Giovanni Domenico Cassini

How to use this course

Giovanni Domenico Cassini was the leading observational astronomer of the later seventeenth century. With better telescopes than anyone else, he found the rotation periods of Jupiter and Mars, four moons of Saturn and the main gap in its rings. His tables of Jupiter’s satellites made it possible to find longitude on land and led, in other hands, to the first measurement of the speed of light. He directed the Paris Observatory for four decades and founded a family that ran it for four generations. He was also a conservative in theory: he doubted the finite speed of light, preferred his own ovals to Kepler’s ellipses for a time, and believed the Earth to be elongated at the poles. This course presents the observer, the organizer and the questions on which he was wrong.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Italy and France used the Gregorian calendar throughout Cassini’s life, so all dates are Gregorian.

Timeline

Perinaldo and Genoa, 1625–1644

Giovanni Domenico Cassini was born on 8 June 1625 at Perinaldo, a hill village above the Ligurian coast near Sanremo. His father, Jacopo Cassini, was from Tuscany; his mother was Julia Crovesi. He was brought up largely by a maternal uncle. After two years of schooling at Vallebone he went to the Jesuit college in Genoa, and then studied at the abbey of San Fruttuoso.

He wrote Latin verse and was drawn to astronomy through astrology, which was then still a respectable learned pursuit and a source of patronage. He read widely in it, and became convinced that its predictions had no foundation. The astronomy behind the horoscopes interested him more than the horoscopes themselves. His skill at calculation brought him to the notice of wealthy patrons in Genoa, one of whom introduced him to a Bolognese senator with an observatory.

Bologna: Panzano and San Petronio, 1644–1664

The Marquis Cornelio Malvasia, a senator of Bologna and an amateur astronomer, was building an observatory at his villa at Panzano near Modena. In about 1644 he invited Cassini to work there, and from 1648 Cassini observed with instruments Malvasia had bought. In Bologna he also learned from the Jesuit astronomers Giovanni Battista Riccioli and Francesco Maria Grimaldi, who were then preparing Riccioli’s great survey of astronomy and his map of the Moon.

In 1650, with Malvasia’s support, the Senate of Bologna appointed him to the principal chair of astronomy at the university, vacant since the death of Bonaventura Cavalieri in 1647. He was twenty-five. He observed the comet of 1652 and wrote a book on it, and in 1662 published solar tables.

The sun in the church

The basilica of San Petronio had an old meridian line laid down by Egnazio Danti in the previous century, but the church had been extended and the line no longer worked. In 1653 Cassini proposed a new and much longer one, running diagonally across the nave between the columns. A small hole high in the roof let a beam of sunlight fall on a brass line set in the floor, and the position of the solar image at noon gave the Sun’s altitude with high precision. The line was completed in 1655. Cassini used it to measure the obliquity of the ecliptic, the length of the year, the effect of refraction, and above all the variation of the Sun’s apparent size through the year (Lesson 04). In 1656 he published the first results as Specimen observationum Bononiensium, dedicated to Queen Christina of Sweden.

Bologna also used him as an engineer. In 1657 he advised the Senate in its long dispute with Ferrara over the course of the river Reno, a question that Pope Alexander VII had to settle. He was later made superintendent of fortifications (1663) and of the waters of the Papal States (1665). He declined the Pope’s suggestion that he take holy orders.

Telescopes, waters and the call to Paris, 1664–1671

From 1664 Cassini had access to the long refracting telescopes of the Roman lens-maker Giuseppe Campani, the best in the world at the time. With them he followed markings on the planets long enough to measure how fast they turned. In July 1664 he began the observations of Jupiter’s spots that gave its rotation period and showed the planet’s flattening. He followed a prominent spot in 1665; credit for first seeing what may be the Great Red Spot is shared with Robert Hooke. In 1666 he found the rotation of Mars. In 1668 he published his Ephemerides Bononienses of Jupiter’s satellites, which made it possible to predict their eclipses (Lesson 06).

That work made his name in Paris. Jean-Baptiste Colbert, Louis XIV’s minister, was gathering foreign scientists for the new Académie Royale des Sciences, founded in 1666, and for an observatory being built on the southern edge of the city. Christiaan Huygens had already come from Holland. Colbert offered Cassini a large pension. Bologna and Pope Clement IX gave him leave for what all expected would be a short visit, and he left Bologna on 25 February 1669. He never returned to his chair.

In Paris he soon gained a decisive voice in the new Observatory. When it opened in 1671 he became its effective director. He took French nationality on 14 July 1673 and became Jean-Dominique Cassini. In 1674 he married Geneviève de Laistre, whose dowry included the château of Thury, the family’s country seat for generations.

A meridian line in a church

For the general reader. A hole in a high roof acts like a pinhole camera. At noon, the Sun’s image falls on the floor at a distance that depends on how high the Sun stands, and its size depends on how far away the Sun is. Cassini turned the nave of San Petronio into an instrument able to measure both, and used it to test whether Kepler was right about the orbit of the Earth.

Geometry of a meridiana

Let the aperture be at height \(H\) above a horizontal floor. At local noon the Sun has altitude \(h=90^\circ-\varphi+\delta\), where \(\varphi\) is the latitude and \(\delta\) the Sun’s declination. The centre of the image lies on the meridian line at distance \(D=H\cot h\) from the point vertically beneath the hole. The beam has length \(L=H/\sin h\), and a Sun of angular diameter \(\theta\) makes an image about \(L\theta\) wide across the line and \(L\theta/\sin h\) long along it.

Since \(\delta=\pm\varepsilon\) at the solstices, adding and subtracting the two altitude formulas gives both results. The winter value is the more sensitive to refraction, which Cassini studied with the same instrument.

The bisection of the eccentricity

Ptolemy and later Copernicus described the Sun’s apparent motion by a circle whose centre is displaced from the Earth. Kepler showed that for the planets the displacement must be split: the centre of the orbit lies halfway between the Sun and the point about which motion appears uniform. For the Earth–Sun orbit this has a measurable consequence.

The observed amplitude is \(A\approx1.92^\circ\approx0.0335\). Kepler predicts the Sun’s diameter changes by about \(\pm1.7\%\) over the year; the simple eccentric predicts \(\pm3.4\%\). The solar image at San Petronio is large enough to tell these apart. Cassini’s measurements came out close to the smaller value, in favour of the bisected eccentricity. It was one of the first direct observational supports for Kepler’s theory of the Earth’s orbit.

The rotation of the planets

For the general reader. Before Cassini nobody knew how long a day lasted on another planet. By watching the same spot return to the same place on the disc, he found that Jupiter turns in under ten hours and Mars in a little more than an Earth day. He also saw that Jupiter is visibly squashed, a fact Newton later explained by its fast spin.

Cassini found Jupiter’s period to be about \(9^{\mathrm h}56^{\mathrm m}\) and that of Mars about \(24^{\mathrm h}40^{\mathrm m}\), within a few minutes of the modern values of \(9^{\mathrm h}55.5^{\mathrm m}\) (System III) and \(24^{\mathrm h}37.4^{\mathrm m}\). Around 1690 he noticed that spots near Jupiter’s equator went round faster than those at higher latitude: the first evidence that a planet’s visible surface does not rotate as a rigid body.

The homogeneous result is Newton’s; the point-mass limit is Huygens’s. Jupiter has \(q\approx0.089\) and observed \(f\approx0.065\), between \(0.045\) and \(0.111\), which reveals a dense core. The same reasoning, applied by Newton to the Earth, predicted the oblate shape that Cassini’s school disputed (Lesson 10).

Jupiter’s satellites, longitude and the speed of light

For the general reader. Galileo had suggested that the moons of Jupiter, disappearing into its shadow at predictable instants, could serve as a clock in the sky visible from anywhere on Earth. Comparing local time with the predicted time of an eclipse gives longitude. Cassini’s tables made the method work on land. Then the eclipses began to arrive early or late, and the reason turned out to be the time light takes to cross space.

Longitude

The Earth turns \(15^\circ\) per hour. If an eclipse of Io is predicted for \(T_0\) in Paris time and observed at local time \(T\), the observer is at longitude \(15^\circ\times(T-T_0)\) east of Paris (with time in hours). Using Cassini’s tables, astronomers of the Academy, Jean Picard among them, fixed the longitudes of places across France in the 1670s and 1680s. The results shrank the outline of the kingdom on the map, especially along the Atlantic coast. The method needed a large telescope on steady ground, so it never worked at sea.

Light time

On 22 August 1676 Cassini announced to the Academy that his eclipse tables would include a new inequality in the motion of the satellites, and he mentioned that it could be explained by a finite speed of light. Ole Rømer, a Danish astronomer working at the Observatory, developed that explanation and predicted that an emergence of Io in November 1676 would be seen about ten minutes later than the old tables predicted. He presented his theory to the Academy on 21 November 1676, estimating that light took about 22 minutes to cross the diameter of the Earth’s orbit. Cassini then turned against the idea. The other satellites did not show the effect clearly, and Io had other irregularities. His 1693 tables used empirical corrections that differed from satellite to satellite. The finite speed of light was generally accepted after James Bradley’s discovery of aberration in 1728.

Saturn: four moons and a gap

For the general reader. Huygens had found Saturn’s largest moon, Titan, and explained its strange appendages as a ring. Cassini, with longer telescopes, found four more moons between 1671 and 1684, and in 1675 saw that the ring was really two rings separated by a dark band. He suggested the ring was a swarm of tiny satellites, which is correct.

Iapetus

Cassini noticed that Iapetus was easy to see when west of Saturn and nearly invisible when east of it. He concluded that one hemisphere is much darker than the other and that the moon always keeps the same face toward Saturn, as our Moon does toward the Earth. Spacecraft images confirmed the two-toned surface in detail.

The division

The Cassini division lies between about 117,600 and 122,200 km from Saturn’s centre. Its inner edge is set by the innermost of the larger moons, Mimas (\(a=185\,539\) km), found by William Herschel in 1789. A ring particle at radius \(r\) has period proportional to \(r^{3/2}\). Where the period is exactly half that of Mimas, the moon pulls the particle the same way at every second conjunction, and the disturbances add up. This 2:1 resonance clears particles from the edge of the B ring.

Mars in 1672 and the distance of the Sun

For the general reader. In September 1672 Mars came close to the Earth. The Academy had sent Jean Richer to Cayenne, in French Guiana, nearly 7,000 km from Paris. By comparing Mars against the stars from both places, Cassini measured its distance, and through Kepler’s third law the distance of the Sun. The answer, about 140 million kilometres, was the first value close to the truth.

The horizontal parallax of a body at distance \(d\) is \(\pi\), with \(\sin\pi=R/d\) for the Earth’s radius \(R\). An observer sees the body displaced from its geocentric position, away from the zenith, by about \(\pi\sin z\), where \(z\) is the zenith distance.

Each observer sees the body depressed toward the southern horizon by \(\pi\sin z_i\) with \(z_i=\varphi_i-\delta\); the star’s parallax is negligible. Paris and Cayenne differ by about 3.6 hours in longitude, so the two meridian crossings are not simultaneous, and Mars’s own motion in the interval had to be removed using observations on several nights.

The Paris–Cayenne campaign led to a solar parallax of about \(9.5''\), against the modern \(8.794''\). In England Flamsteed reached a similar bound from a single station (see the course on Flamsteed). Older values, going back to antiquity, had made the Sun twenty times closer. Halley later proposed the transits of Venus as a more precise method (see the course on Halley).

Cassini’s laws of the Moon’s rotation

For the general reader. The Moon always shows us nearly the same face. Cassini went further. He worked out how its axis is tilted and how that tilt moves, and found three simple rules. They hold not only for the Moon: many moons and planets have settled into the same arrangement, now called a Cassini state.

Since the orbit’s node regresses with a period of 18.6 years, the third law means the spin axis precesses with it, keeping the three normals coplanar. The angle between the Moon’s equator and its orbit is \(I+i\approx1.54^\circ+5.15^\circ\approx6.7^\circ\). The laws were later derived from gravitational dynamics by Lagrange and Laplace; the modern view treats them as a stable equilibrium of the spin axis driven by tides.

Librations

Because rotation is uniform but orbital motion is not, we see a little around the east and west limbs: the libration in longitude. Because the equator is tilted to the orbit, we see alternately over the north and south poles: the libration in latitude.

Ovals and the figure of the Earth

For the general reader. Cassini was an empiricist who distrusted grand theory. He proposed his own curves for planetary orbits in place of Kepler’s ellipses, and he believed his survey of France showed the Earth to be stretched toward the poles, against Newton. Both positions were later abandoned, but the mathematics of his ovals and the methods of his survey lasted.

Cassini ovals

Derivation. Expanding, \(\big((x^2+y^2+a^2)-2ax\big)\big((x^2+y^2+a^2)+2ax\big)=(r^2+a^2)^2-4a^2x^2\). With \(x=r\cos\theta\), \(4a^2r^2\cos^2\theta=2a^2r^2(1+\cos2\theta)\), which gives the polar form. On the \(y\)-axis (\(\theta=90^\circ\)) we need \(r^4+2a^2r^2+a^4-b^4=0\), so \(r^2=-a^2+b^2\): there are points on the \(y\)-axis exactly when \(b\ge a\), which separates the one-loop and two-loop cases.

Cassini studied these curves around 1680 as possible planetary orbits, with the Sun at one focus. For small eccentricity they are close to ellipses, and the observations of his time could not easily distinguish them. Newton’s derivation of the ellipse from the inverse-square law settled the matter.

The length of a degree

From 1683 Cassini measured an arc of the meridian through Paris by triangulation, measuring south while Philippe de La Hire measured north. The work stopped in 1684 for lack of money and resumed in 1700 with his son Jacques, reaching Perpignan. The French results suggested that a degree of latitude grows shorter toward the north.

The meridian radius of curvature is \(M=a(1-e^2)(1-e^2\sin^2\varphi)^{-3/2}\) with \(e^2=2f-f^2\). Expanding to first order gives the formula. Over the short French arc the expected change was tiny and easily swamped by errors. The expeditions to Lapland (1736–37) and Peru settled the matter for Newton (see the courses on Newton and Clairaut).

Director in Paris, 1671–1712

For forty years Cassini ran the Paris Observatory as the centre of French astronomy. Formally he was one academician among others, but the King’s favour and his own energy made him its master. He brought long-focus objectives from Campani and, later, very long aerial telescopes with no tube at all, the lenses mounted on masts. He sent observers abroad: Richer to Cayenne in 1672, and others to the coasts of France to fix longitudes. Jean Picard, who had measured a degree of the meridian near Paris, worked closely with him until Picard’s death in 1682. Ole Rømer worked at the Observatory from 1672 to 1681.

The discoveries of the Paris years were made there: Iapetus and Rhea, the division in Saturn’s ring, Tethys and Dione. In 1679 he presented to the Academy a large map of the Moon. In 1683 he described the zodiacal light, the faint cone of light along the ecliptic after sunset, as sunlight scattered by matter around the Sun. In 1680 and 1681 the young Edmond Halley worked with him on the great comet (see the course on Halley). Cassini also wrote an explanation of Siamese astronomical rules, published in Simon de La Loubère’s account of Siam in 1691.

He took his son Jacques on a journey through Italy in 1695, restoring the San Petronio meridian he had built forty years earlier. From about 1709 Jacques took over more of his duties. By 1711 Cassini was almost blind; the historian of the Academy, Bernard de Fontenelle, wrote that he bore it calmly, sustained by his faith. He died in Paris on 14 September 1712, aged 87.

The Cassini dynasty and legacy

Four generations of Cassinis directed the Paris Observatory: Giovanni Domenico; his son Jacques Cassini (1677–1756), who completed and published the meridian arc and defended an elongated Earth; Jacques’s son César-François Cassini de Thury (1714–1784), who accepted the flattened Earth after the Lapland results and began the great map of France; and his son Jean-Dominique Cassini (1748–1845), under whom the map was completed. The Carte de Cassini, built on a national network of triangles, was the first topographic map of a whole country based on geodetic survey. Giacomo Filippo Maraldi, Cassini’s nephew, also worked at the Observatory.

Historians have judged Cassini in two ways. Jean-Baptiste Delambre, writing in the nineteenth century, portrayed him as authoritarian and quick to take credit. Later historians, such as René Taton, emphasized the observer and organizer, more gifted at seeing than at theory.

Legacy

• Saturn: four moons and the Cassini division. NASA’s Cassini spacecraft, launched in 1997 with the European Huygens probe, orbited Saturn from 2004 to 2017.
• Planetary physics: the first rotation periods of planets, Jupiter’s flattening and differential rotation, and the Cassini states of rotating moons and planets.
• The scale of the solar system: the first reasonably accurate solar parallax.
• Geodesy and cartography: longitude from Jupiter’s satellites, the Paris meridian and the map of France.
• Names: craters on the Moon and Mars, the asteroid 24101 Cassini, and the Cassini ovals of geometry.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Giovanni Domenico Cassini, MacTutor History of Mathematics.
2. Giovanni Domenico Cassini, Wikipedia (with extensive references).
3. Jean Dominique (Giovanni Domenico) Cassini, High Altitude Observatory, NCAR.
4. R. Taton, Cassini, Gian Domenico (Jean-Dominique), Dictionary of Scientific Biography, vol. 3, Scribner, 1971.
5. J. L. Heilbron, The Sun in the Church: Cathedrals as Solar Observatories, Harvard University Press, 1999.

Astronomy

1. Basilica of San Petronio (the meridian line), Wikipedia.
2. Rømer’s determination of the speed of light, Wikipedia.
3. L. Bobis, J. Lequeux, Cassini, Rømer and the velocity of light, Journal of Astronomical History and Heritage 11 (2008).
4. Cassini’s laws, Wikipedia.
5. Cassini oval, Wikipedia.
6. Solar parallax, Encyclopaedia Britannica.
7. C. D. Murray, S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999 (resonances, Cassini states).
8. J. Meeus, Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998 (librations, parallax).