Nicolaus Copernicus

How to use this course

Nicolaus Copernicus put the Sun near the centre of the planetary system and set the Earth moving. That sentence is familiar. Less familiar is why he did it and what it bought him. His models were no more accurate than Ptolemy’s and used about as many circles. What the moving Earth gave him was coherence: one parameter, the size of the Earth’s orbit, fixed the order and the relative distances of every planet. This course explains that geometry, the physical objections he had to answer, and the life of a man who did his astronomy in the spare hours of a busy career in church administration.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Copernicus wrote in the language of Euclid and Ptolemy, with chords and proportions. The course translates freely and says when it does. The companion course on Tycho Brahe continues the story with the observations that tested Copernicus, and the course on Johannes Kepler with the ellipse that replaced his circles.

Timeline

Toruń and Kraków, 1473–1496

Nicolaus Copernicus was born on 19 February 1473 in Toruń, a Hanseatic trading town on the Vistula that had passed to the Polish crown in 1466. His father, Niklas Koppernigk, had come from Kraków and traded in copper; his mother, Barbara Watzenrode, came from a wealthy Toruń merchant family. He was the youngest of four children. The family spoke German, the town lived from trade, and its politics were those of Royal Prussia, a province of the Polish kingdom with its own estates and privileges.

His father died around 1483. The children’s guardian became their mother’s brother, Lucas Watzenrode, a canon who in 1489 became bishop of Warmia (Ermland). Watzenrode was an able and forceful prince-bishop, and he planned church careers for his nephews. Nicolaus attended school in Toruń and, according to MacTutor, the cathedral school at Włocławek from 1488.

Kraków

In the autumn of 1491 Nicolaus and his brother Andreas enrolled at the University of Kraków, then one of the strongest centres of mathematical astronomy in Europe. The arts course included Euclid, the spherical astronomy of Sacrobosco, and the computation of positions from the Alfonsine Tables. Copernicus later owned a Latin Euclid printed at Venice in 1482, the 1492 edition of the Alfonsine Tables and Regiomontanus’s tables of directions. He stayed about four years and left without a degree, which was common for students who meant to study law abroad.

Italy, 1496–1503

In the autumn of 1496 Copernicus began canon law at Bologna, the training his uncle wanted for a future member of the cathedral chapter. At Bologna he also learned Greek and worked closely with Domenico Maria Novara, the professor of astronomy, as an assistant and observer. On 9 March 1497 he watched the Moon pass in front of the bright star Aldebaran. He later used the observation in De revolutionibus. In the same year he succeeded to a canonry in the Warmia chapter at Frombork (Frauenburg), which gave him an income for life.

In the jubilee year 1500 he was in Rome, where he reportedly lectured on mathematics and astronomy, and he observed the lunar eclipse of 5–6 November. In 1501 he returned briefly to Warmia, was formally installed in the chapter, and obtained leave to study medicine at Padua, since the chapter needed a physician. He studied there from 1501 to 1503. On 31 May 1503 he took his doctorate in canon law at Ferrara.

Italy gave him three things: Greek, which opened Ptolemy and the ancient reports of a moving Earth; practice in observation; and medicine, which he practised for the rest of his life for his uncle, his fellow canons and the poor of Warmia.

Lidzbark and the Commentariolus, 1503–1514

Back in Warmia, Copernicus served his uncle at the episcopal castle of Lidzbark (Heilsberg) as secretary and physician, accompanying him to diets and councils. In 1509 he published his only literary work, a Latin translation of the Greek letters of the Byzantine writer Theophylactus Simocatta. He moved to Frombork around 1510, and after his uncle’s death in 1512 he took up the duties of a canon there. He never took holy orders. When a later bishop pressed him in 1531 to become a priest, he declined.

The little commentary

Some time before May 1514 he wrote a short manuscript, known as the Commentariolus, and gave copies to a few readers. It was not printed until 1878. In it he said the old models offended him because they did not respect uniform circular motion, and he set out seven postulates:

1. the heavenly spheres do not all have a single centre;
2. the centre of the Earth is not the centre of the universe, only of gravity and of the Moon’s orbit;
3. all the spheres surround the Sun, which is near the centre of the universe;
4. the Earth–Sun distance is negligible compared with the distance of the fixed stars;
5. the daily motion of the heavens comes from the Earth’s rotation;
6. the Sun’s yearly motion comes from the Earth’s revolution, and the Earth has more than one motion;
7. the apparent stations and retrogradations of the planets come from the Earth’s motion.

The tract then sketched models for each body, with an ordering of planets by period, and promised a larger work with the mathematics. That work took nearly thirty more years. Lessons 04–07 explain what the postulates mean geometrically, and why the fourth and seventh carry most of the weight.

The Ptolemaic inheritance

For the general reader. Ancient astronomers described each planet with circles turning at constant rates. Claudius Ptolemy, writing in Alexandria around AD 150, perfected a toolkit of three devices: the epicycle, the eccentric and the equant. Copernicus kept the first two and rejected the third. To see what he changed, we must first see what he inherited.

Three devices

Work in the plane of the orbit and write positions as complex numbers. Uniform circular motion of radius \(r\) and rate \(\omega\) is \(re^{i(\omega t+\phi)}\). The ancient devices are sums of such terms.

• Eccentric. A circle of radius \(R\) whose centre is displaced by a fixed vector \(c\) from the observer: \(z(t)=c+Re^{i\omega t}\).
• Epicycle. A small circle carried on a large one (the deferent): \(z(t)=Re^{i\omega_1 t}+re^{i\omega_2 t}\).
• Equant. An eccentric circle with centre \(C\), on which the planet moves so that its direction seen from a different point \(Q\) turns uniformly. The planet’s speed along its own circle is then not constant.

The proof is the commutativity of vector addition. The content is physical: the same path can be generated in several ways, so geometry alone cannot tell which circle is “real”. Copernicus used this freedom constantly.

The equant and the objection

For an outer planet Ptolemy used an epicycle whose centre moves on an eccentric deferent, with centre \(C\) at distance \(e\) from the Earth \(O\), and uniform angular motion as seen from the equant \(Q\), at distance \(2e\) on the same line. Placing \(C\) midway between \(O\) and \(Q\) is called bisecting the eccentricity. It fitted the data well. Lesson 06 of the Kepler course shows why: the equant reproduces Kepler’s area law to first order in the eccentricity.

Philosophically, though, the equant broke the rule that celestial motions are uniform and circular about their own centres. A sphere turning uniformly about one point cannot turn uniformly about another. Astronomers of the Maragha school in thirteenth-century Persia, and Ibn al-Shatir in fourteenth-century Damascus, had built equant-free models with extra epicycles. In the Commentariolus Copernicus gives the equant as his starting complaint. Lesson 07 shows how he removed it.

Retrograde motion and the two periods

For the general reader. Seen from the Earth, Mars drifts eastward among the stars, stops, backs up for about two months, stops again and resumes. Ptolemy explained the loop with an epicycle. Copernicus explained it as an effect of perspective: a faster Earth on an inside track overtakes a slower Mars. The same idea turns the timing of the loops into the true periods of the planets.

Relative motion

Let the Earth move on a circle of radius \(a_E=1\) with angular velocity \(n_E\) and a planet on a coplanar circle of radius \(a\) with angular velocity \(n\). The planet seen from the Earth is

Its direction \(\arg w\) is the planet’s longitude against the stars. For an outer planet (\(a>1\), \(n<n_E\)) this is exactly a deferent of radius \(a\) carrying an epicycle of radius 1 that turns at the Earth’s rate, with the epicycle radius always parallel to the Sun–Earth line. That last fact is why the Sun appears in every Ptolemaic model of an outer planet, a coincidence that Ptolemy could state but not explain.

Derivation. At opposition put \(t=0\), so \(w=a-1\) is real and positive. Differentiating, \(\dot w=i(an-n_E)\), so \(\dot\lambda=\operatorname{Im}(\dot w/w)=(an-n_E)/(a-1)\). The orbital speed of the planet \(an\) is less than the Earth’s \(n_E\), so \(\dot\lambda<0\). Copernicus did not have the third law, but he did assume that the more distant planets move more slowly, which is all the argument needs. For inner planets the loop is centred on inferior conjunction.

Synodic and sidereal periods

Observers measure the synodic period \(S\), the time between successive oppositions (or conjunctions with the Sun). The sidereal period \(P\), the time to go round once relative to the stars, is what the heliocentric model needs.

Proof. The angle between the Earth and the planet, seen from the Sun, changes at the rate \(|n_E-n|=2\pi|1/T-1/P|\). A synodic period is the time for this angle to change by \(2\pi\).

With this rule Copernicus could rank all the planets by sidereal period: Mercury, Venus, Earth, Mars, Jupiter, Saturn. In the geocentric system the order of Venus and Mercury relative to the Sun had been a matter of choice. In the heliocentric system period increases with distance without exception.

Measuring the system

For the general reader. Ptolemy could predict where a planet would appear but not how far away it was. Each planet had its own model, scaled as one liked. Copernicus’s great claim, in his dedication to Pope Paul III, was that with the Earth moving the parts were no longer like limbs taken from different bodies. The orbits fit together, and their relative sizes can be measured.

Proof. The geocentric vector is \(w=ae^{int}-e^{in_Et}\). For (a), scale by \(R/a\): the deferent term \(Re^{int}\) carries an epicycle \(-(R/a)e^{in_Et}\), parallel to the Earth–Sun line. For (b), regroup: the deferent term is \(-e^{in_Et}\), which points from Earth to Sun, and the epicycle is \(ae^{int}\). Scaling by \(R\) gives \(r/R=a\). Direction is unchanged by scaling, so only the ratio is observable from Earth.

Ptolemy’s models therefore contained the heliocentric distances all along, each in its own units. Copernicus’s step was to give every model the same unit, the Earth’s orbit. The Almagest’s epicycle radii, on deferents of 60 parts, give the following:

Copernicus derived his values from his own models, re-fitted to old and new observations. The agreement with Ptolemy’s ratios shows that both astronomers were reading the same geometry out of the same sky.

Two direct measurements

Inner planets. The greatest angular distance \(\varepsilon\) of Venus or Mercury from the Sun occurs when the line of sight is tangent to the planet’s orbit. The triangle Sun–planet–Earth then has a right angle at the planet, so \(a=\sin\varepsilon\).

Outer planets. Time the interval \(t\) from opposition to the next quadrature, when the planet is \(90^\circ\) from the Sun. In that time the Earth gains on the planet the angle \(\theta=2\pi t(1/T-1/P)\), seen from the Sun. At quadrature the right angle is at the Earth, so \(a=\sec\theta\).

Abolishing the equant

For the general reader. Copernicus’s second goal, often forgotten, was to make every motion uniform about its own centre. He replaced Ptolemy’s equant with small extra circles, epicyclets, that reproduce its effect. The same device had been found two centuries earlier by Ibn al-Shatir in Damascus, and how Copernicus came to it is still debated.

The equant to first order

Take the deferent radius as 1, the observer (for Copernicus, the centre of the Earth’s orbit, the mean Sun) at 0, the centre at \(e\) and the equant at \(2e\), all on the real axis. Write the planet as \(z=e+e^{i\theta}\). The equant condition is \(\arg(z-2e)=M\), where the mean anomaly \(M\) grows uniformly in time.

Since \(z-2e=e^{i\theta}(1-ee^{-i\theta})\), we get \(M=\theta+\arg(1-ee^{-i\theta})=\theta+e\sin\theta+O(e^2)\), hence \(\theta=M-e\sin M+O(e^2)\). Substituting,

using \(-i\sin M\,e^{iM}=\tfrac12(1-e^{2iM})\).

This is the model of De revolutionibus Book V for the outer planets. At apogee (\(M=0\)) the planet sits on the inner side of the epicyclet, at distance \(1+e\) from the mean Sun; at perigee it sits on the outer side, at distance \(1-e\). In the Commentariolus Copernicus used an equivalent form: a concentric deferent, a first epicycle of radius \(\tfrac32e\) whose radius keeps a fixed direction, and a second of radius \(\tfrac12e\). By the theorem of Lesson 04, these are the same motion. The ratio 1 : 3 between the two small circles is the fingerprint of the equant they replace.

The Tusi couple

Proof. Let the centre of the small circle turn at angle \(\theta\): it is at \(re^{i\theta}\). Rolling without slipping inside a circle of twice the radius makes the small circle turn at angle \(-\theta\) relative to fixed directions. A point starting at \(2r\) is then at \(re^{i\theta}+re^{-i\theta}=2r\cos\theta\), on the real axis.

Al-Tusi introduced the device in his commentary on the Almagest to produce oscillation from uniform rotations. Copernicus used equivalent constructions in his models of Mercury and of the slow, irregular change he believed affected the precession of the equinoxes and the obliquity of the ecliptic, a supposed oscillation called trepidation. His Mercury and lunar models also match those of Ibn al-Shatir. Some historians argue for a lost channel of transmission from the Islamic world; others for independent discovery. No source has been identified.

The physics of a moving Earth

For the general reader. The strongest case against a moving Earth was physical. If the ground turns eastward at hundreds of metres per second, why do stones fall straight down and clouds not race to the west? Ptolemy had raised these objections and Aristotle’s physics gave them force. Copernicus answered them with arguments that were weak by later standards but pointed the right way, and modern mechanics shows exactly how small the effects are.

Objections and replies

Ptolemy argued in the Almagest that a rotating Earth would leave behind anything not attached to it, that bodies thrown upward would land to the west, and that a rotation so fast would tear the Earth apart. In Book I of De revolutionibus Copernicus replied in three ways. Rotation is the natural motion of a sphere, so it does not cause violence, and natural motions do not tear bodies apart. If speed were dangerous, the immense sphere of stars turning daily in the geocentric system would be in far greater danger. And the air near the Earth, with the bodies in it, shares the Earth’s motion. The third reply is the germ of the relativity principle that Galileo developed.

The Earth’s orbital motion is nearly uniform over short times, so it is undetectable by local mechanics, as Copernicus’s argument implicitly requires. The rotation is detectable in principle through the Coriolis and centrifugal terms, but they are small: \(\omega=7.29\times10^{-5}\,\mathrm{s^{-1}}\).

The eastward deflection

Drop a body from rest at height \(h\) at latitude \(\varphi\). To first order in \(\omega\), the Coriolis acceleration is \(-2\boldsymbol\omega\times\mathbf v\) with \(\mathbf v\approx-gt\,\hat{\mathbf z}\). Its eastward component is \(2\omega gt\cos\varphi\). Integrating twice,

The body lands slightly east, not west as Ptolemy’s argument predicts. Its starting point, being farther from the axis, moves eastward faster than the ground.

Three motions, precession and parallax

For the general reader. Copernicus gave the Earth three motions: a daily spin, a yearly orbit, and a slow turning of its axis that keeps it pointing almost at the same stars. The third explains the seasons and the slow drift of the equinoxes. A moving Earth also predicts that nearby stars should appear to shift during the year. None did, and Copernicus had to place the stars at an enormous distance.

The third motion

Copernicus imagined the Earth carried by a sphere, as a planet was carried in older astronomy. A rigid sphere would turn the axis once a year along with the orbital radius. To keep the axis pointing the same way, he added an annual conical motion of the axis in the opposite sense. In modern terms, he described the axis in a frame rotating with the Sun–Earth line.

Angular velocities about a common axis add. Copernicus made the conical motion slightly faster than the annual one, so precession appears as the small difference. We know now that the axis is fixed by inertia, apart from the torque of the Sun and Moon on the equatorial bulge, which produces precession of about \(50.3''\) per year, a cycle of roughly 25,800 years. Copernicus also believed, wrongly, that the rate of precession varied and modelled the variation with the oscillating devices of Lesson 07.

Annual parallax

Derivation. The apparent displacement is \(-\mathbf r_\oplus/d\), the Earth’s position vector projected onto the plane of the sky. The orbit is a circle of radius \(a\) in the ecliptic plane; projected onto a plane inclined at \(90^\circ-\beta\) to it, the circle becomes an ellipse with semi-axes \(a\) and \(a\sin\beta\). Divide by \(d\).

Copernicus’s fourth postulate is the answer to the missing parallax: the stars are so far away that the Earth’s orbit is a point by comparison. The first stellar parallax, about \(0.3''\) for 61 Cygni, was measured by Friedrich Bessel in 1838. Lesson 07 of the Brahe course takes up Tycho’s argument that such distances made the stars absurdly large.

The trigonometry of De revolutionibus

For the general reader. To turn circles into numbers, an astronomer needs tables of chords or sines and rules for solving triangles on the plane and on the sphere. Copernicus wrote his own treatment in Book I, with a table of half-chords, in effect sines. Rheticus thought it useful enough to print separately in 1542, the year before the whole book.

Chords and half-chords

In a circle of radius \(R\), the chord subtending a central angle \(\alpha\) is \(\operatorname{crd}\alpha=2R\sin(\alpha/2)\). Greek astronomy tabulated chords; Indian and Islamic astronomers, and after them Regiomontanus, tabulated half-chords of double arcs, which are sines. Copernicus followed the second practice.

Take \(AC\) a diameter of a circle of radius \(\tfrac12\), with \(\angle BAC=\alpha\) and \(\angle CAD=\beta\) on opposite sides of \(AC\). Then \(AB=\cos\alpha\), \(BC=\sin\alpha\), \(AD=\cos\beta\), \(CD=\sin\beta\), and \(BD=\sin(\alpha+\beta)\) by the extended law of sines. Ptolemy’s theorem becomes \(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\). With half-angle formulas, this generates a complete table from a few known values.

Spherical triangles

Proof sketch. Let the vertices be unit vectors \(\mathbf u,\mathbf v,\mathbf w\). The triple product \(\det(\mathbf u,\mathbf v,\mathbf w)\) equals \(\sin b\,\sin c\,\sin A\): the height of \(\mathbf v\) above the plane of \(\mathbf u,\mathbf w\) is \(\sin c\sin A\), and the area of the parallelogram on \(\mathbf u,\mathbf w\) is \(\sin b\). By symmetry it also equals \(\sin a\sin c\sin B\) and \(\sin a\sin b\sin C\). Divide by \(\sin a\sin b\sin c\).

The basic application: the Sun at ecliptic longitude \(\lambda\), the equinox and the foot of the Sun’s hour circle on the equator form a right spherical triangle with angle \(\varepsilon\) (the obliquity) at the equinox. The law of sines with the right angle gives the Sun’s declination:

Olsztyn, money and war, 1512–1539

At Frombork Copernicus lived in a canon’s house within the fortified cathedral hill and kept his instruments nearby: a quadrant, a triquetrum (a parallactic instrument copied from Ptolemy) and an armillary sphere. More than sixty of his observations are recorded, most made at Frombork between 1512 and 1543.

Around 1514 he was consulted on calendar reform in connection with the Fifth Lateran Council. He replied in writing that the lengths of the year and month were not yet known well enough. In the dedication of 1543 he recalled the question as a spur to his work.

Administrator and defender

From 1516 to 1521 he administered the chapter’s estates around Olsztyn (Allenstein) and Pieniężno (Mehlsack), living in Olsztyn castle, settling peasants on abandoned farms and keeping careful records. The war between Poland and the Teutonic Order reached Warmia in 1519–1521. Copernicus organized the defence of Olsztyn castle against the Order’s troops and afterwards took part in negotiations over the war’s damages.

Money

The Prussian lands suffered from debased coinage issued by several mints. Copernicus wrote on the problem in 1517, presented proposals to the Prussian diet in 1522, and completed the treatise Monetae cudendae ratio in 1526. He argued that the value of money falls as its quantity grows, an early form of the quantity theory of money, and that debased coin drives good coin out of circulation, the principle later named after Thomas Gresham. His proposals for a single, sound Prussian currency were not adopted.

Through these years he revised the planetary theory. Observations of Mars and Saturn in 1512–1515 and of the Sun in 1515 led him to change the models of the Commentariolus, and the manuscript of De revolutionibus grew slowly, with many corrections, in the 1520s and 1530s. Reports of his ideas spread: in 1533 Johann Albrecht Widmanstetter explained them to Pope Clement VII, and in 1536 Cardinal Nikolaus von Schönberg wrote asking for a copy of his work.

Rheticus, the book and its fate

In May 1539 Georg Joachim Rheticus, a 25-year-old professor of mathematics at Wittenberg, arrived at Frombork to learn the new astronomy. He stayed about two years. In 1540 he published at Gdańsk the Narratio prima, a first account of the theory in the form of a letter to his teacher Johann Schöner. Its good reception, and the encouragement of Copernicus’s friend Tiedemann Giese, bishop of Chełmno, persuaded Copernicus to release the full work. Rheticus carried a copy of the finished manuscript to Nuremberg, where the printer Johannes Petreius began work in 1542. Rheticus also printed the trigonometry separately that year.

When Rheticus left Nuremberg for a post in Leipzig, the Lutheran theologian Andreas Osiander took over supervision of the printing. Osiander added an unsigned note to the reader saying that the hypotheses need not be true or even probable, only convenient for calculation. Readers took it for the author’s own view. Kepler named Osiander as its author in print in the Astronomia nova of 1609. Copernicus’s own preface, a dedication to Pope Paul III, follows it and argues the opposite: the moving Earth makes the system a single, coherent whole.

Late in 1542 Copernicus suffered a stroke and was left partly paralysed. He died at Frombork on 24 May 1543. A later account by Giese says he received a printed copy of the book only on his last day. He was buried in Frombork Cathedral. In 2005 a team led by Jerzy Gąssowski located remains beneath the cathedral floor; their identification was announced in 2008, and he was reburied there on 22 May 2010.

Reception

• Astronomers: Owen Gingerich’s census of surviving copies shows that the book was read closely by nearly every astronomer of the period. Most annotated the planetary models and ignored the cosmology. Erasmus Reinhold at Wittenberg computed the Prutenic Tables (1551) from Copernicus’s models, and admired above all the elimination of the equant.
• The moving Earth: few accepted it for decades. Tycho Brahe built an equivalent system with the Earth at rest (see the Brahe course). Michael Maestlin taught it to Kepler, whose Mysterium cosmographicum (1596) defended it on physical as well as geometrical grounds. Galileo’s telescopic discoveries from 1610 made it a public question.
• The Church: on 5 March 1616 the Congregation of the Index suspended the book until corrected. The corrections, issued in 1620, altered a handful of passages presenting the Earth’s motion as fact. Uncorrected editions stayed on the Index until 1758.
• Science: Kepler’s ellipses replaced Copernicus’s circles, and Newton’s dynamics explained why the Sun, not the Earth, is the centre of the motion. But the scale of the solar system, fixed by the Earth’s orbit, and the order of the planets have remained exactly as Copernicus set them out.

Connections

People linked to Nicolaus Copernicus across this collection. A name without a link has no page here yet. See Nicolaus Copernicus in the network of all 337 lives →

Teachers and advisors

• Domenico Maria Novara 1496-1500

Students

• Georg Joachim Rheticus 1539-1541

Influenced by

• Aristarchus of Samos
• Claudius Ptolemy
• Hipparchus
• Jean Buridan
• Nasir al-Din al-Tusi
• Regiomontanus

Influenced

• Galileo Galilei
• Johannes Kepler
• Otto von Guericke
• Simon Stevin
• Tycho Brahe
• Erasmus Reinhold
• Michael Maestlin

Colleagues and collaborators

• Andreas Osiander 1542-1543

Friends

• Tiedemann Giese

Family

• Lucas Watzenrode

Also mentioned in this life

• Apollonius of Perga
• Aristotle
• Friedrich Wilhelm Bessel
• Gaspard-Gustave Coriolis
• Euclid
• Isaac Newton

Fields

• astronomy 50 lives
• mathematics 15 lives
• medicine 8 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Nicolaus Copernicus, MacTutor History of Mathematics.
2. Nicolaus Copernicus, Wikipedia (with extensive references).
3. Georg Joachim Rheticus, Wikipedia.
4. E. Rosen, Copernicus and his Successors, Hambledon Press, 1995.
5. O. Gingerich, The Book Nobody Read, Walker, 2004.
6. D. Sobel, A More Perfect Heaven, Walker, 2011.

Astronomy and mathematics

1. N. Copernicus, De revolutionibus orbium coelestium, Nuremberg, 1543; trans. E. Rosen, On the Revolutions, Johns Hopkins, 1978. See the Wikipedia article on the book.
2. Commentariolus, Wikipedia; E. Rosen, Three Copernican Treatises, Dover, 1959.
3. Copernican heliocentrism, Wikipedia.
4. Tusi couple, Wikipedia.
5. N. M. Swerdlow and O. Neugebauer, Mathematical Astronomy in Copernicus’s De Revolutionibus, Springer, 1984.
6. J. Evans, The History and Practice of Ancient Astronomy, Oxford University Press, 1998.
7. T. S. Kuhn, The Copernican Revolution, Harvard University Press, 1957.
8. G. Saliba, Islamic Science and the Making of the European Renaissance, MIT Press, 2007.