spoon-bending, unidentified flying objects, corn-circles and the like. In addition,
the strong mystical streak in the Harmony ofthe Worldwould nothave appealed to Galileo,
who was attempting to set the whole of natural philosophy on a secure mathematical
foundation.
Another cause ofGalileo's concerns is ofthe greatest interest. Again quoting his words:
It seems to me that one may reasonably conclude that for the maintenance ofperfect order among the
parts ofthe Universe, it is necessary to say that movable bodies are movable only circularly.16
Kepler's assertion that the orbits of the planets were ellipses rather than circles was
intellectually repugnant to Galileo. We can recognise again that this was no more than the

19. 2 .4 Johannes Kepler a nd heavenly harmonies
I
I
3 1
Figure 2. 1 0: The table of contents of Kepler's treatise The Harmony of the World, 1619, Linz.
(From the Crawford Collection, Royal Observatory, Edinburgh.)

20. 32 2 From ptolemy to Kepler - the Copernican revolution
legacy ofAristotelian physics, according to which the only allowable motions are uniform
linear and circular motions. We would now say that this view was an unexamined prejudice
onthepartofGalileo,butweshouldnotdisguisethefactthatwe allmakesimilarjudgements
in our own work.
Eventually Kepler completed the Rudolphine Tables, and they were published in
September 1627. These set a new standard of accuracy in the prediction of solar, lunar
and planetary positions. It is interesting that, in order to simplify the calculations, Kepler
had invented his own form of logarithms. He had seen John Napier's Mirifici Logarith­
morum Canonis Descriptio of 1614 as early as 1617 - this was the first set of tables of
natural logarithms.
Although itwas not recognised at the time, Kepler's three laws ofplanetary motion were
to be crucial for Newton's great synthesis ofthe laws ofgravity andcelestialmechanics. But
there was another giant, whose contributions were even more profound - Galileo Galilei.
He has become a symbol ofthe birth ofmodern science and the struggle against received
dogma. Thetrial ofGalileo strikesrightto the veryheart ofthe modern concept ofscientific
method. But he contributed much more andthese are the topics with which we will grapple
in the next chapter.
2.5 References
1 Pedersen, O. and Pihl, M. (1974). EarlyPhysics andAstronomy, p. 64. London: McDonald
and Co.
2 Pedersen, o. and Pihl, M. (1974). Op. cit., p. 65.
3 Foratranslation, see Duncan, A.M. (1976). Copernicus: On theRevolutionsoftheHeavenly
Spheres. ANew Translationfrom theLatin. London: David and Charles, New York: Barnes
and Noble Books.
4 Hellman, D.C. (1970). Dictionary of Scientific Biography, Vol. 1 1, p. 403. New York:
Charles Scribner's Sons.
5 Dreyer, 1.L.E. (1890). Tycho Brahe. A Picture ofScientific Life and Work in the Sixteenth
Century, pp. 86-7. Edinburgh: Adam and Charles Black.
6 Christianson, 1. (1961). ScientificAmerican, 204, 1 18 (February issue).
7 Thoren, Y.E. (1990). The Lord of Uraniborg. A Biography of Tycho Brahe, pp. 188-9.
Cambridge: Cambridge University Press.
8 Casper, M. (1959). Kepler, trans. C. Doris Hellman, pp. 46-7. London and New York:
Abelard-Schuman.
9 Casper, M. (1959). Op. cit., p. 20.
10 Casper, M. (1959). Op. cit., p. 62.
1 1 Kepler, 1. (1596). From Mysterium Cosmographicum. See Kepleri Opera Omnia,
ed. C. Frisch, Vol. 1, pp. 9ff.
12 Casper, M. (1959). Op. cit., p. 71.
13 Kepler, 1. (1609). From Astronomia Nova. See Johannes Keplers Gesammelte Werke, ed.
M. Casper, Vol. III, p. 178, Munich: Beck (1937).

21. 2.5 References 33
14 Kepler,1. (1610). In Conversationwith Galileos SiderealMessenger,ed. andtrans. E. Rosen
(1965), p. 42. New York and London: Johnson Reprint Co.
15 Galilei, G. (1630). Quoted by Rupert Hall, A. (1970), From Galileo to Newton 1 630-1 720.
The Rise ofModern Science 2, p. 41, London: Fontana Science.
16 Galilei, G. (1632). Dialogues concerning the Two Chief Systems of the World, trans.
S. Drake, p. 32, Berkeley (1953).

22. 34
3 Ga l i l eo a nd the natu re of
the physica I sci ences
3 . 1 I ntroduction
There are three separate but linked stories to be told. The first concerns Galileo as natural
philosopher. UnlikeTycho Brahe the observerandKeplerthemathematician, Galileowas an
experimentalphysicistwhose prime concernwasunderstandingthe laws ofnature in quanti­
tativeterms, from his earliestwritings to his final great treatiseDiscourse andMathematical
Demonstrations concerning Two New Sciences.
The second story is astronomical, and occupies a relatively small, but crucial, period of
Galileo's career, from 1609 to 1612, during which time he made a number offundamental
astronomical discoveries which had a direct impact upon his understanding ofthe physics
ofmotion.
The third story, and the most famous of all, is his trial and subsequent house arrest,
which continues to be the subject of considerable controversy. The scientific aspects of
his censure and subsequent trial are of the greatest interest and strike right at the heart of
the nature ofthe physical sciences. The widespread view is to regard Galileo as the hero
and the Catholic Church as the villain ofthe piece, a source of conservative reaction and
bigoted authority. From the methodologicalpoint ofview Galileo made an logical error, but
the church authorities made a much more disastrous blunder, which has resonated through
science and religion ever since, and which was only officially acknowledged by Pope John
Paul II in the 1980s.
My reasons for devoting a whole chapter to Galileo, his science and his tribulations are
that it is a story which needs to be better known and which has resonances for the way in
which physics as a scientific discipline is carried out today. Galileo's intellectual integrity
and scientific genius are an inspiration - more than anyone else, he created the intellectual
framework for the development ofphysics as we know it.
3.2 Galileo as an experimental physicist
Galileo Galilei was the son of Vincenzio Galileo, a distinguished musician and musical
theorist, and was born in February 1564 in Pisa. In 1587, he was appointed to the chair
of mathematics at the University of Pisa, where he was not particularly popular with his
colleagues. One ofthe main causes was Galileo's opposition to Aristotelian physics, which

23. 3.2 Galileo as an experirnental physicist 35
remained the central pillar ofnatural philosophy. It was apparent to Galileo that Aristotle's
physics was not in accord with the way in which matter actually behaves. For example,
Aristotle's assertion concerning the fall ofbodies ofdifferent weights reads as follows:
If a certain weight moves a certain distance in a certain time, a greater weight will move the same
distance in a shorter time, and the proportion which the weights bear to each other the times too will
bear to one another; for example, if the halfweight covers the distance in x , the whole weight will
cover it in x /2.1
This is just wrong, as could have been demonstrated by a simple experiment - it seems
unlikely that Aristotle evertried the experiment himself. Galileo's objection is symbolised
by the story ofhis dropping different weights from the Leaning Tower ofPisa. Ifdifferent
weights are dropped through the same height, they take the same time to reach the ground
ifthe effects ofair resistance are neglected, as was known to Galileo and earlier writers.
In 1592, Galileo was appointed to the chair ofmathematics at Padua, where he was to
remainuntil 1610. It was during this periodthathe produced his greatest work. Initially, he
was opposed to the Copernican model ofthe solar system but, in 1595, he began to take
it seriously in order to explain the origin ofthe tides in the Adriatic. He observed that the
tides at Venice typically rist:: and fall by about five feet and therefore there must be quite
enormous forces to cause this huge amount ofwater to be raised each half-day at high tide.
Galileo reasoned that ifthe Earth rotated on its own axis and also moved in a circular orbit
about the Sun then the changes in the direction of travel of a point in the surface of the
Earth would cause the sea to slosh about and so produce the effect ofthe tides. This is not
the correct explanation for the tides, but it led Galileo to favour the Copernican picture for
physical reasons.
In Galileo's printed works, the arguments are given entirely in the abstract without ref­
erence in the conventional sense to experimental evidence. Galileo's genius as a pioneer
scientist is described by Stillman Drake in his remarkable book Galileo: PioneerScientist
(1990).1 Drake deciphered Galileo's unpublished notes, which are not set down in any
systematic way, and convincingly demonstrated that Galileo actually carried out the exper­
iments to which he refers in his treatises with considerable experimental skill (Fig. 3.1).
Galileo's task was enormous - he disbelieved the basis ofAristotelian physics, but had
no replacement for it. In the early 1600s, he undertook experimental investigations ofthe
laws offree fall, the motion ofballs rolling down slopes and the motion ofpendulums; his
results clarifiedthe concept ofacceleration for the first time.
A problemwithphysics up to the time ofGalileo was that there was noway ofmeasuring
short time intervals accurately, and so he had to use considerable ingenuity in the design of
his experiments. A very nice example is his experimentto investigate how a ball accelerates
down a slope. He constructed a long shallow slope oflength 2 metres at an angle ofonly
1 .7° to the horizontal and cut a grove in it down which a heavy bronze ball could roll. He
placed little frets onthe slope so thatthere wouldbe a little click astheballpassed overeach
fret. He then adjusted the positions ofthe frets along the slope so that the clicks occurred
at equal time intervals (Fig. 3.2). Drake suggests that he could have equalised the time
intervals to about 1/64 ofa second by singing a rhythmic tune andmaking the clicks occur
at equal beats in the bar. In view ofGalileo's father's profession, this seems quiteplausible.

24. 36 3 Gali leo a nd the nature of the physica l sciences
Figure 3. 1 : Part of Galileo's notes concerning the laws of the pendulum. (From S. Drake, 1 990,
Galileo: Pioneer Scientist, p. 1 9, Toronto: University ofToronto Press.)

25. 3.2 Gal i leo as a n experimental physicist 37
Figure 3.2: How Galileo established the law of motion under uniform acceleration. The numbers
between the frets show their relative positions in order to produce a regular sequence ofclicks.
By this means, he was able to measure the distance travelled as the ball rolled continuously
down the slope and, by taking differences, he could work out the average speed between
successive frets. He foundthatthe speed increased as the oddnumbers 1, 3, 5, 7, . . . in equal
time intervals.
Originally, Galileo had believed that, under constant acceleration, speed is proportional
to distance travelled but, as a result ofthese precise experiments of 1604, he found, rather,
that speed is proportional to time. He now had two relations: the first was the definition of
speed, x = v t, and the second related speed to time under constant acceleration, v = at.
Now, there isno algebra in Galileo's publishedworks and the differential calculus hadyetto
be discovered. Suppose the speeds ofa uniformly accelerated sphere are measured at times
0, 1, 2, 3, 4, 5 seconds (Fig. 3.2). Assume the sphere starts from rest at time 0. The speeds
atthe above times will be, say, 0, 1, 2, 3, 4, 5, . . . cm s-1 , an acceleration of 1 cm s-2. How
far has the sphere travelled after 0, 1, 2, 3, 4, 5 seconds?
At zero time, no distance has been travelled. Between ° and 1 s, the average speed is
0.5 cm S-l and so the distance travelled must be 0.5 cm. In the next interval, between
1 and 2 s, the average speed is 1 .5 C1TI S-l and so the distance traveiled in that interval is
1.5 cm; the total distance travelled from the position ofrest is now 0.5 + 1.5 = 2 cm. In the
following interval, the average speed is 2.5 cn1 s-l, the distance travelled is 2.5 cm and the
total distance is 4.5 C1TI, and so on. We thus obtain a series of distances, 0, 0.5, 2, 4.5, 8,
12.5, . . . cm, which can be 'written in cm as
This is Galileo's famous time-squared law for uniformly accelerated motion, expressed
algebraically as
(3.2)
This result represented a revolution in thinking about the nature ofaccelerated lTIotion and
led directly to the Newtonian revolution.
Galileo did not stop there but went on to carry out two further brilliant experiments. He
next studied the question offree fall, natTIely, ifan object is dropped from a given height,
how long does it take it to hit the ground? He used a form ofwater clock to measure time
intervals accurately. Waterwas allowed to pour out ofa tube at the bOtt01TI ofa large vessel,
kept full; the an10unt of water which flowed out was a measure ofthe tilTIe interval. By
dropping objects from different heights, Galileo established that freely falling objects obey
the time-squared law - in other words, when objects fall freely they experience a constant
acceleration, the acceleration due to gravity.

26. 38 3 Gali leo and the nature of the physica l sciences
Figure 3.3: How Galileo established the theorem known by his name.
Having established these two results, he sought a relation between them. The desired
relation, Galileos theorem, is very beautiful. Suppose a body is dropped freely through a
certaindistanceI, which is representedbythe length AB inFig. 3.3. Constructacirclewhose
diameter is AB. Now suppose the body slides without friction down an inclined plane and,
for convenience, the top ofthe plane is placed at the point A. Galileo's theorem states that:
The time it takes a body to slide down the slope from the point A to the point C, where the slope cuts
the circle, is equal to the time it takes the body to fall freely from A to B.
In other words, the time ittakes a body to fall along any chordofa circle is the same as the
time it takes the body to fall freely down the diameter ofthe circle. The component ofthe
acceleration due to gravity is g sinex as the body slides down the slope; the component of
acceleration perpendicular to the slope is zero (Fig. 3.3).
Now, any triangle constructed onthe diameter ofa circle and with its thirdpoint lying on
the circle is a right-angled triangle. Therefore, we can equate angles, as shown in Fig. 3.3,
from whichitis apparent that AC/AB, the ratio ofthe distances travelled, is equal to sinex .
Since, for equal times, the distance travelled is proportional to the acceleration, x = �at2,
this proves Galileo's theorem.
The next piece of genius was to recognise the relation between these deductions and
the properties ofswinging pendulums. As a youth, Galileo is said to have noticed that the
period ofthe swing ofa chandelier suspended in a church is independent ofthe amplitude
ofits swing. Galileo made use ofhis law ofchords ofa circle to explain this observation.
Ifthe pendulum is long enough, the arc AC described by the pendulum is almost exactly
equal to the chord across the circlejoining the extreme point ofswing ofthe pendulum to
the lowest point (Fig. 3.4). Inverting Fig. 3.3, it is therefore obvious why the period ofthe
pendulum is independent of the amplitude of its swing - according to Galileo's theorem,
the time to travel along any chord drawn to A will be the same as the time it takes the body
to fall freely down twice the length ofthe pendulum. This is really brilliant physics.
What Galileo hadachieved was to put into mathematical form the nature ofacceleration
under gravity. This had immediate practical application, because he could now work out
the trajectories ofprojectiles. They travel with constant speed parallel to the ground and are
accelerated by gravity in the vertical direction. For the first time, he was able to work out
the parabolic paths ofcannon balls and other projectiles (Fig. 3.5).

27. 3.2 Gal ileo as an experimental physicist
B
/ 1
/ 1
/ I
I I
/ I
/ I
/ I
/ I
/ I
/ I
/ I
/ I
/ I
39
Figure 3.4: How Galileo showed that the period ofa long pendulum is independent ofthe amplitude
ofthe swing. Note the relation to Fig. 3.3.
Figure 3.5: A page from Galileo's notebooks showing the trajectories ofprojectiles under the com­
bination ofacceleration under gravity and constant horizontal speed. (From S. Drake, 1 990, Galileo:
Pioneer Scientist, p. 1 07, Toronto: University ofToronto Press.)

28. 40 3 Gal i leo and the nature of the physica l sciences
Galileo began writing a systematic treatment ofall these topics, showing how they could
all be understood on the basis ofthe law ofthe constant acceleration; in 1610, in his own
words, he was planning to write:
. . . three books on mechanics, two with demonstrations of its principles, and one concerning its
problems; and though other men have written on the subject, what has been done is not one quarter
ofwhat I write, either in quantity or otherwise.2
Later he writes in the same vein:
. . . three books on local motion - an entirely new science in which no one else, ancient or modern,
has discovered any of the most remarkable laws which I demonstrate to exist in both natural and
violent movement; hence I may call this a new science and one discovered by me from its very
foundations.3
The publication ofthese discoveries was delayed until the 1620s and 1630s. He was diverted
fromthistaskbynews ofthe inventionofthetelescope. This wasthebeginning ofhis serious
study ofastronomy.
3.3 Galileo's telescopic discoveries
The invention ofthe telescope is attributed to the Dutch lens-grinder Hans Lipperhey, who
in October 1608 applied to Count Maurice ofNassau for a patent for a device which could
make distant objects appear closer. Galileo heard of this invention in July 1609 and set
about building one for himself. By August, he had succeeded in constructing a telescope
which magnified nine times, a factor three better than that patented by Lipperhey. This
greatly impressed the Venetian Senate, who understoodthe importance ofsuch a device for
a maritime nation. Galileo was immediately given a lifetime appointment at the University
ofPadua at a vastly increased salary.
By the end of 1609, he had made a number of telescopes of increasing magnifying
power, culminating in a telescope with a magnifying power of 30. In January 1610, he
first turned his telescopes on the skies and immediately there came a flood ofremarkable
discoveries. These were rapidly published in March 1610 in his Sidereus Nuncius or The
SiderealMessenger.4 In summary, the discoveries were:
(i) the Moon is mountainous rather than a perfectly smooth sphere (Fig. 3.6(a));
(ii) the MilkyWay consists ofvastnUlnbers ofstars ratherthanbeing aunifonn distribution
oflight (Fig. 3.6(b));
(iii) Jupiter has four satellites, whose motions can be followed over several complete orbits
in a matter ofweeks (Fig. 2.9).
The book caused a sensation throughout Europe and Galileo won immediate international
fame. These discoveries demolished a number of Aristotelian precepts which had been
accepted over the centuries. For example, the resolution ofthe Milky Way into individual
stars was quite contrary to the Aristotelean view. In the satellites of Jupiter, Galileo saw a
prototype for the Copernican picture of the Solar System. The immediate effect of these

29. (a)
(b) �
..
*' '*
'*
*�.�
*
�
)E.:tt: '*
,*.
'* �
"* ,;If
* **
:tf *"'
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if
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-* ..� Jf *
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*:$.
* li( �
�
� �
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-}If
�:-
Figure 3.6: (a) Galileo's drawing ofthe Moon as observed through his telescope. (b) Galileo's sketch
of the region of sky in the vicinity of Orion's belt, showing the resolution of the background light
into faint stars. (From G. Galilei, 16 1 0, Sidereus Nuncius, Venice. See also the translation by A. van
HeIden, 1 989, Chicago: University ofChicago Press.)

30. 42 3 Galileo and the nature of the physica l sciences
J) J) 1) « cc cr J) D 0 o (] ([6 5 4 I
1 2 I
� <--M-- �SU N
4 I 1
5
VENUS
1
, 5 4
5
" J /
-0-/ "
I SUN
f
1 2
I
A
ARTH
'-- y�
VENUS
1
Figure 3.7: Illustrating the phases ofVenus, according to the geocentric and heliocentric pictures of
the structure of the Solar System. (From A. van HeIden, 1 989, Sidereus Nuncius, or The Sidereal
Messenger, p. 1 08, Chicago: University of Chicago Press.)
discoveries was that Galileo was appointed Mathematician and Philosopher to the Grand
Duke ofTuscany, Cosimo de Medici, to whom the Sidereus Nuncius was dedicated.
Later in 1610, he made two other crucial telescopic discoveries:
(iv) the rings of Saturn, which he took to be close satellites ofthe planet;
(v) the phases ofthe planet Venus.
This last discovery was ofthe greatest importance. When Venus is on the far side ofits orbit
with respect to the Earth, it appears circular but when it is on the same side ofthe Sun as
the Earth, it looks like a crescent Moon. This was interpreted as evidence in favour ofthe
Copernican picture because it is explained completely naturally ifVenus andthe Earthboth
orbit the Sun, the latterbeing the source oftheir illumination (Fig. 3.7). If, however, Venus
moved on an epicycle about a circular orbit around the Earth and the Sun moved on a more
distant sphere then the pattern ofillumination relative to the Earth would be quite different,
as illustrated in Fig. 3.7. In 161 1 these great discoveries were presented by Galileo to the
Pope and several cardinals, who were all favourably impressedby them. Galileo was elected
to the Academia Lincei.
3.4 The trial of Galileo - the hea rt of the matter
Before recounting the events which led up to Galileo's appearance before the Inquisition
and his conviction for the second most serious crime in the papal system ofjustice, let us
summarise briefly some ofthe different facets ofthe debate between the Ptolemaeans, the

31. 3.4 The trial of Gal i leo -- the heart of the matter 43
Copernicans and the church authorities; Finocchiaro provides an excellent summary in his
documentary history The GalileoAffair (1989).5 The established laws ofphysics remained
in essence Aristotelian and only a few adventurous spirits doubted the basic correctness of
the Ptolemaic system. There wereproblellls withthe Copernicanpicture and so Galileo had
to become involved in these issues because they undermined his new-found understanding
ofthe laws ofmotion.
3.4.1 The issues
The physical issues centred on these questions. First, does the Earth rotate on its axis with
respect to the fixed stars? Second, do the Earth and the planets orbit the Sun? Specifically,
is the Earth in motion? This latter concept was referred to as the geokinetic hypothesis.
Finocchiaro summarises the pre-Galilean objections to this hypothesis underfive headings.
(i) The deception ofthesenses. None ofour senses gives us any evidence thatthe Earth
is moving in an orbit about the Sun. Ifthis were a fact ofnature, surely it would be of such
importance that our senses w'Ould make us aware ofit.
(ii) Astronomicalproblems. First, the heavenly bodies were believed to be composed of
different forms ofmatter fromthematerial ofthe Earth. Second, Venus should showphases
similar to the Moon ifit were in orbit about the Sun. Third, ifthe Earth moved, why didn't
the stars exhibitparallaxes?
(iii) Physical arguments. These were largely based upon Aristotelian physics and we
have encountered some ofthem already.
(a) If the Earth moves, falling bodies should not fall vertically. Many counter-examples
could be given - rain falls vertically, objects thrown vertically upwards fall straight
down again, and so on. This was in contrast to the trajectory ofan object dropped from
the top ofa ship's mast when a ship is in motion. In this case, the object does not fall
vertically downwards, with respect to an observer on the shore.
(b) Projectiles fired in the direction of rotation of the Earth and in the opposite direction
would have different trajectories. No such difference had been observed.
(c) Objects placed on a rotating potter's wheel are flung offifthey are not held down. This
was calledthe extrudingpowerofwhirling, what is nowknown as the centrifugalforce.
The same phenomenon should occur ifthe Earth is in a state ofrotation, butwe are not
flung offthe surface ofthe Earth.
(d) Next, therewerepurelyphilosophical arguments. Two forms ofmotion, uniformmotion
in a straight line and uniform circular motion, were thought to be the only 'natural'
motions which objects could have. Objects must either fall in a straight line to the
centre of the Universe or be in a state of uniform circular motion. We have already
discussed the question v{hether objects fall towards the centre ofthe Earth or towards
the Sun. Furthermore, according to Aristotelianphysics, a simple body couldhave only
one natural motion. But, according to Copernicus, objects dropped on Earth have three
motions - downward motion under free fall, motion due to the rotation ofthe Earth on
its axis and motion in a circular orbit about the Sun.
(e) Finally, if Aristotelian physics was to be rejected, what was there to replace it? The
Copernicans had to provide a better theory and none was available.

32. 44 3 Galileo and the nature of the physica l sciences
(iv) TheAuthority ofthe Bible. There are no absolutely unambiguous statements in the
Bible that assert that the Earth is stationary at the centre of the Universe. According to
Finocchiaro, the most relevant statements6 are as follows:
(a) Psalm 104:5. '0 Lord my God . . . who laid the foundations ofthe Earth, that it should
not be removed forever.'
(b) Ecclesiastes 1 :5. 'The Sun also riseth, andthe Sun goeth down, and hasteth to the place
where he ariseth.'
(c) Joshua 10:12,13. 'Then spake Joshua to the Lord in the day when the Lord delivered
up the Amorites before the children ofIsrael, and he said in the sight ofIsrael, "Sun,
stand thou still upon Gibeon; and thou, Moon, in the valley ofAjalon." And the Sun
stood still, and the Moon stayed, until the people had avenged themselves upon their
enemies.'
These are rather oblique references andit is intriguing thatthe Protestants were much more
virulently anti-Copernican than the Catholics because oftheir belief in the literal truth of
the Bible. The Catholic theologians took a more sophisticated and flexible interpretation of
holy writ. However, the concept that the Earth is stationary at the centre ofthe Universe
had also been the conclusion ofthe ChurchFathers-the saints, theologians and churchmen
who codified Christianity. To quote Finocchiaro,
The argument claimed that all Church Fathers were unanimous in interpreting relevant Biblical
passages . . . in accordance with the geostatic view; therefore the geostatic system is binding on all
believers, and to claim otherwise (as Copernicus did) is heretical.7
(v) The mostinterestingargumentfromourperspective concernsthehypotheticalnature
ofCopernican theory. It strikes at the very heart ofthe nature ofthe natural sciences. The
crucialpoint is howwe express statements concerningthe success ofthe Copernicanmodel.
A correct statementis: 'Ifthe Earthrotates onits axis andmoves ina circularorbit aboutthe
Sun, and ifthe other planets also orbit the Sun, then we can describe simply and elegantly
the observed motions of the Sun, Moon and planets on the celestial sphere.' What we
cannot do logically is to reverse the argument and state that because the planetary motions
are explained simply and elegantly by the Copernican hypothesis therefore the Earth must
rotateandmoveinacircularorbitaboutthe Sun. This is anelementaryerroroflogic,because
there might well be quite different reasons why the Copernican model was successful.
The key point is the difference between induction and deduction. Owen Gingerich8 gives a
pleasant example. A deductive sequence ofarguments might run:
(a) Ifit is raining, the streets are wet.
(b) It is raining.
(c) Therefore, the streets are wet.
There is no problem here. But, now reverse (b) and (c) and we get into trouble.
(a) Ifit is raining, the streets are wet.
(b) The streets are wet.
(c) Therefore, it is raining.

33. 3 .4 The trial of G a l i leo -- the hea rt of the matter 45
This second line of reasoning is obviously false since the streets could be the streets of
Venice, or could have been newlywashed. In otherwords, you cannot prove anything about
the absolute truth of statements in this way. This type of reasoning, in which we attempt
to find general laws fronl specific pieces of evidence is called induction. All the physical
sciences areto agreater orlesserextentbasedoninduction, and so physical lawsnecessarily
have a provisional, hypothetical nature. This was seen as in marked contrast to the absolute
certainty ofGod's wordas containedinthe holy scriptures and its interpretation as dogmaby
the Church Fathers. According to Owen Gingerich,8 this was the issue of substance which
led to the trial and censure of Galileo - thejuxtaposition ofthe hypothetical world picture
of Copernicus with the truth as revealed in the Bible.
3.4.2 The Galileo affair
Prior to his great telescopic discoveries of 1610-1 1, Galileo was at best a cautious Coper­
nican, but it gradually became apparent to him that his new understanding of the nature
of motion eliminated the physical and astronomical problems listed under (ii) and (iii)
above. The new evidence was consistent with the Copernican model; specifically, there are
mountains on the Moon,just as there are on Earth, suggesting that the Earth and the Moon
are similar bodies and the phases ofVenus are exactly as would expected according to the
Copernicanpicture. Thus, thephysical and astronomical objections to Copernicanisln could
be discarded, leaving only the logical and theological problems to be debated.
As the evidence beganto accumulate in favour ofCopernicanism, conservative scientists
andphilosophers had to rely more andmore upon the theological, philosophical and logical
arguments. In December 1613, the Grand Duchess Dowager Christina asked Castelli, one
ofGalileo's friends andcolleagues, aboutthe religious objections to the motion oftheEarth.
Castelli responded to the satisfaction ofboth the Duchess and Galileo, but Galileo felt the
need to set out the arguments in more detail. He suggested that there were three fatal flaws
in the theological argument. To quote Finoccharo:
First, it attempts to prove a conclusion [the Earth's rest] on the basis ofa premise [the Bible's commit­
ment to the geostatic system] which can only be ascertained with a knowledge ofthat conclusion in
the first place . . . the business ofBiblical interpretation is dependent on physical investigation, and to
base a controversial physical conclusion on the Bible is to put the cart before the horse. Second, the
Biblical objection is a non sequitur, since the Bible is an authority only in matters offaith and morals,
not in scientific questions . . . Finally, it is questionable whether the Earth's motion really contradicts
the Bible.9
This letterto the GrandDuchess Christinacirculatedprivatelyand came into the hands of
the conservatives. Sennons were delivered attacking the heliocentric picture and accusing
its proponents ofheresy. In March 1615, the Dominican friar Tommaso Caccini, who had
alreadypreached against Galileo, laid aformal charge ofsuspicion ofheresyagainst Galileo
before the Roman Inquisition. This charge was less severe than that offormal heresy, but
was still a serious one. The Inquisition nlanual stated that, 'Suspects of heresy are those
who occasionallyutterpropositionsthatoffendthelisteners . . . Thosewho keep, write, read,
or give others to read books forbidden in the Index . . .' Further, there were two types of

34. 46 3 Galileo a nd the nature of the physica l sciences
*
suspicion ofheresy, vehement andslight suspicion ofheresy, the formerbeing considerably
more serious than the latter. Once an accusation was made, there was a formal procedure
which had to be followed.
Galileo responded by seeking the support of his friends and patrons and circulated
three long essays privately. One of these repeated the arguments concerning the validity
ofthe theological arguments and became known as Galileos letter to the GrandDuchess
Christina; the revised version was expanded from eight to forty pages. By good fortune,
a Neapolitan friar, Paolo Antonio Foscarini, published a book in the same year arguing in
detailthat a moving Earth was compatible with the Bible. In December 1615, after a delay
dueto illness, Galileo himselfwentto Rome to clear his name andto prevent Copernicanism
being condemned.
So far as Cardinal Roberto Bellarmine, the leading Catholic theologian ofthe day, was
concerned, the main problem concerned the hypothetical nature ofthe Copernican picture.
Here are his words, written on 12 April 1615 to Foscarini, after the Letter to Christina was
circulating in Rome.
. . . it seems to me that Your Paternity [Foscarini] and Mr Galileo are proceeding prudently by limiting
yourselves to speaking suppositionally* and not absolutely, as I have always believed that Copernicus
spoke. For there is no danger in saying that, by assuming the earth moves and the sun stands still,
one saves all the appearances better than by postulating eccentrics and epicycles is to speak well;
and that is sufficient for the mathematicians. However, it is different to want to affirm that in reality
the Sun is at the centre of the world and only turns on itself without moving from east to west, and
the Earth is in the third heaven and revolves with great speed around the Sun; this is a very dangerous
thing, likely not only to irritate all scholastic philosophers and theologians, but also to harm the Holy
Faith by rendering Holy Scripture false.Io
Behindtheseremarks is aperfectlyvalidcriticism ofGalileo's supportforthe Copernican
picture. It is not correct to state, as Galileo did, that the observation of the phases of
Venus proves that the Copernican picture is necessarily correct; for example, in Tycho's
cosmology, in which the planets orbit the Sun but the Sun together with the planets orbit the
Earth (Fig. 2.7), exactly the same phases ofVenus would be observed as in the Copernican
picture. According to Gingerich, this was Galileo's crucial logical error. Strictly speaking,
he could only make a hypothetical statement.
The findings ofthe Inquisition were favourable to Galileo personally - he was acquitted
of the charge of suspicion of heresy. However, the Inquisition also asked a committee of
1 1 consultants for an opinion on the status of Copernicanism. On 16 February 1616, it
reportedunanimously thatCopernicanismwas philosophically and scientifically untenable
and theologically heretical. This erroneous judgement was the prime cause ofthe subse­
quent condemnation of Galileo. It seems that the Inquisition had misgivings about this
outcome because it issued no formal condemnation. Instead, it issued two milder instruc­
tions. First, GalileowasgivenaprivatewarningbyCardinalBellarmineto stop defendingthe
Copernican world picture. Exactlywhatwas said is a matter ofcontroversy, but Bellarmine
reported back to the Inquisition that the warning had been issued and that Galileo had
accepted it.
This word is often translated hypothetically.

35. 3 . 5 The trial of Gali leo 47
ThesecondresultwasapublicdecreebytheCongregationoftheIndex. First, itreaffirmed
that the doctrine of the Earth's motion was heretical; second, Foscarini's book was con­
demnedandprohibitedbybeingplaced onthe Index; third, Copernicus'sDeRevolutionibus
was suspendeduntil a few offendingpassages were amended; fourth, all similarbooks were
subject to the same prohibition.
Rumours circulatedthat Galileo hadbeentriedandcondemnedbythe Inquisition and, to
counteract these, Bellarmine issued a brief statement to the effect that Galileo had neither
been tried nor condemned, but that he had been informed of the Decree of Index and
instructed not to hold or defend the Copernican picture. Although he had been personally
exonerated, the result was a defeat for Galileo.
3 . 5 The trial o f Galileo
For the next seven years, Galileo kept a low profile and complied with the Papal instruc­
tion. In 1623, Gregory XV died and his successor, Cardinal Maffeo Barbarini, was elected
Pope Urban VIII. He was Florentine and took a more relaxed view of the interpretation
ofthe scriptures than his predecessor. An admirer ofGalileo, he adopted the position that
Copernicanism could be discussed hypothetically and might well prove to be of great value
inmaking astronomicalpredictions. Galileohad six conversations withUrbanVIIIin Spring
1624 and came to the conclusion that Copernicanism could be discussed, provided that it
was only considered hypothetically.
Galileo returned to Florence and immediately set about writing the Dialogue on the
Two Chief World Systems, Ptolemaic and Copernican. He believed he had made every
effort to comply with the wishes ofthe censors. The preface was writtenjointly by Galileo
and the censors and, after senne delay, the great treatise was published in 1632. Galileo
wrote the book in the form ofa dialogue between three speakers, Simplicio defending the
traditional AristotelianandPtolelnaic positions, Salviati defending the Copernicanposition
and Sagredo an uncommitted observer and man ofthe world. Consistently, Galileo argued
thatthepurpose was notto makejudgelnents, buttopass oninformationandenlightenment.
The book was published with full papal authority.
The Two Chief World Systems was well received in scientific circles, but very soon
complaints and rumours began to circulate in Rome. A document dated February 1616,
almost certainly a fabrication, was found in which Galileo was specifically forbidden from
discussingCopernicanisminanyfonn. Bynow, CardinalBellarminehadbeendead 1 1 years.
In fact, in his book Galileo had not treated the Copernican model hypothetically at all, but
ratheras afactofnature- Salviati is Galileo speakinghisownmind. The Copernican system
was portrayed in a lnuch more favourable light than the Ptolemaic picture, contradicting
Urban VIII's conditions for discussion ofthe two systems ofthe world.
The pope was forced to take action - papal authority was being undermined at a tilne
whenthe Counter-refornlationandthereassertion ofthatauthoritywereparamountpolitical
considerations. Galileo, now 68 years old and in poor health, was ordered to come to Rome
underthethreatofarrest. Theresultofthetrialwas aforegone conclusion. Intheend, Galileo
pleaded guilty to a lesser charge onthe basis that, ifhe had violatedthe conditions imposed

36. 48 3 Gal i leo and the nature of the physica l sciences
v
(a) (b)
v
)
(c)
Figure 3.8: (a) Dropping an object from the top of a mast in a ship that is stationary in the frame of
reference S. (b) Dropping an object from the top of a mast in a moving ship, viewed in the frame of
reference Sf of the ship. (c) Dropping an object from the top ofa mast in a moving ship, as observed
from the frame ofreference S. The ship moves to the lighter grey position during the time the object
falls.
uponhimin 1616, he haddone so inadvertently. Thepope insistedupon interrogation under
the formal threat oftorture. On 22 June 1633, he was found guilty of 'vehement suspicion
ofheresy' and was forced to make a public abjuration, the proceedings being recorded in
the Book ofDecrees.
I do not hold this opinion of Copernicus, and I have not held it after being ordered by injunction to
abandon it. For the rest, here I am in your hands; do as you please.I I
Galileo eventually returned to Florence where he remained under house arrest for the rest
ofhis life - he died in Arcetri on 9 January 1642.
With indomitable spirit, Galileo set about writing his greatest work, Discourses and
MathematicalDemonstrations on Two NewSciencesPertainingtoMechanics andtoLocal
Motion, normally known as simply Two New Sciences. In this treatise, he brought together
the understanding of the physical world which he had gained over a lifetime. The funda­
mental insights concern the second new science - the analysis ofmotion.
3.6 Galilean relativity
The ideas expounded in Two NewScienceshadbeen inhis mind since 1608. One ofthem is
what is now called Galilean relativity. Relativity is often thought ofas something invented
by Einstein in 1905, but this does not do justice to Galileo's great achievement. Suppose
an experiment is carried out on the shore and then on a ship moving at a constant speed.
If the effect of air resistance is neglected, is there any difference in the outcome of any
experiment? Galileo answers firmly, 'No, there is not.'
Therelativity ofmotion is vividly illustrated as mentioned earlier, by dropping an object
from the top of a ship's mast (Fig. 3.8). Ifthe ship is stationary, the object falls vertically
downwards. Now suppose the ship is moving. If the object is again dropped from the

37. 3 .6 Galilean relativity
z
y S y
'
Sf
v
o 0' I
�--------�=========5� x
x
,
Z
49
Figure 3.9: Illustrating two Cartesian frames ofreference moving atrelative velocity v inthedirection
ofthe positive x -axis in 'standard configuration'.
top of the mast, it again falls vertically downwards according to an observer on the ship.
However, a stationary observer sitting on the shore notes that, relative to the shore, the path
is curved (Fig. 3.8(c)). The reason is that, relative to the shore, the object has two separate
components to its motion - vertical acceleration downwards due to gravity and uniform
horizontal motion due to the motion ofthe ship.
This leads naturally to the concept offi"ames ofreference. When the position of some
object in three-dimensional space is n1easured, we can locate it by its coordinates in some
rectangular coordinate system (Fig. 3.9). The point P has coordinates x , y, z in this sta­
tionary frame ofreference, which we will call S. Now, suppose the ship moves along the
positive x-axis at some speed v. We can set up another rectangular frame of reference S'
on the ship, in which the coordinates ofthe point P are x
'
, y', z' . It is now straightforward
to relate the coordinates in these two frames ofreference. Ifthe object is stationary in the
frame S then x is a constant but the value ofx
'
changes as x
'
== x - vt, where t is the time,
assumingthatthe origins ofthe two frames ofreference are coincident at t == O. The values
ofy and y' remain the Saine in S and S', as do z and z'. Also, time is the same in the two
frames ofreference. We have derived a set ofrelations between the coordinates of objects
in the frames S and S':
x
'
== x - vt, (3.3)
These are known as the Galilean transformations between the frames S and S'. Frames of
reference which move at constant relative speed to one another are called inertial frames
ofreference. Galileo's great insight can be sUlnmarised by stating that the laws ofphysics
are the same in every inertialframe ofreference. As a corollary of this insight, Galileo
was the first to establish the law ofcomposition ofvelocities - if a body has con1ponents
ofvelocity in two different directions then the motion ofthe body can be found by adding

38. 50 3 Galileo a nd the nature of the physical sciences
together the separate effects ofthese motions. This was how he showed that the trajectories
ofcannon-balls and missiles are parabolae (Fig. 3.5).
In Two NewSciences Galileo describedhis discoveries concerning the nature ofconstant
acceleration, the motion ofpendulums and free fall under gravity. Finally, he stated his law
ofinertia, which asserts that a body moves at a constant velocity unless some impulse or
force causes it to change that velocity - notice that it is now velocity rather thanjust speed
which is constant, becausethe direction ofmotion does not change inthe absence offorces.
This is sometimes referred to as the conservation ofmotion - in the absence offorces, the
separate components ofthe velocity remain unaltered. The word inertia is used here in the
sense that it is a property ofthe body which resists change ofmotion. This law will become
Newton's first law ofmotion. It can be appreciated why the Earth's motion caused Galileo
no problems. Because ofhis understanding ofGalilean relativity, he realised that the laws
of physics would be the same whether the Earth were stationary or moving at a constant
speed.
3.7 Reflections
We cannot leave this study without reflecting on the methodological and philosophical
implications ofthe Galileo case. There is no questionnowthatthe Church made anerrorin
condemning the new physics ofCopernicus and Galileo. It was a further 350 years before
Pope John Paul II admittedthatan error had been made. InNovember 1979 onthe occasion
ofthe centenary ofthe birth ofAlbert Einstein, John Paul II stated that Galileo '. . . had to
suffer a great deal - we cannot conceal the fact - at the hands ofmen and organisms ofthe
Church.' He went on to assert that '. . . in this affair the agreements between religion and
science aremore numerous andabove all more importantthan the incomprehensions which
led to the bitter andpainful conflict that continued in the course ofthe following centuries.'
For scientists, the central issue is the nature of scientific knowledge and the concept
oftruth in the physical sciences. Part of Cardinal Bellarmine's argument is correct. What
Copernicus had achieved was a model that was much more elegant and economical for
understanding the motions ofthe Sun, Moon and planets than the Ptolemaic picture, but in
what sense was it the truth? If one were to put in enough effort, a Ptolemaic model of
the Solar System could be created today which would replicate exactly the motions ofthe
planets on the sky, but it would be of enormous complexity and provide little insight into
the underlying physics which describes their motions. The value ofthe new model was not
only that it provided a vastlyimproved framework for understanding the observed motions
ofthe celestial bodies but also that, inthe hands ofNewton, itwas to become the avenue for
obtaining avery much deeperunderstanding ofthe laws ofmotion in general, leading to the
unification of celestial physics, the laws ofmotion and the law of gravity. A scientifically
satisfactorymodelhasthe capabilitynot only ofaccountingeconomicallyfora large number
ofdisparateobservational andexperimentalphenomenabutalsoofbeing extendabletomake
quantitative predictions about apparently unrelated phenomena.
Notice that I use the word model in describing this process rather than asserting that
it is in any sense truth. Galileo's enormous achievement was to realise that the models to

39. 3 .7 Reflections 5 1
describe nature could be put on a rigorous mathematical basis. In one ofhis most famous
remarks, he stated in his treatise Il Saggiatore (TheAssayer) of 1624:
Philosophy is written in this very great book which always lies before our eyes (I mean the Universe),
but one cannot understand it unless one first learns to understand the language and recognise the
characters in which it is written. It is writteninmathematical language and the characters are triangles,
circles and other geometrical figures; without these means it is humanly impossible to understand a
word of it; without these there is only clueless scrabbling around in a dark labyrinth.1 2
This is often abbreviated to the statement that
The Book ofNature is written in mathematical characters.
This was the great achievement ofthe Galilean revolution. The apparentlyelementary facts
established by Galileo required an extraordinary degree ofimaginative abstraction. Matter
does not obey the apparently simple laws ofGalileo - there is always friction, experiments
canonlybe carriedoutwith limitedaccuracyandoftendonotwork. Itneeds deep insightand
imagination to sweep away the unnecessary baggage and appreciate the basic simplicity in
thewaymatterbehaves. Themodernapproachto science isnomorethantheformalisation of
the process begunby Galileo.. Ithas been calledthe hypothetico-deductive method, whereby
hypotheses are made andconsequences deducedlogically from them. A model is acceptable
so long as it does not run significantly counter to the waymatter is observed to behave. But
lllodeis are only valid within well-defined regions ofparameter space. Profesionals become
very attached to them and the remarks by Dirac and Douglas Gough quoted in Chapter 1
describe the need to be satisfied with approximate theories and the 'pain' experienced on
being forced to give up a cherished prejudice.
It is rare nowadays for religious dogma to impede progress in the physical sciences.
However, scientific prejudice and dogma are the common currency of scientific debate.
There is nothing particularly disturbing about this so long as we recognise what is going
on. A scientific prejudice becomes embodied in a model, whichprovides a framework for
carrying forward the debate and for suggesting experiments and calculations which can
provide tests ofthe self-consistency ofthe model. We will find many examples throughout
this book where the 'authorities' and 'receivedwisdom' were barriers to scientific progress.
It takes a great deal ofintellectual courage andperseverance to standup to what is normally
an overwhelming weight of conservative opinion. It is not just whimsy that leads us to
use pontifical language to describe some ofthe bandwagons which can dominate areas of
enquiry in the physical sciences. In extrellle cases, through scientific patronage scientific
doglllahas attained an authorityto the exclusion ofalternative approaches. One ofthe most
disastrous exampleswastheLysenko affairintheUSSRshortlyafterthe SecondWorldWar,
where Communist political philosophy strongly impacted the biological sciences, resulting
in a catastrophe forthese sciences in the Soviet Union.
Let me give two topical examples. It is intriguing how the idea of inflation during the
very early stages of expansion ofthe Universe has attained the status of 'received dogma'
among certain sections of the cosmological cOllllllunity. There are good reasons why this
idea shouldbetakenseriously� aswillbediscussedinChapter 19. There is, however,nodirect
experimental evidence for the actual physics which could cause the inflationary expansion

40. 52 3 Gali leo a nd the nature of the physica l sciences
ofthe early Universe. Indeed, a common procedure is to work backwards and 'derive' the
physics ofinflation from the need to account for the features ofthe Universe as we observe
it today. Then, theories of particle physics need to be found which can account for these
forces. There is the obvious dangerofendingupwithbootstrapped self-consistencywithout
any independent experimental check of the theory. Maybe this is the best one can do, but
some ofus will maintain a healthy scepticism until there are more independent arguments
which support the conjecture ofinflation.
The same methodology has occurred in the theory ofelementary particles with the de­
velopment ofstring theory. The creationofself-consistent quantum field theories involving
one-dimensional objects rather than point particles has been a quite remarkable achieve­
ment. The latestversions ofthese theories involvethe quantisation ofgravity as an essential
ingredient. Yet, they have not resulted in predictions which can be tested experimentally.
Nonetheless, this is the area into which many ofthe most distinguishedtheorists have trans­
ferred all their efforts. It is taken as an article offaith that this is the most promising way of
tackling these problems, despite the fact that it might well prove very difficult to find any
experimental or observational tests of the theory in the foreseeable future.
3.8 References
1 Drake, S. (1990). Galileo: PioneerScientist, p. 63. Toronto: University ofToronto Press.
2 Drake, S. (1990). Gp. cit., p. 83.
3 Drake, S. (1990). Gp. cit., p. 84.
4 Galilei, G. (1610). Sidereus Nuncius, Venice. See the translation by A. van HeIden (1989),
Sidereus Nuncius or The SiderealMessenger, Chicago: University ofChicago Press.
5 Finocchiaro, M.A. (1989). The GalileoAffair.ADocumentaryHistory. Berkeley: University
ofCalifornia Press.
6 Finocchiaro, M.A. (1989). Gp. cit., p. 24.
7 Finocchiaro, M.A. (1989). Gp. cit., p. 24.
8 Gingerich, o. (1982). ScientificAmerican, 247, 1 18.
9 Finocchiaro, M.A. (1989). Gp. cit., p. 28.
10 Finocchiaro, M.A. (1989). Gp. cit., p. 67.
1 1 Finocchiaro, M.A. (1989). Gp. cit., p. 287.
12 Sharratt, M. (1994). Galileo:DecisiveInnovator,p. 140. Cambridge: CambridgeUniversity
Press.