Galileo Galilei

How to use this course

Galileo is remembered for two things that are easy to separate and should not be. One is the trial of 1633, a conflict between a scientist and the Roman Church that has been retold, and often distorted, for four centuries. The other is a body of physics: the law of fall, the parabolic path of projectiles, the relativity of uniform motion, the pendulum, the first telescopic survey of the sky, and the first mathematical theory of the strength of beams. This course treats both, and keeps them in contact, because the physics is what made the trial matter.

Each technical lesson opens with a paragraph for the general reader, then states results precisely, derives them in modern notation and, where useful, shows how Galileo argued with proportions instead of equations. Exercises have folded solutions. The lessons run roughly in the order Galileo worked: hydrostatics (1586), motion (1590–1609), astronomy (1609–1613), and the final synthesis in Two New Sciences (1638).

On the trial, Lessons 11 and 12 rely on the published documents and on the standard historical studies listed under Sources. Where accounts disagree or a story is legend, the text says so.

Timeline

Pisa and Florence, 1564–1589

Galileo Galilei was born in Pisa on 15 February 1564, the first of six children of Vincenzo Galilei and Giulia Ammannati. Pisa then belonged to the Grand Duchy of Tuscany, ruled from Florence by the Medici. Vincenzo was a lutenist, composer and polemical music theorist. He tested the ancient numerical theory of consonance with weighted strings and found that pitch does not follow the simple ratios the Pythagorean tradition assumed: for a string of fixed length, the frequency goes as the square root of the tension. Galileo grew up watching a father settle a theoretical argument by experiment.

The family moved to Florence in 1572. Galileo joined them about two years later, was tutored there, and was then sent to the monastery of Vallombrosa, where he became a novice. His father removed him. In 1581 Vincenzo sent him back to Pisa to study medicine, a profession that paid far better than music or mathematics.

Mathematics instead of medicine

At Pisa Galileo heard Ostilio Ricci, a mathematician attached to the Tuscan court, teach Euclid. In the summer of 1583 Ricci persuaded Vincenzo to let his son study mathematics. Galileo read Euclid and Archimedes in Niccolò Tartaglia’s Italian editions, and Archimedes became his model: a mathematician who proved physical results from a few clear postulates. Viviani, his first biographer, dated to these years the story of the swinging lamp in Pisa Cathedral, timed against the pulse. The story comes only from Viviani, and the earliest written trace of Galileo’s interest in pendulums is later.

He left Pisa around 1585 without a degree and lived by private teaching in Florence and Siena. In 1586 he wrote La Bilancetta, a short treatise describing a precise balance for determining specific gravity by Archimedes’ method (Lesson 04). In 1587 he visited Christopher Clavius, the leading Jesuit mathematician, at the Collegio Romano, and began corresponding with him and with the Marquis Guidobaldo del Monte, an expert in mechanics who became his patron. In 1588 the Florentine Academy invited him to lecture on the location, shape and size of Dante’s Inferno, a geometrical exercise on a literary text. He also wrote on the centres of gravity of solids. These works earned him a reputation, and in 1589, with Guidobaldo’s support, he was appointed to the chair of mathematics at Pisa.

The Pisa chair and De motu, 1589–1592

The Pisa chair was poorly paid and of low status beside philosophy and medicine. Its holder taught Euclid and the astronomy of the Sphere. Galileo’s real work in these years was a critique of Aristotle’s theory of motion, written in Latin manuscripts now called De motu (On Motion), drafted around 1590 and never published.

Aristotle held that heavy bodies fall with a speed proportional to their weight and inversely proportional to the resistance of the medium. Galileo rejected this, but not yet for the reasons he later gave. In De motu he treated fall with an Archimedean model: a body falls with a speed set by the excess of its density over the density of the medium, so that two bodies of the same material fall at the same speed whatever their size. He also argued that there is no natural lightness, only less heaviness. The model is wrong, since it predicts uniform speed of fall, but it already uses Archimedes as a weapon against Aristotle and already says that size does not matter.

The Leaning Tower

Viviani wrote that Galileo dropped bodies of different weights from the Leaning Tower in front of students and professors. Galileo himself never describes such a demonstration, and most historians treat it as unproven; Stillman Drake argued that something of the kind did take place. Drops from towers had been made by others, including Simon Stevin in Delft in the 1580s. Whatever happened at Pisa, the arguments that convinced Galileo were the thought experiments and the inclined-plane measurements of Lesson 05.

His father died in 1591, leaving Galileo responsible for his mother, his brother Michelangelo and the dowries of his sisters. The Pisa salary could not meet these obligations. In 1592, again with Guidobaldo’s help, he obtained the chair of mathematics at the University of Padua, at a substantially higher salary.

Padua, 1592–1610

Padua was the university of the Venetian Republic, prosperous and relatively free of clerical control. Galileo gave his inaugural lecture on 7 December 1592 and stayed eighteen years, later calling them the best of his life. He taught geometry, astronomy and mechanics, took private pupils as boarders, and ran a workshop that produced instruments for sale.

The best known of these was the geometric and military compass, a hinged sector engraved with scales that let gunners, surveyors and merchants compute square roots, proportions and charges without arithmetic. He developed it in the late 1590s, and the instrument-maker Marc’Antonio Mazzoleni built more than a hundred under his direction. He published a printed manual for it, and in 1607 he took legal action against Baldassarre Capra, who had published a Latin plagiarism of it; Capra was expelled from Padua. In these years he also built a thermoscope, an air thermometer without a fixed scale, and obtained a Venetian patent for a horse-driven water pump.

In Padua he formed a long relationship with Marina Gamba, a Venetian woman whom he never married. They had three children: Virginia (1600), Livia (1601) and Vincenzo (1606). The daughters later entered the convent of San Matteo at Arcetri, as Sister Maria Celeste and Sister Arcangela. Maria Celeste’s letters to her father survive and are a principal source for his later life.

Motion and the Copernican question

In a letter to Johannes Kepler in the late 1590s Galileo said privately that he had long accepted the Copernican system, but he did not teach it. In 1604 a new star appeared, and he lectured to large audiences arguing that, since it showed no parallax, it lay beyond the Moon, against the Aristotelian doctrine of an unchanging heaven. The same year, in a letter to the Venetian friar Paolo Sarpi, he stated that distances fallen from rest grow as the squares of the times. Over the next years he tested this with rolling balls and timed descents (Lessons 05 and 06), work he would not publish for three decades.

The telescope

In May 1609 Sarpi told Galileo of a Dutch spyglass. Galileo worked out its principle, ground his own lenses, and on 25 August 1609 demonstrated an instrument of about eight or nine powers to Venetian senators, who could see ships hours before they reached port. The Senate confirmed his chair for life with a large salary increase, with the condition that it would not rise again. By the end of 1609 he had a twenty-power instrument pointed at the Moon. Lesson 09 follows what he saw. The discoveries made him famous across Europe within months, and in the summer of 1610 he accepted a post as Chief Mathematician of the University of Pisa, without teaching duties, and Philosopher and Mathematician to Grand Duke Cosimo II de’ Medici. In September 1610 he left Padua for Florence.

Archimedean beginnings: hydrostatics

For the general reader. A legend says that Archimedes discovered how to detect a fraudulent gold crown while getting into his bath. The young Galileo thought the story as usually told was crude and proposed a precise method: weigh the object in air and in water. Twenty-six years later he used the same principle to defeat Aristotelian philosophers in a public dispute over why ice floats.

La Bilancetta (1586)

Let a body weigh \(W_a\) in air and \(W_w\) when suspended in water. The loss \(W_a-W_w\) equals the weight of water of the same volume, so the specific gravity is

Galileo’s instrument was a balance with a long arm wound with fine wire, so that the position of the counterweight could be read by counting turns. For an alloy of two metals of specific gravities \(s_1\gt s_2\), volumes add, so a mass fraction \(x\) of the first metal gives

Floating bodies (1612)

In 1611 a dispute arose in Florence with the Aristotelian Lodovico delle Colombe and others, who held that ice floats because of its flat shape and that shape decides whether a body sinks. Galileo answered with the Discourse on Bodies that Stay atop Water (1612). Ice floats because it is less dense than water, as shown by the fact that it rises when pushed under. He also explained a striking consequence of the principle.

Falling bodies and the inclined plane

For the general reader. Drop a stone and it speeds up. Galileo asked exactly how. His answer, that speed grows in proportion to time and distance in proportion to the square of time, was the first mathematical law of motion confirmed by measurement. Because free fall was too fast to time, he slowed it down by rolling balls along a gentle ramp.

Against Aristotle

In Two New Sciences (Day One) Galileo’s spokesman Salviati offers a thought experiment. Suppose heavier bodies fall faster. Tie a heavy stone to a light one. The light one should retard the heavy one, so the pair falls slower than the heavy stone alone. But the pair is heavier than the heavy stone, so it should fall faster. The assumption contradicts itself. In a vacuum, Galileo concluded, all bodies would fall with the same speed; in a medium, differences come from resistance and buoyancy.

The definition of natural acceleration

Galileo first tried the hypothesis that speed grows in proportion to distance fallen, and in the 1604 letter to Sarpi he derived the correct distance law from it by a flawed argument. In Two New Sciences he rejects that hypothesis and defines uniformly accelerated motion as motion whose speed acquires equal increments in equal times: \(v\propto t\). The definition is chosen, he says, because it is the simplest one nature might use, and it must then be tested.

The mean speed theorem was known to fourteenth-century scholars at Merton College and to Nicole Oresme, as a statement about abstract qualities. Galileo proved it with a triangle whose area represents distance: speed rises linearly along one side, and the triangle has the same area as a rectangle of half the height. In modern terms, with \(v=gt\),

The inclined plane

Galileo assumed that a body descending a smooth incline acquires the same speed as one falling vertically through the same height. Then the component of gravity along an incline of angle \(\theta\) is \(g\sin\theta\), and motion along it is again uniformly accelerated, with a smaller constant:

His described experiment used a beam about twelve cubits long with a straight groove lined with parchment, and a hard bronze ball. Time was measured by collecting water from a thin jet into a cup and weighing it. He reported that distances went as the squares of the times for every inclination tried, with discrepancies too small to matter. Repeated with modern equipment, the method gives this accuracy.

What he did not know is that a rolling ball stores part of its energy in rotation. For a solid sphere rolling without slipping, \(\tfrac12 mv^2+\tfrac12 I\omega^2=mgh\) with \(I=\tfrac25 mr^2\) and \(\omega=v/r\) gives \(v^2=\tfrac{10}{7}gh\). The acceleration is reduced by the constant factor \(5/7\), so the times-squared law survives untouched. Only Galileo’s attempt to infer \(g\) itself would have been affected.

Projectile motion

For the general reader. Before Galileo, gunners’ manuals drew a cannonball’s path as a straight line, then a curve, then a vertical drop. Galileo showed that if air is ignored the whole path is a single curve, the parabola, because the ball does two simple things at once: it keeps its horizontal speed and it falls exactly as a dropped ball does.

Derivation

With launch speed \(v_0\) at angle \(\theta\) above the horizontal, and \(y\) measured upward,

This is a parabola. Setting \(y=0\) gives the range \(R=\dfrac{v_0^2\sin2\theta}{g}\), maximal at \(\theta=45^\circ\) and symmetric under \(\theta\mapsto90^\circ-\theta\). Galileo reached the same results with Apollonius’s geometry of conic sections, and Day Four includes tables of ranges and heights for elevations in steps of one degree. He was aware of two idealizations: air resistance, which matters a great deal for fast projectiles, and the convergence of vertical lines toward the Earth’s centre, which he judged negligible over gunnery distances.

Folio 116v

Among Galileo’s working papers in Florence is a sheet on which Stillman Drake recognized the record of an experiment, dated by him to about 1608. A ball rolls down an incline from a height \(h\) above a table, is deflected to move horizontally, and leaves the table edge. Galileo recorded the horizontal distance \(d\) to the landing point for five heights, in his unit of length, the punto (about 0.94 mm):

The fall from the table takes the same time for every launch, so \(d\) is proportional to the horizontal speed, and the speed acquired from height \(h\) goes as \(\sqrt h\). The prediction is \(d\propto\sqrt h\). The hero illustration plots exactly this test.

Relativity and inertia

For the general reader. If the Earth spins and circles the Sun, why don’t we feel it? Galileo’s answer, given in a famous passage about a ship, is that uniform motion cannot be detected from inside. Everything on board, butterflies, dripping water, fish in a bowl, behaves the same whether the ship is moored or sailing smoothly. This principle of relativity runs from Galileo through Newton to Einstein.

The ship

In the Second Day of the Dialogue Concerning the Two Chief World Systems (1632), Salviati asks the reader to shut himself below deck in a large ship with flies, butterflies, a bowl of fish and a dripping bottle. Observe everything with the ship at rest; then let the ship move at any speed, as long as the motion is uniform and does not fluctuate. No change will be seen in any of the effects, and from them one cannot tell whether the ship is moving or standing still.

Differentiating, \(\dot{\mathbf x}'=\dot{\mathbf x}-\mathbf u\) and \(\ddot{\mathbf x}'=\ddot{\mathbf x}\). Forces that depend only on relative positions and relative velocities are unchanged, so \(m\ddot{\mathbf x}=\mathbf F\) holds in \(S'\) whenever it holds in \(S\).

Answering the stationary-Earth arguments

Aristotelians argued that a stone dropped from a tower on a rotating Earth would land far to the west of its foot, and that a ball dropped from a ship’s mast lands behind the mast when the ship moves. Galileo replied that the stone shares the motion of the tower, and that a ball dropped from the mast of a moving ship lands at its foot. Experiment, he said, would confirm this, though his opponents had asserted the contrary without trying it. The same principle explains why birds and clouds are not left behind.

Galileo’s inertia

Galileo saw that a ball on a perfectly smooth horizontal plane, once set moving, would move forever. But “horizontal” for him meant a surface everywhere equidistant from the Earth’s centre, so his preserved motion was circular, which suited his view of planetary motion. René Descartes and then Isaac Newton stated inertia as uniform motion in a straight line. Galileo’s projectile analysis used the straight-line version locally.

The tides

Galileo believed the tides gave a physical proof of the Earth’s motion, and devoted the Fourth Day of the Dialogue to it. The combination of daily rotation and annual revolution makes points on the Earth speed up and slow down, and he argued this sloshes the seas. The argument violates his own principle of relativity, since the Earth’s orbital motion is very nearly uniform over a day, and it predicts one high tide per day, not two, and ignores the Moon. Kepler had already linked the tides to the Moon; Galileo dismissed that as occult.

The pendulum

For the general reader. A pendulum seems to swing in the same time whether its swing is wide or narrow. Galileo noticed this, used it to time experiments, and in his last years designed a clock around it. He was nearly right: the timing is constant only for small swings, a refinement found by Christiaan Huygens.

Small oscillations

For a bob on a light rod of length \(L\), \(\ddot\theta=-(g/L)\sin\theta\). For small \(\theta\), \(\sin\theta\approx\theta\) and

which contains (i) and (ii). For finite amplitude, energy conservation gives the exact period as an elliptic integral,

where \(K(k)=\int_0^{\pi/2}(1-k^2\sin^2\phi)^{-1/2}\,d\phi\). The period grows with amplitude. Huygens showed in 1673 that a bob constrained to a cycloid is exactly isochronous, and built clocks on that basis.

Chords of a circle

Proof. A chord from the top making angle \(\varphi\) with the vertical has length \(2R\cos\varphi\), and the acceleration along it is \(g\cos\varphi\). From \(2R\cos\varphi=\tfrac12(g\cos\varphi)t^2\), \(t=2\sqrt{R/g}\), independent of \(\varphi\). The case of chords ending at the lowest point is the same figure reflected.

Galileo then showed that descent along two successive chords inscribed in a quarter circle is faster than along the single chord, and concluded, without proof, that the circular arc itself is the path of quickest descent. That is false. Johann Bernoulli posed the brachistochrone problem in 1696, and its solution is the cycloid, the same curve as Huygens’s isochrone.

The pendulum clock

Late in life, already blind, Galileo designed an escapement to keep a pendulum swinging and count its beats, and explained the design to his son Vincenzo and to Viviani. Vincenzo began building it but died in 1649 before it was finished. Huygens built the first working pendulum clock independently in 1656–57.

Telescopic astronomy

For the general reader. In a few months of 1609 and 1610, with a tube and two lenses, Galileo saw mountains on the Moon, countless stars in the Milky Way, and four moons circling Jupiter. Later in 1610 he saw Venus go through phases like the Moon. None of this proved that the Earth moves, but it destroyed the old picture of a perfect heaven and made the Ptolemaic arrangement of the planets untenable.

The instrument

A Galilean telescope combines a weak convex objective of focal length \(f_o\) with a strong concave eyepiece of focal length \(-f_e\), separated by \(f_o-f_e\). The image is upright and the angular magnification is \(M=f_o/f_e\). The field of view is small because the exit pupil lies inside the instrument, and Galileo’s lenses suffered from spherical and chromatic aberration. He nevertheless reached about 20× for the observations of Sidereus Nuncius and later about 30×.

Sidereus Nuncius (Venice, 13 March 1610)

The book, dedicated to Cosimo II, reported four things. The Moon is rough and mountainous, with bright points standing in the dark part beyond the terminator. The telescope reveals many times more stars than the naked eye, for instance in Orion and the Pleiades. The Milky Way and several nebulous patches resolve into crowds of faint stars. And four “Medicean stars” accompany Jupiter: first seen on 7 January 1610, followed night after night, and recognized by 15 January as bodies circling the planet. Simon Marius observed them independently in the same weeks. Kepler endorsed the book within weeks in his Conversation with the Starry Messenger and confirmed the moons himself later in 1610.

Galileo observed lit points more than one twentieth of the lunar diameter beyond the terminator. Taking the diameter as 2000 Italian miles, \(d=100\) and \(R=1000\) give \(h\approx5\), so he concluded that some lunar mountains rise more than four miles.

The moons of Jupiter

A miniature solar system was a direct answer to the objection that the Moon would be left behind if the Earth moved: Jupiter, which moves on any theory, keeps its moons. By 1612 Galileo had estimated their periods, and he proposed using their eclipses as a clock for finding longitude. With modern values their orbits obey Kepler’s third law, \(a^3/T^2=GM_J/4\pi^2\).

The phases of Venus

In the Ptolemaic system Venus moves on an epicycle whose centre stays on the line from Earth to Sun, and in the usual ordering of the spheres Venus always lies between the Earth and the Sun. It would then never show more than a crescent. From the autumn of 1610 Galileo watched Venus pass from a small round disc, through half phase, to a large crescent. In December 1610 he sent Kepler a Latin anagram claiming the discovery; its solution, sent weeks later, read “the mother of loves imitates the shapes of Cynthia”. Full phases mean Venus goes round the Sun. This refuted Ptolemy but not the system of Tycho Brahe, in which the planets circle the Sun while the Sun circles a fixed Earth.

Saturn and sunspots

In July 1610 Saturn appeared to him as three bodies, a large one with two attendants; in 1612 the attendants vanished. The ring, then seen edge-on, was explained by Huygens in 1659. Galileo also observed sunspots and, in the Letters on Sunspots published by the Lincei in 1613, argued from their foreshortening near the limb that they lie on the Sun’s surface and that the Sun rotates in about a month, against the Jesuit Christoph Scheiner, who took them for small satellites. The priority dispute with Scheiner was long and bitter.

Strength of materials

For the general reader. Why can’t there be a horse the size of a house, or a ship ten times larger built exactly like a smaller one? Galileo found that weight grows with the cube of size but strength only with the square of size, so large structures are proportionally weaker. This scaling argument, from the first science of Two New Sciences, founded the theory of the strength of materials and explains the shapes of bones.

The book

Discorsi e dimostrazioni matematiche intorno a due nuove scienze was written during the house arrest and printed in Leiden by Louis Elzevir in 1638, beyond the reach of the Inquisition. The same three speakers as the Dialogue discuss, over four days, the resistance of solids to breaking (Days One and Two) and local motion (Days Three and Four). Day One ranges widely: cohesion and the vacuum, why a suction pump cannot lift water more than about eighteen cubits, a paradox of infinite sets (there are as many squares as positive integers, though most integers are not squares), an attempt to measure the speed of light with lanterns, the pendulum and the vibrating string. A further day on the force of percussion was published only in 1718.

The cantilever

Day Two considers a prism fixed horizontally in a wall, of breadth \(b\), depth \(h\) and length \(L\), with a load \(P\) at its free end. Galileo treats the beam as a lever pivoting about the bottom edge of the section at the wall, with the whole cross-section resisting in tension, each fibre pulling with the same stress \(\sigma\). The resultant tension acts at mid-depth, so balance of moments gives

What later theory changed

A real beam bends: fibres above the neutral axis stretch and those below compress, and in the elastic range stress varies linearly across the section. The Euler–Bernoulli theory, completed in the eighteenth century with contributions from Edme Mariotte, Antoine Parent and Charles-Augustin de Coulomb, gives the moment at which the outer fibres reach stress \(\sigma\):

Galileo’s coefficient is three times too large, but every ratio he deduced is correct, because both formulas scale as \(b h^2\). His proportional reasoning was robust exactly where his physical model was weak.

Bones and hollow tubes

Galileo drew a small bone and a bone three times longer, thickened to do the same job for a proportionally larger animal; the larger is grossly stout. He also showed that, for the same amount of material and length, a hollow tube resists breaking better than a solid rod, in the ratio of their diameters, and pointed to bird bones and plant stalks.

Florence, Rome and the Copernican question, 1610–1632

As court philosopher, Galileo was free of teaching and expected to bring glory to the Medici. In March 1611 he went to Rome. The Jesuit astronomers of the Collegio Romano, asked by Cardinal Robert Bellarmine whether the new observations were real, confirmed them and held a session in his honour. Prince Federico Cesi made him a member of the Accademia dei Lincei, whose members coined the word “telescope” at a banquet that spring. The Lincei published his Letters on Sunspots in 1613, and he signed his works thereafter as a Lincean.

Science and Scripture

In December 1613 his former student Benedetto Castelli, now professor at Pisa, was questioned at the Medici table by the Grand Duchess Christina about passages of Scripture that seemed to contradict the Earth’s motion. Galileo answered in a letter to Castelli, later expanded as the Letter to the Grand Duchess Christina (1615). He argued that Scripture speaks to ordinary people about salvation, not about astronomy; that two truths cannot contradict; and that where a physical conclusion is demonstrated, interpreters should adjust their reading. In February 1615 the Dominican Niccolò Lorini sent a copy of the letter to Castelli to the Roman Inquisition, and in March Tommaso Caccini, who had preached against Galileo in Florence, made a deposition against him. In April 1615 Bellarmine wrote to the Carmelite Paolo Antonio Foscarini that the Earth’s motion could be discussed as a hypothesis, but not asserted as true unless it were demonstrated.

1616

Galileo went to Rome in December 1615 to argue his case. On 24 February 1616 the Inquisition’s theological consultants declared that the Sun’s being motionless at the centre was foolish and absurd in philosophy and formally heretical, and that the Earth’s motion was at least erroneous in faith. On 26 February, by order of Pope Paul V, Bellarmine summoned Galileo and told him to abandon the opinion. A note in the Inquisition file, unsigned by Galileo or by Bellarmine, records a stricter injunction by the Commissary General: not to hold, teach or defend it in any way, verbally or in writing. On 5 March the Congregation of the Index suspended Copernicus’s De revolutionibus until corrected and banned Foscarini’s book; Galileo was not named. In May Bellarmine gave Galileo a certificate stating that he had not abjured or been penanced, only informed of the decree.

The Assayer and the Dialogue

A dispute over the comets of 1618 with the Jesuit Orazio Grassi produced Il Saggiatore (The Assayer, 1623), a polemic remembered for its claim that the book of nature is written in the language of mathematics, and for its distinction between primary qualities of bodies and secondary qualities such as colour and taste. In August 1623 Galileo’s admirer Cardinal Maffeo Barberini became Pope Urban VIII, and the Lincei dedicated the book to him. In 1624 Galileo had six audiences with the new pope and understood that he might write about the Copernican system as long as he treated it as a hypothesis.

The result was the Dialogue Concerning the Two Chief World Systems, finished in 1630. Three characters discuss the question over four days: Salviati, the Copernican; Sagredo, an intelligent layman; and Simplicio, an Aristotelian. After a complicated licensing process, with conditional approval from Rome and the printing moved to Florence, the book came off the press of Giovanni Battista Landini in February 1632. Although it ended with a reminder that God could produce the tides in ways unknown to us, it was plainly an argument for Copernicus, and that closing argument, which Urban had urged on Galileo, is given to Simplicio. Contemporary accounts report that the pope took offence.

The trial of 1633, Arcetri, legacy

The proceedings

In the summer of 1632 Urban VIII stopped the sale of the Dialogue and appointed a special commission to examine it. The commission found the 1616 file note, and in September the case was sent to the Inquisition. Galileo, aged 68 and unwell, asked to be tried in Florence; he was refused, and when physicians certified that he was too ill to travel, the Inquisition warned that he would be brought in chains if necessary. He left Florence on 20 January 1633 and, after a plague quarantine, reached Rome on 13 February, lodging with the Tuscan ambassador at the Villa Medici.

The legal question was narrow. Galileo was not charged with heresy for heliocentrism as such. The charge was that he had disobeyed the 1616 injunction by writing a book that defended the condemned opinion. At his first deposition, on 12 April 1633, before the Commissary General Vincenzo Maculani, Galileo produced Bellarmine’s certificate, which said nothing of “in any way whatever teaching”, and claimed that the book did not defend Copernicus. He was held for the next weeks in rooms within the Inquisition’s building. After a private conversation with Maculani, he admitted on 30 April that in places he had made the Copernican arguments seem stronger than he intended, out of vanity. On 10 May he submitted a written defence, pleading his age and health. On 21 June he was examined on his intention under the formal threat of torture, which was not carried out, and he maintained that since 1616 he had not held the opinion.

The sentence

On 22 June 1633, at the Dominican convent of Santa Maria sopra Minerva, the sentence was read. Galileo was found “vehemently suspected of heresy”, a technical category in canon law, for having held and defended the opinions that the Sun is at the centre and does not move, that the Earth moves, and that an opinion may be held as probable after being declared contrary to Scripture. He was to abjure, curse and detest these errors; the Dialogue was prohibited by public edict; he was condemned to formal imprisonment at the Inquisition’s pleasure; and as a penance he was to recite the seven penitential psalms once a week for three years. Seven of the ten cardinals of the tribunal signed the sentence. Galileo knelt and read the abjuration the same day. The words “and yet it moves” first appear in print more than a century later and are almost certainly legend.

The imprisonment was commuted at once to confinement at the Villa Medici, then to the custody of Archbishop Ascanio Piccolomini in Siena, who received him warmly. In December 1633 he was allowed to return to his villa at Arcetri, above Florence, under house arrest for life, forbidden to receive visitors without permission or to publish.

Arcetri

The villa was near the convent of his daughter Maria Celeste, who died on 2 April 1634. Galileo turned back to the work on motion and materials he had set aside a quarter century earlier. The manuscript of Two New Sciences reached Louis Elzevir in Leiden, and the book appeared in July 1638. By then Galileo had lost the sight of both eyes. Visitors were allowed under supervision; the young poet John Milton, on his Italian journey of 1638–39, later wrote that he had visited the old astronomer as a prisoner of the Inquisition. In 1639 the teenage Vincenzo Viviani came to live with him as assistant, and in 1641 Evangelista Torricelli joined them. Galileo died at Arcetri on 8 January 1642, aged 77. He was buried without a monument in Santa Croce in Florence. In 1737 his remains were moved to a tomb in the main body of the church, built with funds left by Viviani.

The long aftermath

The Index of 1758 dropped the general prohibition of books teaching the Earth’s motion, and the edition of 1835 no longer listed the Dialogue or De revolutionibus. In 1979 Pope John Paul II asked for a new study of the case; a commission formed in 1981 reported in 1992, and on 31 October 1992 he acknowledged that the theologians who condemned Galileo had erred in reading the structure of the physical world through the literal sense of Scripture. Historians continue to debate the weight of the 1616 file note, the role of Urban VIII’s personal anger, and the political situation of a pope at war with critics during the Thirty Years’ War. The documents are published, and the reader can judge them directly (see Finocchiaro under Sources).

Legacy

• Mechanics: the law of fall, the composition of motions, the parabola and relativity are the foundation on which Huygens and Newton built. Newton’s first two laws generalize Galileo’s results.
• Method: idealized mathematical laws, tested by experiments designed to remove disturbing factors, became the model of physics.
• Astronomy: the telescope turned astronomy into an observational science. Jupiter’s four large moons are now called the Galilean satellites; NASA’s Galileo spacecraft orbited Jupiter from 1995 to 2003, and Europe’s satellite navigation system bears his name.
• Engineering: scaling laws and the \(bh^2\) rule for beams remain in every structures course.
• Science and authority: the trial became the standard case study, and the standard myth, of conflict between science and religion. Albert Einstein called Galileo the father of modern physics.

Sources and further reading

Biography and the trial

1. J. J. O’Connor and E. F. Robertson, Galileo Galilei, MacTutor History of Mathematics.
2. P. Machamer and D. M. Miller, Galileo Galilei, Stanford Encyclopedia of Philosophy.
3. The Galileo Project, Rice University, Chronology of Galileo’s life.
4. Galileo Galilei and Galileo affair, Wikipedia (with extensive references).
5. Museo Galileo, Florence, The sentence of condemnation of Galileo.
6. D. Linder, The trial of Galileo: chronology and documents, Famous Trials.
7. M. A. Finocchiaro (ed. and trans.), The Galileo Affair: A Documentary History, University of California Press, 1989; and Retrying Galileo, 1633–1992, 2005.
8. A. Fantoli, Galileo: For Copernicanism and for the Church, 3rd ed., Vatican Observatory Publications, 2003.
9. J. L. Heilbron, Galileo, Oxford University Press, 2010.
10. S. Drake, Galileo at Work: His Scientific Biography, University of Chicago Press, 1978.
11. D. Sobel, Galileo’s Daughter, Walker, 1999 (based on the letters of Maria Celeste).
12. V. Viviani, Racconto istorico della vita di Galileo Galilei, written 1654.

Galileo’s works and the physics

1. G. Galilei, Sidereus Nuncius, Venice, 1610. English trans. A. Van Helden, University of Chicago Press, 1989. See Sidereus Nuncius.
2. G. Galilei, Dialogue Concerning the Two Chief World Systems, Florence, 1632. English trans. S. Drake, University of California Press, 1953.
3. G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden, 1638. English trans. H. Crew and A. de Salvio, Macmillan, 1914; S. Drake, University of Wisconsin Press, 1974. See Two New Sciences.
4. G. Galilei, Le Opere di Galileo Galilei, Edizione Nazionale, ed. A. Favaro, 20 vols, Florence, 1890–1909.
5. S. Drake and J. MacLachlan, Galileo’s discovery of the parabolic trajectory, Scientific American 232 (March 1975). Data reproduced in G. Cowan’s exercise file.
6. Phases of Venus and Galilean invariance, Wikipedia.
7. S. P. Timoshenko, History of Strength of Materials, McGraw-Hill, 1953 (Galileo’s beam theory and its successors).
8. T. B. Settle, An experiment in the history of science, Science 133 (1961), on reproducing the inclined-plane experiment.