Posted on Jul 19, 2020

# July 19, 1595: Kepler’s Insight Leading to Mysterium Cosmographicum

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On July 19, 1595, Astronomer Johannes Kepler had an epiphany and developed his theory of the geometrical basis of the universe while teaching in Graz. From the article:

"July 19, 1595: Kepler’s Insight Leading to Mysterium Cosmographicum

Today, we think of physics and astronomy as being inexorably linked, but this was not the case in the 16th century, when the former was deemed natural philosophy, while the latter was linked to mathematics and liberal arts. One scientist who helped break down that barrier was Johannes Kepler.

Born in December 1571, just west of modern-day Stuttgart, Kepler was the grandson of a former lord mayor, the youngest of four children. By the time of his birth, the family’s financial circumstances were much reduced, and his father eked out a living as a mercenary, abandoning the family when young Johannes was just five years old. His mother was a healer and herbalist — a dangerous profession in that superstitious age.

A bout with smallpox crippled his hands and hampered his vision, but Kepler showed a gift for mathematics early on. He fell in love with astronomy at age six, when his mother took him out to watch a comet streak across the night sky. He experienced his first lunar eclipse a few years later. He studied philosophy and theology at the University of Tübingen, where he gained a reputation as a skilled astrologer — still considered a legitimate branch of astronomy at the time — and embraced the then-relatively-new Copernican heliocentric system for the motion of the planets, eventually becoming a math and astronomy teacher in Graz.

On July 19, 1595, while lecturing on the periodic conjunction of Jupiter and Saturn, Kepler had an insight: There might be a geometric underpinning to the universe. He worked out a scheme in which the five Platonic solids — octahedron, icosahedron, dodecahedron, tetrahedron, and cube — could be encased within spheres and then nested within each other. This produced six layers, which in Kepler’s view corresponded to the six planets known at the time (Mercury, Venus, Earth, Mars, Jupiter and Saturn). He even worked out a preliminary formula connecting the size of each planet’s orb to how long each took to complete one orbit around the sun, although he later discarded it in favor of something more precise.

This became the basis for one of his earliest treatises, Mysterium Cosmographicum, among the first defenses of the nascent Copernican system to appear in print. His labors over its publication nearly ended his engagement to a rich young widow named Barbara Müller. Her father initially opposed the match, despite Kepler’s noble birth, because the astronomer was so poor, but eventually relented. Kepler married Müller in April, 1597. The marriage was not a particularly happy one, and Müller died of spotted fever in 1611. (His second marriage, to Susanna Reuttinger in 1613, proved more successful.)

Mysterium Cosmographicum was an unusual scientific treatise, given the inclusion of a detailed chapter attempting to reconcile the Bible with the geocentrism of Copernicus. Those passages were removed before the book was published late in 1596. This work cemented his reputation. He sent copies to several noted colleagues, including Danish astronomer Tycho Brahe, who initially offered a rather harsh critique of the German scientist’s model.

Kepler was keen to address those criticisms and make further progress on his ideas. Since Brahe had amassed far more accurate observational data from his private observatory than that readily available to Kepler, he visited Brahe in Prague early in 1600, studying Brahe’s data on Mars to test the theory laid out in the Mysterium Cosmographicum. The following year, he moved his entire family to Brahe’s observatory — in part because he had been banished from Graz for refusing to convert to Catholicism.

When Brahe died soon after, Kepler succeeded him as imperial mathematician; his Lutheran faith was tolerated in the Prague court. His duties mostly consisted of providing horoscopes to the emperor. Kepler despised much of astrology, dismissing it as “evil-smelling dung,” but — a creature of his era — he believed a scientific approach to the subject could be useful. He later published a treatise attempting to find middle ground between the overselling of astrology and what he thought was a kneejerk rejection of it by many scientists.

Brahe’s observational data proved invaluable to Kepler’s research, which expanded to include the laws of optics. In 1604, he published Astronomiae Pars Optica, which helped lay the foundation of modern optics. Those insights proved useful when he studied the newly invented telescopes used by Galileo and designed an improved Keplerian model using two convex lenses, rather than one convex and one concave. Kepler also witnessed a supernova in October 1604, which became the basis for De Stella Nova, published two years later.

Kepler also tried his hand at more fanciful writing, penning an allegory called Somnium (The Dream) in 1611 — arguably the earliest work of science fiction, since it centered on a trip to the moon and speculated about what astronomy would be like if conducted on another planet.

Many years later, Somnium would be used as evidence in his mother’s 14-month imprisonment and trial for witchcraft; it described a woman who summons a demon for help in mixing potions. (He revised the work after her acquittal to make the allegorical aspects crystal clear for the too-literal minded.)

Of course, Kepler is best known for his Laws of Planetary Motion. The planets move in elliptical orbits around the sun (Law of Ellipses); if you draw an imaginary line from the center of the sun to the center of a given planet, that line will sweep out equal areas in equal amounts of time (the Law of Equal Areas); and the ratio of the squares of any two planetary periods is equal to the ratio of the cubes of their average distances from the sun (the Law of Harmonies). The first two laws formed the basis for his treatise Astronomia Nova, and all three appeared in his most influential work, the Epitome Astronomiae Copernicanae.

Kepler died on November 15, 1630, and while the location of his grave has been lost over the intervening centuries, the epitaph he penned for himself survived:

Mensus eram coelos, nunc terrae metior umbras:

Mens coelestis erat, corporis umbra jacet.

I measured the skies, now the shadows I measure

Skybound was the mind, earthbound the body rests.

©1995 - 2020, AMERICAN PHYSICAL SOCIETY

APS encourages the redistribution of the materials included in this newspaper provided that attribution to the source is noted and the materials are not truncated or changed.

Editor: David Voss

Staff Science Writer: Michael Lucibella

Art Director and Special Publications Manager: Kerry G. Johnson

Publication Designer and Production: Nancy Bennett-Karasik."

"July 19, 1595: Kepler’s Insight Leading to Mysterium Cosmographicum

Today, we think of physics and astronomy as being inexorably linked, but this was not the case in the 16th century, when the former was deemed natural philosophy, while the latter was linked to mathematics and liberal arts. One scientist who helped break down that barrier was Johannes Kepler.

Born in December 1571, just west of modern-day Stuttgart, Kepler was the grandson of a former lord mayor, the youngest of four children. By the time of his birth, the family’s financial circumstances were much reduced, and his father eked out a living as a mercenary, abandoning the family when young Johannes was just five years old. His mother was a healer and herbalist — a dangerous profession in that superstitious age.

A bout with smallpox crippled his hands and hampered his vision, but Kepler showed a gift for mathematics early on. He fell in love with astronomy at age six, when his mother took him out to watch a comet streak across the night sky. He experienced his first lunar eclipse a few years later. He studied philosophy and theology at the University of Tübingen, where he gained a reputation as a skilled astrologer — still considered a legitimate branch of astronomy at the time — and embraced the then-relatively-new Copernican heliocentric system for the motion of the planets, eventually becoming a math and astronomy teacher in Graz.

On July 19, 1595, while lecturing on the periodic conjunction of Jupiter and Saturn, Kepler had an insight: There might be a geometric underpinning to the universe. He worked out a scheme in which the five Platonic solids — octahedron, icosahedron, dodecahedron, tetrahedron, and cube — could be encased within spheres and then nested within each other. This produced six layers, which in Kepler’s view corresponded to the six planets known at the time (Mercury, Venus, Earth, Mars, Jupiter and Saturn). He even worked out a preliminary formula connecting the size of each planet’s orb to how long each took to complete one orbit around the sun, although he later discarded it in favor of something more precise.

This became the basis for one of his earliest treatises, Mysterium Cosmographicum, among the first defenses of the nascent Copernican system to appear in print. His labors over its publication nearly ended his engagement to a rich young widow named Barbara Müller. Her father initially opposed the match, despite Kepler’s noble birth, because the astronomer was so poor, but eventually relented. Kepler married Müller in April, 1597. The marriage was not a particularly happy one, and Müller died of spotted fever in 1611. (His second marriage, to Susanna Reuttinger in 1613, proved more successful.)

Mysterium Cosmographicum was an unusual scientific treatise, given the inclusion of a detailed chapter attempting to reconcile the Bible with the geocentrism of Copernicus. Those passages were removed before the book was published late in 1596. This work cemented his reputation. He sent copies to several noted colleagues, including Danish astronomer Tycho Brahe, who initially offered a rather harsh critique of the German scientist’s model.

Kepler was keen to address those criticisms and make further progress on his ideas. Since Brahe had amassed far more accurate observational data from his private observatory than that readily available to Kepler, he visited Brahe in Prague early in 1600, studying Brahe’s data on Mars to test the theory laid out in the Mysterium Cosmographicum. The following year, he moved his entire family to Brahe’s observatory — in part because he had been banished from Graz for refusing to convert to Catholicism.

When Brahe died soon after, Kepler succeeded him as imperial mathematician; his Lutheran faith was tolerated in the Prague court. His duties mostly consisted of providing horoscopes to the emperor. Kepler despised much of astrology, dismissing it as “evil-smelling dung,” but — a creature of his era — he believed a scientific approach to the subject could be useful. He later published a treatise attempting to find middle ground between the overselling of astrology and what he thought was a kneejerk rejection of it by many scientists.

Brahe’s observational data proved invaluable to Kepler’s research, which expanded to include the laws of optics. In 1604, he published Astronomiae Pars Optica, which helped lay the foundation of modern optics. Those insights proved useful when he studied the newly invented telescopes used by Galileo and designed an improved Keplerian model using two convex lenses, rather than one convex and one concave. Kepler also witnessed a supernova in October 1604, which became the basis for De Stella Nova, published two years later.

Kepler also tried his hand at more fanciful writing, penning an allegory called Somnium (The Dream) in 1611 — arguably the earliest work of science fiction, since it centered on a trip to the moon and speculated about what astronomy would be like if conducted on another planet.

Many years later, Somnium would be used as evidence in his mother’s 14-month imprisonment and trial for witchcraft; it described a woman who summons a demon for help in mixing potions. (He revised the work after her acquittal to make the allegorical aspects crystal clear for the too-literal minded.)

Of course, Kepler is best known for his Laws of Planetary Motion. The planets move in elliptical orbits around the sun (Law of Ellipses); if you draw an imaginary line from the center of the sun to the center of a given planet, that line will sweep out equal areas in equal amounts of time (the Law of Equal Areas); and the ratio of the squares of any two planetary periods is equal to the ratio of the cubes of their average distances from the sun (the Law of Harmonies). The first two laws formed the basis for his treatise Astronomia Nova, and all three appeared in his most influential work, the Epitome Astronomiae Copernicanae.

Kepler died on November 15, 1630, and while the location of his grave has been lost over the intervening centuries, the epitaph he penned for himself survived:

Mensus eram coelos, nunc terrae metior umbras:

Mens coelestis erat, corporis umbra jacet.

I measured the skies, now the shadows I measure

Skybound was the mind, earthbound the body rests.

©1995 - 2020, AMERICAN PHYSICAL SOCIETY

APS encourages the redistribution of the materials included in this newspaper provided that attribution to the source is noted and the materials are not truncated or changed.

Editor: David Voss

Staff Science Writer: Michael Lucibella

Art Director and Special Publications Manager: Kerry G. Johnson

Publication Designer and Production: Nancy Bennett-Karasik."

#### July 19, 1595: Kepler’s Insight Leading to Mysterium Cosmographicum

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Episode 21: Kepler's Three Laws - The Mechanical Universe

Episode 21. Kepler's Three Laws: The discovery of elliptical orbits helps describe the motion of heavenly bodies with unprecedented accuracy. “The Mechanical...

Thank you my friend SGT (Join to see) for making us aware that on July 19, 1595, Astronomer Johannes Kepler "had an epiphany and developed his theory of the geometrical basis of the universe while teaching in Graz."

Episode 21: Kepler's Three Laws - The Mechanical Universe

"Episode 21. Kepler's Three Laws: The discovery of elliptical orbits helps describe the motion of heavenly bodies with unprecedented accuracy.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.

The series was originally produced as a broadcast telecourse in 1985 by Caltech and Intelecom, Inc. with program funding from the Annenberg/CPB Project.

The online version of the series is sponsored by the Information Science and Technology initiative at Caltech"

https://www.youtube.com/watch?v=uJvOGp1wzTI

Images

1. Portrait of Johannes Kepler. Engraving by Frederick Mackenzie, 1787(88)-1854, published by Charles Knight, 1834.

2. Table 3 in Mysterium Cosmographicum, with Kepler's model illustrating the intercalation of the five regular solids between the imaginary spheres of the planets (cf. KGW 1, pp. 26–27).

"The Cosmic Bowl. This engraving, which Kepler made at the age of 25, depicts his view of the universe based on the application of regular polyhedra. He inserts an octahedron between the orbits of Mercury and Venus, an icosahedron between Venus and Earth, a dodecahedron between Earth and Mars, a tetrahedron between Mars and Jupiter, and a cube between Jupiter and Saturn. Extensive calculation led Kepler to the view that “symmetrical shapes can be placed so precisely inside each other to separate the relevant orbits that if a layman were to ask how the heavens are supported to prevent them from falling, the answer would be simple.” Not content with this explanation of celestial harmony, Kepler had the idea of developing it into an artefact and devised a “cosmic bowl” which would actually dispense beverages representing the symmetry of the polyhedra and the harmony of the planets. In February 1596 he went to visit his patron, Duke Frederick of Wurtemberg, to ask him to authorise the construction of a model of the universe in the form of a bowl. The planets would be made of precious stones – a diamond for Saturn, a sapphire for Jupiter, a pearl for the moon, etc. – and the beverages contained in each planetary sphere would be fed through invisible pipes to seven taps around the rim of the bowl. Mercury would provide brandy, Venus mead, Mars a strong vermouth, and so on. The project was too costly and was never realised. Johannes Kepler, Prodromus Dissertationum Cosmographicarum, Continens Mysterium Cosmographicum de Admirabiii Proportione Orbium Coeslestium, Tubingen, G. Gruppenbach, 1596."

3. Kepler's first law of ellipse and second law of areas (modern representation with greatly exaggerated eccentricity).

4. The Eccentric Orbit of Mars. In the first edition of The New Astronomy Kepler suggested a new way of describing the solar system based on the observations of Tycho Brahe.

"n the first edition of The New Astronomy Kepler suggested a new way of describing the solar system based on the observations of Tycho Brahe. A fervent believer in the heliocentric view of the cosmos, Kepler paid particular attention to the orbit of Mars, which he attempted to explain in terms of the various systems of the world available to him. He calculated its sidereal period (the time taken for the planet to complete a single orbit of the sun and return to its exact starting point) from the numerous tables of measurements drawn up by Tycho Brahe: the period was 687 days. He first drew a circle to represent the earth’s orbit (which it virtually is) and marked on it the position of the earth – the observation point — at two distinct dates. Then he drew a straight line across the earth’s ecliptic plane to indicate a sighting of Mars. Detailed analysis of Tycho Brahe’s records enabled him to find five pairs of sightings of the planet at intervals of exactly 687 days. Some points where the pairs of lines met all lay inside the circumference of a circle. So he searched for an explanation which would satisfy his quest for perfection: an ellipse has a geometric centre and two foci. His conclusion was unequivocal: “A perfect ellipse is the only possible shape.”

Johannes Kepler, Astronomia Nova, 1609.

Paris, BNF, Rare Books Archive, gV. 454, p. 149"

Background from {[https://plato.stanford.edu/entries/kepler/]}

"Stanford Encyclopedia of Philosophy

Johannes Kepler

First published Mon May 2, 2011; substantive revision Thu May 21, 2015

Johannes Kepler (1571–1630) is one of the most significant representatives of the so-called Scientific Revolution of the 16th and 17th centuries. Although he received only the basic training of a “magister” and was professionally oriented towards theology at the beginning of his career, he rapidly became known for his mathematical skills and theoretical creativity. As a convinced Copernican, Kepler was able to defend the new system on different fronts: against the old astronomers who still sustained the system of Ptolemy, against the Aristotelian natural philosophers, against the followers of the new “mixed system” of Tycho Brahe—whom Kepler succeeded as Imperial Mathematician in Prague—and even against the standard Copernican position according to which the new system was to be considered merely as a computational device and not necessarily a physical reality. Kepler's complete corpus can be hardly summarized as a “system” of ideas like scholastic philosophy or the new Cartesian systems which arose in the second half of the 17th century. Nevertheless, it is possible to identify two main tendencies, one linked to Platonism and giving priority to the role of geometry in the structure of the world, the other connected with the Aristotelian tradition and accentuating the role of experience and causality in epistemology. While he attained immortal fame in astronomy because of his three planetary laws, Kepler also made fundamental contributions in the fields of optics and mathematics. To the little-known facts regarding Kepler’s indefatigable scientific activity belong his efforts to develop different technical devices, for instance, a water pump, which he tried to patent and apply in different practical contexts (for the documents, see KGW 21.2.2., pp. 509–57 and 667–691). To his contemporaries he was also a famous mathematician and astrologer; for his own part, he wanted to be considered a philosopher who investigated the innermost structure of the cosmos scientifically.

• 1. Life and Works

• 2. Philosophy, theology, cosmology

• 3. The five regular solids

• 4. Epistemology and philosophy of sciences

o 4.1 Realism

o 4.2 Causality

o 4.3 Philosophy of mathematics

o 4.4 Empiricism

• 5. Copernicanism reformed and the three planetary laws

• 6. Optics and metaphysics of light

• 7. Harmony and Soul

• Bibliography

o A. Primary Sources

o B. Secondary Literature

• Academic Tools

• Other Internet Resources

• Related Entries

________________________________________

1. Life and Works

Johannes Kepler was born on December 27, 1571 in Weil der Stadt, a little town near Stuttgart in Württemberg in southwestern Germany. Unlike his father Heinrich, who was a soldier and mercenary, his mother Katharina was able to foster Kepler's intellectual interests. He was educated in Swabia; firstly, at the schools Leonberg (1576), Adelberg (1584) and Maulbronn (1586); later, thanks to support for a place in the famous Tübinger Stift, at the University of Tübingen. Here, Kepler became Magister Artium (1591) before he began his studies in the Theological Faculty. At Tübingen, where he received a solid education in languages and in science, he met Michael Maestlin, who introduced him to the new world system of Copernicus (see Mysterium Cosmographicum, trans. Duncan, p. 63, and KGW 20.1, VI, pp. 144–180).

Before concluding his theology studies at Tübingen, in March/April 1594 Kepler accepted an offer to teach mathematics as the successor to Georg Stadius at the Protestant school in Graz (in Styria, Austria). During this period (1594–1600), he composed many official calendars and prognostications and published his first significant work, the Mysterium Cosmographicum (= MC), which catapulted him to fame overnight. On April 27, 1597 Kepler married his first wife, Barbara Müller von Mühleck. As a consequence of the anti-Protestant atmosphere in Graz and thanks also to the positive impact of his MC on the scientific community, he abandoned Graz and moved to Prague in 1600, to work under the supervision of the great Danish astronomer Tycho Brahe (1546–1601). His first contact with Tycho was, however, extremely traumatic, particularly because of the Ursus affair (see below Section 4.1). After Tycho's unexpected death on October 1601 Kepler succeeded him as Imperial Mathematician. During his time in Prague, Kepler was particularly productive. He completed his most important optical works, Astronomiae pars optica (=APO) and Dioptrice (=D), published several treatises on astrology (De fundamentis astrologiae certioribus, Antwort auf Roeslini Diskurs; Tertius interveniens), discussed Galileo's telescopic discoveries (Dissertatio cum nuncio sidereo), and composed his most significant astronomical work, the Astronomia nova (=AN), which contains his first two laws of planetary motion.

On August 3, 1611 Kepler's wife, Barbara Müller, died. In 1612 he moved to Linz, in Upper Austria, and became a professor at the Landschaftsschule. There, he served as Mathematician of the Upper Austrian Estates from 1612 to 1628. In 1613, he married Susanne Reuttinger, with whom he had six children. In 1615, he completed the mathematical works Stereometria doliorum and Messekunst Archimedis. At the end of 1617 Kepler successfully defended his mother, who had been accused of witchcraft. In 1619 he published his principal philosophical work, the Harmonice mundi (=HM), and wrote, partially at the same time, the Epitome astronomiae copernicanae (=EAC). In 1624 Kepler continued his investigations on mathematics, publishing his work on logarithms (Chilias logarithmorum…).

In pursuit of an accurate printer for the Tabulae Rudolphinae, he moved to Ulm near the end of 1626 and remained there until the end of 1627. In July 1628 he went to Sagan to enter the service of Albrecht von Wallenstein (1583–1634). He died on November 15, 1630 at Regensburg, where he was to present his financial claims before the imperial authorities (for Kepler's life, Caspar's biography (1993) is still the best work. KGW 19 contains biographically relevant documents).

2. Philosophy, theology, cosmology

There is probably no such thing as “Kepler's philosophy” in any pure form. Nevertheless, many attempts to deal with the “philosophy of Kepler” have been made, all of which are very valuable in their own way. Some studies have concentrated on a particular text (see, for instance, Jardine 1988, for the Defense of Tycho against Ursus), or have followed some particular ideas of Kepler over a longer period of his life and scientific career (see, for instance, Martens 2000, on Kepler's theory of the archetypes). Others have tried to determine from a philosophical point of view his place in the development of the astronomical revolution from the 15th to the 17th centuries (Koyré 1957 and 1961) or in the more general context of the scientific movement of the 17th century (Hall, 1963 and especially Burtt, 1924). Still others have discussed a long list of philosophical principles operating in Kepler's scientific world, and have claimed to have found, by means of such an analysis, compelling evidence for the interaction between science, philosophy, and religion (Kozhamthadam, 1994). If, in the particular case of Kepler, philosophy is immediately related to astronomy, mathematics and, finally, “cosmology” (a notion which arises much later), the core of these speculations is to be sought in the spectrum of problems with which he dealt in his Mysterium Cosmographicum and Harmonice Mundi (on this topic, Field 1988 is one of the most representative works on Kepler). In addition, because of the particular circumstances of his life and his fascinating personality and genius, the literature on Kepler is extremely wide-ranging, covering a spectrum from literary pieces like Max Brod's Tycho Brahes Weg zu Gott (1915)—though still not free from mistakes concerning Kepler—to general introductions in the genre of historical novels, and even fictional stories and charlatanry on astrology or, running for some years now, portraying him as the assassin of Tycho Brahe. However, according to recent reports, it is still a matter of controversy whether Tycho was assassinated at all (see the report at phys.org).

Kepler mastered, like the best scientists, the most complicated technical issues, especially in astronomy, but he always emphasized his philosophical, even theological, approach to the questions he dealt with: God manifests himself not only in the words of the Scriptures but also in the wonderful arrangement of the universe and in its conformity with the human intellect. Thus, astronomy represents for Kepler, if done philosophically, the best path to God (see Hübner 1975; Methuen 1998 and 2009; Jardine 2009). As Kepler at the core of his greatest astronomical work confesses (AN, Part II, chap. 7, KGW 6, p. 108, Engl. trans., p. 183), at the beginning of his career he “was able to taste the sweetness of philosophy … with no special interest whatsoever in astronomy.” And, even in his later work, after having calculated many ephemerides and different astronomical data, Kepler writes in a letter of February 17, 1619 to V. Bianchi: “I also ask you, my friends, that you do not condemn me to the treadmill of mathematical calculations; allow me time for philosophical speculation, my only delight!” (KGW 17, let. N° 827, p. 327, lin. 249– 51).

Especially where Kepler deals with the geometrical structure of the cosmos, he always returns to his Platonic and Neoplatonic framework of thought. Thus, the polyhedral hypothesis (see Section 3 below) he postulated for the first time in his MC represents a kind of “formal cause” constituting the foundational structure of the universe. In addition, an “efficient cause,” which realizes this structure in the corporeal world, is also needed. This is, of course, God the Creator, who accomplished His work according to the model of the five regular polyhedra. Kepler reinterprets the traditional statements about the Creation as an image of the Creator giving to the ancient ideas a more systematic and a quantitative character. Even the doctrine of the Trinity could be geometrically represented, taking the center for the Father, the spherical surface for the Son, and the intermediate space, which is mathematically expressed in the regularity of the relationship between the point and the surface, for the Holy Spirit. In Kepler's model, we have to be able to reduce all appearances to straightness and curvature as providing the foundation for the geometrical structure of the world's creation. The very first category, through which God produced a fundamental similitude of the created World to Himself, is that of quantity (see MC, chapter 2, KGW pp. 23–26). Furthermore, quantity was also introduced into the human soul for the specific purpose that this fundamental symmetry could be apprehended and known scientifically.

This kind of speculation also belongs to the basic principles of Kepler's philosophical optics. In Chapter 1 of APO (“On the Nature of Light”), Kepler gives a new account of this “Trinitarian Cosmogony.” As he admits in a letter to Thomas Harriot (1560–1621), his approach here is more theological than optical (KGW 15, let. 394, p. 348, lin. 18). Similar to his speculations in MC, Kepler explains again the symmetry between God and the Creation, but now he goes slightly beyond the limits of a theologico-geometrical reflection. Firstly, he seems to assume that the bodies of the world were provided in the Creation with some powers which enable them to exceed their geometrical limits and to act on other bodies (magnetic power is a good example of this). Secondly, the principle of symmetry introduced into matter constitutes “the most excellent thing in the whole corporeal world, the matrix of the animate faculties, and the chain linking the corporeal and spiritual world” (APO, Engl. trans., p. 19). Thirdly, as expressed by Kepler in a wonderful, long Latin sentence, with multiple subordinations, this principle “has passed over into the same laws (in leges easdem) by which the world was to be furnished” (ibid., p. 20; the original passage is in KGW 2, p. 19: the marginal note in the edition is “lucis encomium”). Finally, these reflections are concluded with a remark, in which—as with Copernicus, Marsilio Ficino, and others—the central position of the Sun is legitimated because of its function in spreading light and, indirectly, life. Similar speculations are still present in EAC (KGW 7, pp. 47–48 and 267). It is also worth noting that these speculations are of vital importance to the special way in which Kepler conceived of astrology (see, for instance, De fundamentis astrologiae certioribus with Engl. trans. and commentary in Field 1984).

3. The five regular solids

Philosophical, geometrical and even theological speculations related to the five regular polyhedra, the cube or hexahedron, the tetrahedron, the octahedron, the icosahedron and the dodecahedron, were known at least from the time of the ancient Pythagoreans. Since Plato's Timaeus, these five geometrical solids played a leading role, and for the later tradition they became known as the “five Platonic solids”.

Plato establishes at the physical and chemical level a correspondence between them and the five elements—earth, water, air, fire and ether—and tries to provide this correspondence with geometrical grounds. A further source of historically decisive importance is the fact that the five regular polyhedra are treated in Euclid's Elements of Geometry, a work that for Kepler, especially in the Platonic approach of Proclus, has a central position. At the very beginning of HM Kepler complains about the fact that the modern philosophical and mathematical school of Peter Ramus (1515–1572) had not been able to understand the architectonic structure of the Elements, which are crowned with the treatment of the five regular polyhedra. In addition, a revival of Platonic philosophy was taking place in Kepler's time and inspired not only philosophers and mathematicians but also architects, artists and illustrators (see Field 1997).

Amidst this general interest in the regular polyhedra during the Renaissance, Kepler was specifically concerned with their application in the resolution of a cosmological problem, namely the reality of the Copernican system (see Section 5 below). To achieve this goal, he introduced his polyhedral hypothesis already in MC, where he looked for an “a priori” foundation of the Copernican system (see Aiton 1977, Di Liscia 2009). The background for such an approach seems to be that the “a posteriori” way, which according to Kepler was taken over by Copernicus himself, cannot lead to a necessary affirmation of the reality of the new world system, but only to a probable, and hence to an “instrumental”, representation of it as a computational device. This is the “Osiander” or “Wittenberg Interpretation” of Copernicus which Kepler directly attacked not only in his MC but also later in his AN (see Westman 1972 and 1975). In MC, he claimed to have found an answer to the following three main questions: 1) the number of the planets; 2) the size of the orbits, i.e., the distances; 3) the velocities of the planets in their orbits. By referring to the polyhedral hypothesis (see Figure 1), Kepler found a definitive and simple answer to the first question. By intercalating the polyhedra between the spheres which carry the planets, one must inevitably finish with the sphere of Saturn surrounding the cube—there are no more polyhedra to be intercalated and, as remains the standard case for Renaissance and modern astronomy, there are no more planets to be carried by the spheres. It is absolutely decisive for the consistency of the argument that the necessity of the hypothesis is guaranteed by the fact that it already exists as a mathematical demonstration (by Euclid, Elements XIII, prop. 18, schol.), according to which there are only five regular (“Platonic” for the tradition) polyhedra. For the second and third questions the answer is, of course, not as evident as for the first one. However, Kepler was able to show that distances which are derived from the geometrical model of the five regular bodies fit much better with the Copernican system than with that of Ptolemy. The answer to the third question needs, in addition, the introduction of a notion of power that emanates from the Sun and extends to the outer limit of the universe (see Stephenson 1987, pp. 9–20).

In HM, Kepler continues his investigations of the polyhedra at a cosmological as well as a mathematical level. In the second book, dealing with “congruence” (which here does not mean, as it does today, “the same size and shape”, but the property of figures filling the surface together with other regular polygons – on the plane—or, to build closed geometric solids – hence, in space), Kepler made new mathematical discoveries working with tessellations. He discovered two new solids, the so-called “small and great stellated dodecahedrons”.

The treatment of the regular polyhedra constitutes one of the two principal pillars of HM, Book 5 (chapters 1–2), where the third law is formulated (see Caspar in KGW 6, Nachbericht, p. 497, and Engl. trans. p. xxxiii). Following his approach from MC and complementing it with occasional references to his EAC, Kepler makes use again of the Platonic solids to determine the number of the planets and their distances from the Sun. Meanwhile, he has learned that the application of his old polyhedral hypothesis has limits. As he tells us in a footnote in the second edition of his MC from 1621, he was earlier convinced of the possibility of explaining the eccentricities of the planetary orbits by values derived “a priori” from this hypothesis (MC, Engl. trans., p. 189). Now, with access to the observational data of Tycho, Kepler had to exclude this explanation and look for another. And this is one of the most significant achievements of his basic harmonies (which in turn are derived from the regular polygons), constituting the second great pillar of Book 5.

4. Epistemology and philosophy of sciences

Almost all of Kepler's scientific investigations reflect a philosophical background, and many of his philosophical questions find their final answer, even if they are of scientific interest, in the realm of theology. From a very modern point of view, one could highlight Kepler's epistemological thought in terms of four different items: realism; causality; his philosophy of mathematics; and his—own particular—empiricism.

4.1 Realism

Realism is a constant and integral part of Kepler's thought, and one which appears in sophisticated form from the outset. The reason for this is that his realism always runs parallel to his defense of the Copernican worldview, which appeared from his first public pronouncements and publications.

Many of Kepler's thoughts about epistemology can be found in his Defense of Tycho against Ursus or Contra Ursum (=CU), a work which emerged from a polemical framework, the plagiarism conflict between Nicolaus Raimarus Ursus (1551–1600) and Tycho Brahe: causality and physicalization of astronomical theories, the concept and status of astronomical hypotheses, the polemic “realism-instrumentalism”, his criticism of skepticism in general, the epistemological role of history, etc. It is one of the most significant works ever written on this subject and is sometimes compared with Bacon's Novum organum and Descartes' Discourse on Method (Jardine 1988, p. 5; for an excellent new edition and complete study of this work see Jardine / Segonds 2008).

The focus of the epistemological issues could be ranked mutatis mutandi with modern discussion surrounding the scientific status of astronomical theories (however, as Jardine has pointed out, it would be sounder to read Kepler's CU more as a work against skepticism than in the context of the modern realism/instrumentalism polemic). For Pierre Duhem (1861–1916), for instance, the position of Andreas Osiander, which was adopted by Ursus and which was, according to Duhem, naively criticized by Kepler in his MC, represents the modern approach known as “instrumentalism”. According to this epistemological position, held by Duhem himself, scientific theories are not to be closely linked to the concepts of truth and falsehood. Hypotheses and scientific laws are nothing more than “instruments” for describing and predicting phenomena (seldom for explaining them). The aim of physical theories is not to offer a causal explanation or to study the causes of phenomena, but simply to represent them. In the best-case scenario, theories are able to order and classify what is decisive for their predictive capacity (Duhem 1908, 1914).

Contrary to Tycho and Kepler, Ursus held a fictionalist position in astronomy. Yet in the very beginning of his work De hypothesibus, Ursus makes a clear declaration about the nature of astronomical theories, which is very similar to the approach suggested by Osiander in his forward to Copernicus' De revolutionibus: a hypothesis is a “fictitious supposition”, introduced just for the sake of “saving the motions of the heavenly bodies” and to “calculate them” (trans. Jardine 1988, p. 41)

Following his approach in MC and anticipating the opening pages of his later AN (see particularly AN, II.21: “Why, and to what extent, may a false hypothesis yield the truth?” Engl. trans., pp. 294–301), Kepler addresses the question of Copernicanism and its reception by thinkers such as Osiander, who emphasized that the truth of astronomical hypotheses cannot necessarily be deduced from the correct prediction of astronomical facts. According to this interpretation, Copernican hypotheses are not necessarily true even if they are able to save the phenomena, otherwise one would commit a fallacia affirmationis consequentis. However, according to Kepler, “this happens only by chance and not always, but only when the error in the one proposition meets another proposition, whether true or false, appropriate for eliciting the truth” (trans. Jardine, p. 140). To be noted is that, as Jardine (2005, p. 137) has pointed out, the modern scientific realist departs from a real independent world, while Kepler's notion of truth presupposes that neither nature nor the human mind are independent of God's mind (Jardine 2005, p. 137).

4.2 Causality

The reality of astronomical hypotheses—and hence the superiority of the Copernican world system—implied a physicalization of astronomical theories and, in turn, an accentuation of causality. Despite Kepler's criticism of Aristotle, this aspect can actually be considered the realization in the field of astronomy of the old Aristotelian ideal of knowledge: “knowledge” means to grasp the causes of the phenomena.

Thus, on the one hand, “causality” is a notion implying the most general idea of “actual scientific knowledge” which guides and stimulates each investigation. In this sense, Kepler already embarked in his MC on a causal investigation by asking for the cause of the number, the sizes and the “motions” (= the speeds) of the heavenly spheres (see Section 3 above).

On the other hand, “causality” implies in Kepler, according to the Aristotelian conception of physical science, the concrete “physical cause”, the efficient cause which produces a motion or is responsible for keeping the body in motion. Original to Kepler, however, and typical of his approach is the resoluteness with which he was convinced that the problem of equipollence of the astronomical hypotheses can be resolved and the consequent introduction of the concept of causality into astronomy – traditionally a mathematical science. This approach is already present in his MC, where he, for instance, relates for the first time the distances of the planets to a power which emerges from the Sun and decreases in proportion to the distance of each planet, up to the sphere of the fixed stars (see Stephenson 1987, pp. 9–10).

One of Kepler's decisive innovations in his MC is that he replaced the “mean Sun” of Copernicus with the real Sun, which was no longer merely a geometrical point but a body capable of physically influencing the surrounding planets. In addition, in notes to the 1621 edition of MC Kepler strongly criticizes the notion of “soul” (anima) as a dynamical factor in planetary motion and proposes to substitute “force” (vis) for it (see KGW 8, p. 113, Engl. trans. p. 203, note 3).

One of the most important philosophical aspects of Kepler's Astronomia Nova from 1609 (=AN) is its methodological approach and its causal foundation (see Mittelstrass 1972). Kepler was sufficiently conscious of the change of perspective he was introducing into astronomy. Hence, he decided to announce this in the full title of the work: Astronomia Nova, Aitiologetos, seu physica coelestis, tradita commentariis de motibus stellae Martis. Ex observationibus G. V. Tychonis Brahe: New Astronomy Based upon Causes or Celestial Physics Treated by Means of Commentaries on the Motions of the Star Mars from the observations of Tycho Brahe … (trans. Donahue). In the introduction to AN Kepler insists on his radical change of view: his work is about physics, not pure kinematical or geometrical astronomy. “Physics”, as in the traditional, Aristotelian understanding of the discipline, deals with the causes of phenomena, and for Kepler that constitutes his ultimate approach to deciding between rival hypotheses (AN, Engl. trans., p. 48; see Krafft 1991). On the other hand, since his celestial physics uses not only geometrical axioms but also other, non-mathematical axioms, the knowledge obtained often has a kernel of guesswork.

In the third part of AN, chapters 22–40, Kepler deals with the path of the Earth and intends to offer a physical account of the Copernican theory. By so doing he includes the idea that a certain notion of power should be made responsible for the regulation of the differences in velocities of the planets, which in turn have to be established in relation to the planets' distances. Now, the Copernican planetary theory departs from the general principle that the Earth moves regularly on an eccentric circle. For Kepler, on the contrary, the planets are moved irregularly, and the slower they are moved, the greater their distance is from the center of power, the Sun. Addressing the physical aspects of his new astronomy, he deals in chapters 32–40, perhaps the most idiosyncratic of the work, with his notion of motive power. Here, he combines different approaches and sources, sometimes producing—for the purpose of simplifying the whole geometrical construction of geometrical astronomy by introducing a power causing motions—a new confusion at the dynamical level. To begin with, it is not always absolutely clear what kind of power Kepler has in mind. He inclines, above all, to the idea of a magnetic power residing in the Sun, but he also mentions light and, at least indirectly, gravity (which he does not bring into operation in the central chapters of the AN but which is to a certain extent implied in his explanations using the model of the balance and which he surely accepts as true for the Sun-Moon system, as he explains in the general introduction). Secondly, it is not always clear what this power is and how it acts, especially when he is speaking merely analogically, “as if” (particularly in the case of light). Essentially, Kepler breaks down the motions of the planets into two components. On the one hand, the planets move around the Sun—at this state of the discussion—circularly. On the other hand, they exhibit a libration on the Sun-planet vector. The rotation of the Sun is responsible for the motion of the planets. Irradiating from the rotating Sun is a power which spreads at the ecliptic plain. This power diminishes with distance to the source of the power, that is, to the Sun. A decisive work for Kepler's development in his physical astronomy is William Gilbert's (1544–1603) De magnete (London, 1600), a work which also intends to offer a new physics for the new Copernican cosmology and which surely influenced Kepler's thoughts about this power. One of the main problems was, of course, how to apply the general principles of magnetism to planetary motion, first to explain the difference in velocity on a circular path, and later to give an account of the motion on an ellipse. Kepler conceives of a model with parallel magnetic fibers which links the Sun with the planets in such a way that the rotation of the Sun causes the motion of the planets around it. The fibers are born in the planets parallel and perpendicular to the lines of apsides by a kind of “animal power”. The planets themselves are polarized, that is, with one pole they are attracted to the Sun, with the other pole they are pushed away from it. This explains very well the direction of planetary motion: the planets all move in one direction because the Sun rotates in that direction. Nevertheless, a further problem still seems to remain unresolved: according to Kepler's explication, the planets should move around the Sun as fast as the Sun itself rotates, which is not the case. This phenomenon can be explained by referring to a property of matter, which for Kepler has an axiomatic character: the inclinatio ad quietem, that is, the tendency to rest (see especially AN, chap. 39; KGW 3, p. 256). As a consequence, the planets are moved around the Sun slower than they would be if the power of the Sun were at work alone.

Kepler's causal approach is above all present in his Epitome, a voluminous work which exercised a considerable influence on the later development of astronomy. In the second part of Book 4, he deals with the motion of the world's parts. Not the two first laws but rather the third law, which he had recently announced in his HM, is Kepler's starting point; for this law, rather than a calculational device for the path of one planet, represents a general cosmological statement, and thus it is more convenient for his approach here. At the same time, it should be pointed out that the third law is not necessarily the best point of departure for a dynamical, causal approach to motion, as Kepler intends here; for, in comparison with the previous causal approaches, the question of the location of the cause of power responsible for the production of motion remains relevant. The spheres, which in the traditional view transported the planets, had been abolished since the time of Tycho. Furthermore, Kepler is clearly against the “moving intelligences” of the Aristotelian tradition. The fact that the orbits are elliptical and not circular, shows that the motions are not caused by a spiritual power but rather by a natural one, which is internal to the composition of matter. The planets themselves are provided with “inertia”, a property, as Kepler understood it, that inhibits motion and represents an impediment to it. The motive power (vix motrix) comes indeed from the Sun, which sends its rays of light and power in all directions. These rays are captured by the planets. Kepler, however, tries to explain this behavior of the planets less through astrology and much more through magnetism (a physical phenomenon which was by no means clearly understood in his time). Firstly, the Sun rotates and, by so doing, sets in motion the planets around it. Secondly, since the planets are poles of magnets and the Sun itself acts with magnetic power, the planets are, at different parts of their orbits, either attracted or repelled; in this way the elliptical path is causally produced. Kepler partially gives up the mechanical approach by postulating a soul in the Sun which is responsible for its regular motion of rotation, a motion on which, finally, the entire system depends. In fact, the planets are also supposed by Kepler to rotate and are therefore provided with “a sort of soul” or some such principle which produces the rotation.

In addition to astronomy and cosmology Kepler expanded his causal approach to include the fields of optics (see Section 6 below) and harmonics (Section 7 below).

4.3 Philosophy of mathematics

Beyond his own original talent, it is clear that Kepler was trained in mathematics from his earliest studies at Tübingen. At least officially, his positions at Graz, Prague, Linz, Ulm and Sagan can be characterized as the typical professional occupations of a mathematician in the broadest sense, i.e., including astrology and astronomy, theoretical mechanics and pneumatics, metrology, and every topic that could in some way be related to mathematics. Besides the field of astronomy and optics, where mathematics is ordinarily applied in different ways, Kepler produced original contributions to the theory of logarithms and above all within his favorite field, geometry (especially with his stereometrical investigations). Thus, on account of his natural predilection and talent and the importance of mathematics, particularly of geometry, for his thought, it is not surprising to find many different passages in his works where he articulated his philosophy of mathematics. However, Kepler's principal exposition on this topic is to be found in his HM, a work in which the first two books are purely mathematical in content. As he himself declares, in HM he played the role “not of a geometer in philosophy but of a philosopher in this part of geometry” (KGW 6, p. 20, Engl. trans., p. 14).

While in philosophical questions related to mathematics, Proclus and Plato were Kepler's most important inspirational sources, he did not always see Plato and Aristotle as completely opposed, for the latter—in Kepler's interpretation—also accepted “a certain existence of the mathematical entities” (KGW 14, let. N° 226, p. 265; see Peters, p. 130). To a great extent Kepler understood his mathematical investigations of HM as a continuation of Euclid's Elements, especially of the analysis of irrationalities in Book 10. The central notion that he works out here is that of “constructability”. According to Kepler, each branch of knowledge must, in principle, be reducible to geometry if it is to be accepted as knowledge in the strong sense (although, in the case of the physics, this condition is, as the AN emphasizes, only a necessary and not a sufficient condition). Thus, the new principles he was elaborating over the years in astrology were geometrical ones. A similar case occurs with the basic notions of harmony, which, after Kepler, could be reduced to geometry. Of course, not every geometrical statement is equally relevant and equally fundamental. For Kepler, the geometrical entities, principles and propositions which are especially fundamental are those that can be constructed in the classical sense, i.e., using only ruler (without measurement units) and compass. On this are based further notions according to different degrees of “knowability” (scibilitas), which begins with the circle and its diameter. Once again, Kepler understood this within the framework of his cosmological and theological philosophy: geometry, and especially geometrically constructible entities, have a higher meaning than other kinds of knowledge because God has used them to delineate and to create this perfect harmonic world. From this point of view, it is clear that Kepler defends a Platonist conception of mathematics, that he cannot assume the Aristotelian theory of abstraction and that he is not able to accept algebra, at least in the way he understood it. So, for instance, there are figures that cannot be constructed “geometrically”, although they are often assumed as safe geometrical knowledge. The best example of this is perhaps the heptagon. This figure cannot be described outside of the circle, and in the circle its sides have, of course, a determinate magnitude, but this is not knowable. Kepler himself says that this is important because here he finds the explanation for why God did not use such figures to structure the world. Consequently, he devotes many pages to discussing the issue (KGW 6, Prop. 45, pp. 47–56, see also KGW 9, p. 147). Certainly for a geometer like Kepler, approximations constitute – as mathematical theory—a painful and precarious way to progress. The philosophical background for his rejection of algebra seems to be, at least partially, Aristotelian in some of its basic suppositions: geometrical quantities are continuous quantities which therefore cannot be treated with numbers that are, in the inverse, discrete quantities. But the difference from the Aristotelian ideal of science remains an important one: for Aristotle, a crossover between arithmetic and geometry is allowed only in the case of the “middle sciences”, while for Kepler all knowledge must be reduced to its geometrical foundations.

4.4 Empiricism

A general presentation of Kepler's philosophical attitude and principles is not complete without reference to his link to the world of experience. For, despite his mainly theoretical approach in the natural sciences, Kepler often emphasized the significance of experience and, in general, of empirical data. In his correspondence there are many remarks about the significance of observation and experience, as for instance in a letter to Herwart von Hohenburg from 1598 (KGW 13, let. N° 91, lines 150–152) or from 1603 to Fabricius (KGW 14, let. N° 262, p. 191, lines 129–130), to mention only two of his most important correspondents. Looking for empirical support for the Copernican system, Kepler compares different astronomical tables in his MC, and in AN he makes extensive use of Tycho's observational treasure trove. In MC (chapter 18) he quotes a long passage from Rheticus for the sake of rhetorical support when, as was the case here, the data of the tables he used did not fit perfectly with the calculated values from the polyhedral hypothesis. In this passage, the reader learns that the great Copernicus, whose world system Kepler defends in MC, said one day to Rheticus that it made no sense to insist on absolute agreement with the data, because these themselves were surely not perfect. After all, it is questionable whether Kepler, using for instance the Prutenic Tables (1551) of Erasmus Reinhold (1511–1553), had access to complete and correct empirical information to confirm the Copernican hypothesis in grand style, as he claimed (for an analysis of Reihold's tables and their influence see Gingerich 1993, pp. 205–255).

The situation changed completely when Kepler came into contact at Prague with Tycho's observations (which, as Kepler often reports, were seldom at his disposal). However, a change of attitude is evident in AN, where he used Tycho's observations without restriction (which is something he makes clear in the work's title). In part 2 (chap. 7–21), he presents the “vicarious hypothesis”, which in the end he refutes. This hypothesis represents the best result which can be reached within the limits of traditional astronomy. This works with circular orbits and with the supposition that the motion of a planet appears regular from a point on the lines of apsides. Against the traditional method, here, Kepler does not cut the eccentricity into equal parts but leaves the partition open. To check his hypothesis, he needs observations of Mars in opposition, where Mars, the Earth, and the Sun are at midnight on the same line. From Tycho, he “inherited” ten such observations between the years 1580 and 1600, and to them he added another two for 1602 and 1604. In chapters 17–21, Kepler carries out an observational and computational check of his vicarious hypothesis. On the one hand, he points out that this hypothesis is good enough, since the variations of the calculated positions from the observed positions fall within the limits of acceptability (2 minutes of arc). In fact, Kepler presents this hypothesis as the best hypothesis which can be proposed within the framework of a “traditional astronomy”, as opposed to his new astronomy, which he will offer in the following parts of the work. On the other hand, this hypothesis can be falsified if one takes the observations of the latitudes into consideration. Further calculations with these observations produce a difference of eight minutes, something that cannot be assumed because the observations of Tycho are reliable enough. Kepler's famous sentence runs: “these eight minutes alone will have led the way to the reformation of all of astronomy” (AN, KGW 3, p. 286; Engl. trans., p. 286). There seems to be agreement that Kepler's AN contains the first explicit consideration of the problem of observational error (for this question see Hon 1987 and Field 2005).

Kepler also gave an important place to experience in the field of optics. As a matter of fact, he began his research on optics because of a disagreement between theory and observation, and he made use of scientific instruments he had designed himself (see, for instance, KGW 21.1, p. 244). Recent research on the problem of the camera obscura and the “images in the air” shows, however, the limits of a traditional approach to Kepler's optics following the main current of the history of physics. Rather, his notion of experimentum needs to be contextualized within the social practices and epistemological commitments of his time (see Dupré 2008).

Finally, it should be mentioned that a similar significance is assigned to experience and empirical data in Kepler's harmonic-musical and astrological theories, two fields which are subordinated to his greater cosmological project of HM. For astrology, he uses meteorological data, which he recorded for many years, as confirmation material. This material shows that the Earth, as a whole living being, reacts to the aspects which occur regularly in the heavens. In his musical theory Kepler was a modern thinker, especially because of the role he gave to experience. As has been noted (Walker, 1978, p. 48), Kepler made acoustic experiments with a monochord long before he wrote his HM. In a letter to Herwart von Hohenburg (KGW 15, ep. 424, p. 450), he describes how he checked the sound of a string at different lengths, establishing in which cases the ear judges the sound to be pleasurable. Kepler does not accept that this limitation is founded on arithmetical speculations, even if this was already assumed by Plato, whom he often follows, and by the Pythagoreans. On the basis of his experiments, Kepler found that there are other divisions of the string that the ear perceives as consonant, i.e., thirds and sixths.

If cosmology is the main framework of Kepler's interest, there is no doubt that, as Field has pointed out, he “felt the need to seek observational support for his model of the Universe” (Field 1988, p. 28; see also Field 1982).

5. Copernicanism reformed and the three planetary laws

Today Kepler is remembered in the history of sciences above all for his three planetary laws, which he produced in very specific contexts and at different times. While it is questionable whether he would have understood these scientific statements as “laws”—and it is even arguable that he used this term with a different meaning than we do today—it seems to be clear that all three laws (as a linguistic convention, we may continue to use the term) suppose some fundamentals of Kepler's philosophy: (a) realism, (b) causality, (c) the geometric structure of the cosmos. Besides this, it should be remarked that the common denominator of all three laws is Kepler's defense of the Copernican worldview, a cosmological system which he was not able to defend without reforming it radically. It is noteworthy that already at the very beginning of his career Kepler vehemently defended the reality of the Copernican worldview in a way that he characterized, taking over the terminology from the standard Aristotelian epistemology, as “a priori” (see above Section 3 above and Di Liscia 2009).

The first two laws were published initially in AN (1609), although it is known that Kepler had arrived at these results much earlier. His first law establishes that the orbit of a planet is an ellipse with the Sun in one of the foci (see Figure 2). According to the second law, the radius vector from the Sun to a planet P sweeps out equal areas, for instance SP1P2 and SP3P4,in equal times. The planet P is therefore faster at perihelion, where it is closer to the Sun, and slower at aphelion, where it is farther from the Sun. In accordance with his dynamical approach, Kepler first found the second law and, then, as a further result because of the effect produced by the supposed force, the elliptical path of the planets (for the two first planetary laws see especially Aiton 1973, 1975a, Davis 1992a-e, and 1998a; Donahue 1994; Wilson 1968 and 1972).

Perhaps the most significant impact of Kepler's two laws can be found by considering their cosmological consequences. The first law abolishes the old axiom of the circular orbits of the planets, an axiom which was still valid not only for pre-Copernican astronomy and cosmology but also for Copernicus himself, and for Tycho and Galileo. The second law breaks with another axiom of traditional astronomy, according to which the motion of the planets is uniform in swiftness. The Ptolemaic tradition in astronomy was, of course, aware of this difficulty and applied a particularly effective device for saving the “appearance” of acceleration: the equant. Copernicus, for his own part, insisted on the necessity of the axiom of uniform circular motion. Ptolemy's equant was understood by Copernicus as a technical device based on the violation of this axiom. Kepler, on the contrary, affirms the reality of changes in the velocities of the planetary motions and provides a physical account for them. After struggling strenuously with established ideas which were located not only in the tradition before him but also in his own thinking, Kepler abandoned the circular path of planetary motion and in this way initiated a more empirical approach to cosmology (though see Brackenridge 1982).

Kepler published the third law, the so-called “harmonic law”, for the first time in his Harmonice mundi (1619), i.e., ten years later. In his Epitome, he provided a more systematic approach to all three laws, their grounds and implications (see Davis 2003; Stephenson 1987). In Book 5, chapter 3, as point 8 of 13 (KGW 6, p. 302; Engl. trans., pp. 411–12), Kepler expresses, almost accidentally, his fundamental relationship connecting elapsed times with distances, which in modern notation could be expressed as:

(T1/T2)2 = (a1/a2)3

with T1 and T2 representing the periodic times of two planets and a1 and a2 the length of their semi-major axes. A further formulation of this relationship, which is often found in the literature, is: a3/T2 = K, which expresses with K that the relationship between the third power of the distances and the square of the times is a constant (however, see Davis 2005, pp. 171–172; for the third planetary law see especially Stephenson 1987). As a consequence of the third law, the time a planet takes to travel around the Sun will significantly increase the farther away it is or the longer the radius of its orbit. Thus, for instance, Saturn's sidereal period is almost 30 years, while Mercury needs fewer than 88 days to go around the Sun. For the history of cosmology, it is important to make clear that the third law fulfils Kepler's search for a systematic representation and defense of the Copernican worldview, in which planets are not absolutely independent of each other but integrated in a harmonic world system.

6. Optics and metaphysics of light

Kepler contributed to the special field of optics with two seminal works, the Ad Vitellionem paralipomena (=APO) and the Dioptrice (=DI), the latter motivated in large part by the publication in 1610 of Galileo's Sidereal Messenger (Sidereus Nuncius). In his Conversation with the Sidereal Messenger (Dissertatio cum Nuncio Sidereo … a Galillaeo Galilaeo, KGW 4, pp. 281–311), he supported the factual information given by Galileo, indicating at the same time the necessity of giving an account of the causes of the observed phenomena. The background for his investigation into optics was undoubtedly the different particular questions of astronomical optics (see Straker 1971). In this context he concentrated his efforts on an explanation of the phenomena of eclipses, of the apparent size of the Moon and of atmospheric refraction. Kepler investigated the theory of the camera obscura very early and recorded its general principles (see commentary by M. Hammer in KGW 2, pp. 400–1 and Straker 1981). In addition, he worked intensively on the theory of the telescope and invented the refracting astronomical or ‘Keplerian’ telescope, which involved a considerable improvement over the Galilean telescope (see especially DI, Problem 86, KGW 4, pp. 387–88). Besides these impressive contributions, Kepler expanded his research program to embrace mathematics as well as anatomy, discussing for instance conic sections and explaining the process of vision (see Crombie 1991 and especially Lindberg 1976b).

In Chapter 1 of APO (“On the Nature of Light”), Kepler expounds 38 propositions concerning different properties of light: light flows in all directions from every point of a body's surface; it has no matter, weight, or resistance. Following—but also inverting—the Aristotelian argument for the temporality of motion, he affirms that the motion of light takes place not in time but in an instant (in momento). Light is propagated by straight lines (rays), which are not light itself but its motion. It is important to note that although light travels from one body to another, it is not a body but a two-dimensional entity which tends to expand to a curved surface. The two-dimensionality of light is probably the main reason why it is incorporeal. Motion in general plays a significant role in Kepler's philosophy of light. For Straker, the supposed link between optics and physics (especially in Prop. 20, where the mechanical analyses are introduced) “reveals the full extent of his commitment to a mechanical physics of light” (Straker 1971, p. 509).

Two questions are intensively discussed by modern specialists. Firstly, to what extent is the attribution of a mechanistic approach to Kepler justified? Secondly, how should one determine his place in the history of sciences, especially in the field of optics: do the main lines of thought in Kepler's optics indicate a continuity or rather a rupture with tradition? There are well–grounded arguments for different positions on both questions. For Crombie (1967, 1991) and Straker, Kepler develops a mechanical approach, which can be particularly appreciated in his explanation of vision using the model of the camera obscura. Besides this, Straker stresses that Kepler's basic mechanicism is also powerfully assisted by his conception of light as a non-active, passive entity. In addition, the concept of motion and the explanations using the model of the balance are indicative of a commitment to mechanicism (Straker 1970, pp. 502–3). On the contrary, Lindberg (1976a), who supports the “continuity side” of the dispute, has quite convincingly showed that, for Kepler, light has a constructive and active function in the universe, not only in optics but also in astrology, astronomy, and natural philosophy (for Kepler's criticism of the medieval tradition see also Chen-Morris / Unguru, 2001).

7. Harmony and Soul

From a philosophical point of view, Kepler considered the HM to be his main work and the one he most cherished. Containing his third planetary law, this work represents definitively a seminal contribution to the history of astronomy. But he did not reduce his long prepared project to an astronomical investigation—his first thoughts on the notion of “harmony” arose already in 1599, although he did not publish his work until 1619—but instead extensively discussed its mathematical foundations and its philosophical implications, including astrology, natural philosophy and psychology. Thus, Kepler's third planetary law appears in a context which goes far beyond astronomy and to a great extent takes up again the perspective of his youthful MC.

According to Kepler, it is necessary to distinguish “sensible” from “pure” harmony. The first is to be found among natural, sensible entities, like sounds in music or rays of light; both could be in proportion to one another and hence in harmony. He resolves this matter by combining three of the Aristotelian categories: quantity, relation and, finally, quality. Through the function of the category of relation Kepler passes over to the active function of the mind (or soul). It turns out that two things can be characterized as harmonic if they can be compared according to the category of quantity. But the fact that at least two things are needed shows that the property of “being harmonic” is not a property of an isolated thing. Furthermore, the relationship between the things cannot be found in the things themselves either; rather, it is produced by the mind: “in general every relation is nothing without mind apart from the things which it relates, because they do not have the relation which they are said to have unless the presence of some mind is assumed, to relate one to another” (KGW 6, p. 212; Engl. trans., p. 291). This process takes place through the comparison of different sensible things with an archetype (archetypus) present in the mind.

The next central question directly concerns gnoseology, for Kepler gives a psychological account of the path followed by sensible things into the mind. He resumes the scholastic species theory: immaterial species radiate from the sensible things and affect the sense organs by acting firstly on the “forecourts” and then on the internal functions. They arrive at the imagination and from there go over to the sensus communis, so that, according to the traditional teaching, the sensible information received is now able to be processed and used in statements. From here onwards, the sensible things are “preserved in the memory, brought forth by recollection, [and] distinguished by the higher faculty of the soul” (Caspar 1993, p. 269, cf. KGW 6, p. 214, lines 18–23; Engl. trans., p. 293). While harmony arises as an activity of the soul/mind consisting in relating quantitatively, Kepler adds, taking over the Aristotelian doctrine of categories, that harmony is a “qualitative relation” as well, involving the “quality of shape, being formed from the regular figures”, which provides the grounds for comparison (KGW 6, p. 216, lines 37–41; Engl. trans., p. 296). If this is how “things”, i.e., sensible entities, find their way into the soul in order to be compared, it by no means represents—as Kepler admitted—a sufficient explanation of how non-sensible things, i.e., mathematical entities, find their way into the mind. How do they come into the soul? Kepler accepted Aristotle's criticism of Pythagorean philosophy concerning numbers: both Kepler and Aristotle are convinced that numbers constitute ontologically a lower class within mathematical entities (for Kepler, they are derived from geometrical entities). Nevertheless, Aristotle's philosophy is insufficient to grasp the essence of mathematics. By aligning himself with Proclus, from whom he quotes a long passage of his Commentary on Euclides, Kepler defends Plato's theory of anamnêsis against Aristotle's doctrine of the tabula rasa. His discussion lies at the origin of the classical debate between empiricism and rationalism which was to dominate the philosophical scene for generations to come. A connection with idealism is, of course, apparent (see, for instance, Caspar 1993, Engl. trans., p. 269), and it is a fact that Kepler was positively received within German Idealism of the 19th century. Historically, however, it seems to be more accurate to link his position with the philosophical tradition of St. Augustine.

Besides psychology and gnoseology, the other main spectrum of questions Kepler deals with in Book 4 is his theory of “aspects”, i.e., astrology (HM, IV, Cap. 4–7), a further field of application of his psychology and further evidence of the role of geometry in his philosophy. The “aspects”, i.e., the angles between the planets, Moon and Sun, are all he wishes to save from the old astrology, which he harshly criticizes; for the aspects are or can be reduced to geometrical structures, the archetypes, which can be recognized by the soul. According to Kepler, there is no “mechanical influence” of the heavens (stars and constellations are not relevant in his astrology) which exerts a determining effect on the Earth and on human life. Rather, both the Earth and human beings, ultimately, like all other living entities, are provided with a soul in which the geometrical archetypes are present. By the formation of an aspect in the heavens, symmetry arises and stimulates the soul of the Earth or of human beings. “The Earth,” Kepler writes, responds to “what the aspects whistle” (KGW 11.2, p. 48; for his astrology see especially Field 1984 and Rabin 1997, Boner 2005 and 2006).

Bibliography

A. Primary Sources

Complete Editions

• Joannis Kepleri Astronomi Opera omnia, ed. Ch. Frisch, vols. 1–8, 2; Frankfurt a.M. and Erlangen: Heyder & Zimmer, 1858–1872.

• Johannes Kepler Gesammelte Werke, herausgegeben im Auftrag der Deutschen Forschungsgemeinschaft und der Bayerischen Akademie der Wissenschaften, unter der Leitung von Walther von Dyck und Max Caspar, Bd. 1–21.2.2; München: C.H. Beck'sche Verlagsbuchandlung, 1937–2009 (apparently finished) [ = KGW].

Selected English Translations of Individual Works

• Mysterium cosmographicum: Trans. A. M. Duncan, The secret of the universe Translation by A.M. Duncan / Introduction and commentary by E. J. Aiton, New York: Abaris, 1981.

• Apologia Tychonis contra Ursum: Trans. N. Jardine, The Birth of History and Philosophy and Science: Kepler's Defense of Tycho against Ursus with Essays on its Provenance and Significance, Cambridge: Cambridge University Press, 1984 (with corrections 1988).

• Ad Vitellionem paralipomena. Astronomiae pars optica: Trans. W. H. Donahue, Johannes Kepler. Optics: Paralipomena to Witelo & Optical Part of Astronomy, trans. William H. Donahue, New Mexico: Green Lion Press, 2000.

• Dissertatio cum Nuncio Sidereo and Narratio de Observatis Jovis Satellitibus: Trans. E. Rosen, Kepler's Conversation with Galileo's Sideral Messenger, New York and London: Johnson Reprint Corporation, 1965.

• De fundamentis astrologiae certioribus: = On Giving Astrology Sounder Foundations: Trans. J.V. Field, “A Lutheran Astrologer: Johannes Kepler”, Archive for History of Exact Sciences, 31/3 (1984): 189–272.

• Astronomia nova: Trans. W. H. Donahue, New Astronomy, Cambridge, New York: University Press, 1992; and Selections from Kepler's Astronomia Nova: A Science Classics Module for Humanities Studies, trans. W. H. Donahue, Santa Fe, NM: Green Lion Press, 2005.

• Harmonices mundi libri V: Trans. E. J. Aiton, A. M. Duncan, J.V. Field, The Harmony of the World. Philadelphia: American Philosophical Society (Memoirs of the American Philosophical Society), 1997.

• Epitome astronomiae copernicanae: Partial trans. C. G. Wallis, Epitome of Copernican Astronomy: IV and V, Chicago, London: Encyclopaedia Britannica (Great Books of the Western World, Volume 16], 1952, pp. 839–1004.

• Strena seu de nive sexangula: Trans. C. Hardie, The Six-Cornered Snowflake, Oxford: Clarendon Press, 1966.

• Somnium seu de astronomia lunari: Trans. E. Rosen, Kepler's Somnium: The Dream, or Posthumous Work on Lunar Astronomy, Madison: University of Wisconsin Press, 1967.

• Letters (selection): Trans. C. Baumgardt; introd. A. Einstein: Johannes Kepler: Life and Letters, New York: Philosophical Library, 1951.

B. Secondary Literature

• Aiton, E.J., 1973, “Infinitesimals and the Area Law”, in F. Krafft, K. Meyer, and B. Sticker (eds.), 1973: Internationales Kepler-Symposium Weil der Stadt 1971, Hildesheim: Gerstenberg, pp. 285–305.

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• –––, 2001, “Theological Foundations of Kepler's Astronomy”, Osiris, 16: 88–113.

• Bialas, V., 1990, “Keplers komplizierter Weg zur Wahrheit: von neuen Schwierigkeiten, die ‘Astronomia Nova’ zu lesen”, Berichte zur Wissensschaftsgeschichte, 13/3: 167–176.

• Beer, A., and P. Beer (eds.), 1975, Kepler Four Hundred Years, Oxford: Pergamon Press.

• Bennett, B, 1998, “Kepler's Response to the Mystery: A New Cosmographical Epistemology”, in J. Folta (ed.), Mysterium Cosmographicum 1596–1956, Prague, pp. 49–64.

• Boockmann, F., D.A. Di Liscia, and H. Kothmann (eds.), 2005: Miscellanea Kepleriana, [Algorismus 47], Augsburg: Erwin Rauner Verlag.

• Boner, P.J., 2005, “Soul-searching with Kepler: An analysis of Anima, in his astrology”, Journal for the History of Astronomy 36/1: 7–20.

• –––, 2006, “Kepler's Living Cosmology: Bridging the Celestial and Terrestrial Realms”, Centaurus, 48: 32–39.

• –––, 2013, Kepler’s Cosmological Synthesis: Astrology, Mechanism and the Soul, Leiden: Brill.

• Brackenridge, J.B., 1982, “Kepler, elliptical orbits, and celestial circularity: a study in the persistence of metaphysical commitment. I”, Annals of Science 39/2: 117–143 and II in Annals of Science, 39/3: 265–295.

• Buchdahl, G., 1972, “Methodological aspects of Kepler's theory of refraction”, Studies in History and Philosophy of Science, 3: 265–298.

• Burtt, E.A., 1924, The Metaphysical Foundations of Modern Physical Science, New York: Kegan Paul; 2nd rev. ed., 1954, Garden City, NY: Doubleday, reprinted 2003, New York: Dover Publications.

• Caspar, M., 1993, Johannes Kepler, New York: Dover Publications.

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• Crombie, J.A., 1967, “The Mechanistic Hypothesis, and the Study of Vision: Some Optical Ideas as a Background to the Invention of the Microscope”, in S. Bradbury and G.L.E. Turner (eds.), Historical Aspects of Microscopy, Cambridge: Heffer (for the Royal Microscopical Society), pp. 3–112.

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• –––, Cohen, B., 1985, The Birth of a New Physics, New York and London: W.W. Norton.

• Davis, A.E.L., 1975, “Systems of Conics in Kepler's Work”, Vistas in Astronomy, 18: 673–685.

• –––, 1992a, “Kepler's resolution of individual planetary motion”, Centaurus, 35/2: 97–102.

• –––, 1992b, “Kepler's ‘distance law’ – myth not reality”, Centaurus, 35/2: 103–120.

• –––, 1992c, “Grading the eggs (Kepler's sizing-procedure for the planetary orbit)”, Centaurus, 35/2: 121–142.

• –––, 1992d, “Kepler's road to Damascus”, Centaurus, 35/2: 143–164.

• –––, 1992e, “Kepler's physical framework for planetary motion”, Centaurus, 35/2: 165–191.

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• –––, 1983, “Cosmology in the work of Kepler and Galileo”, P. Galluzzi (ed.) Novità celesti e crisi del sapere, [Supplement volume to Annali del Istituto e Museo di Storia della Scienza di Firenze], Florence: Giunti Barbèra, pp. 207–215

• –––, 1984, “A Lutheran Astrologer: Johannes Kepler”, Archive for History of Exact Science, 31: 189–272.

• –––, 1988, Kepler's Geometrical Cosmology, London and Chicago: Athlone and University of Chicago Press.

• –––, 1997, “Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler”, Archive for History of Exact Sciences, 50/3–4: 241–289.

• –––, 1998, “Kepler's mathematization of cosmology”, Acta historiae rerum naturalium necnon technicarum, 2: 27–48.

• –––, 2005, “Tycho Brahe, Johannes Kepler and the concept of error”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.), pp. 143–155.

• –––, 2009, “Kepler's Harmony of the World”, in Richard L. Kremer and Jarosław Włodarczyk (eds.), Johannes Kepler: From Tübingen to Zagan (Proceedings of a conference held in Zielona Gora in June 2008), Studia Copernicana, 42: 11–28.

• Folta, J. (ed.), 1998, Mysterium cosmographicum 1596–1996, Prague [Acta Historiae rerum naturalium necnon technicarum, New Series, Vol. 2].

• Gingerich, Owen, 1993, The Eye of Heaven. Ptolemy, Copernicus, Kepler, New York: American Institute of Physics.

• Granada, M.A., 2008, “Kepler and Bruno on the Infinity of the Universe and of Solar Systems”, Journal for the History of Astronomy, 39: 1–27.

• Granada, M.A., and E. Mehl (eds.), 2009, Nouveau ciel, nouvelle terre / La révolution copernicienne en Allemagne 1530–1630, Paris: Belles Lettres.

• Hall, Rupert, 1963, From Galileo to Newton 1630–1720, New York: Collins.

• Holton, G., 1956, “Johannes Kepler's universe: its physics and metaphysics”, American Journal of Physics, 24: 340–351.

• Hübner, J., 1975, Die Theologie Johannes Keplers zwischen Orthodoxie und Naturwissenschaft, Tübingen: J.C.B. Mohr.

• Hon, G., 1987, “On Kepler's awareness of the problem of experimental error”, Annals of Science, 44/6, 545–591.

• Itokazu, A.G., 2009, “On the Equivalence of the Hypotheses in Part 1 of Johannes Kepler's New Astronomy”, Journal for the History of Astronomy, 50: 173–190.

• Jardine, N., 1988, The Birth of History and Philosophy and Science: Kepler's Defense of Tycho against Ursus with Essays on its Provenance and Significance, Cambridge: Cambridge University Press, 1984 (with corrections 1988).

• –––, 2005, “The Many Significances of Kepler's Contra Ursum”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.), pp. 129–142.

• –––, 2009, “Kepler, God, and the Virtues of Copernican Hypotheses”, in M. A. Granada and E. Mehl (eds.), pp. 269–277.

• Jardine, N., and A. Segonds, 1999, “Kepler as Reader and Translator of Aristotle”, in C. Blackwell and S. Kusukawa, eds., Philosophy in the Sixteenth and Seventeenth Centuries: Conversations with Aristotle, Aldershot: Ashgate, pp. 206–233.

• Jardine, N., and A. Segonds, 2001, “A Challenge to the Reader: Ramus on Astrologia, without Hypotheses”, in M. Feingold, J.S. Freedman, and W. Rother (eds.), The Influence of Petrus Ramus. Studies in Sixteenth and Seventeenth Century Philosophy and Sciences, Basel: Schwabe, pp. 248–266.

• Jardine, N., and A. Segonds, 2008, La Guerre des Astronomes: La querelle au sujet de l'origine du système géo-hèliocentrique à la fin du XVIe siècle (Science et humanisme, 9–10), Paris: Belles Lettres, 2 volumes.

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Episode 21: Kepler's Three Laws - The Mechanical Universe

"Episode 21. Kepler's Three Laws: The discovery of elliptical orbits helps describe the motion of heavenly bodies with unprecedented accuracy.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.

The series was originally produced as a broadcast telecourse in 1985 by Caltech and Intelecom, Inc. with program funding from the Annenberg/CPB Project.

The online version of the series is sponsored by the Information Science and Technology initiative at Caltech"

https://www.youtube.com/watch?v=uJvOGp1wzTI

Images

1. Portrait of Johannes Kepler. Engraving by Frederick Mackenzie, 1787(88)-1854, published by Charles Knight, 1834.

2. Table 3 in Mysterium Cosmographicum, with Kepler's model illustrating the intercalation of the five regular solids between the imaginary spheres of the planets (cf. KGW 1, pp. 26–27).

"The Cosmic Bowl. This engraving, which Kepler made at the age of 25, depicts his view of the universe based on the application of regular polyhedra. He inserts an octahedron between the orbits of Mercury and Venus, an icosahedron between Venus and Earth, a dodecahedron between Earth and Mars, a tetrahedron between Mars and Jupiter, and a cube between Jupiter and Saturn. Extensive calculation led Kepler to the view that “symmetrical shapes can be placed so precisely inside each other to separate the relevant orbits that if a layman were to ask how the heavens are supported to prevent them from falling, the answer would be simple.” Not content with this explanation of celestial harmony, Kepler had the idea of developing it into an artefact and devised a “cosmic bowl” which would actually dispense beverages representing the symmetry of the polyhedra and the harmony of the planets. In February 1596 he went to visit his patron, Duke Frederick of Wurtemberg, to ask him to authorise the construction of a model of the universe in the form of a bowl. The planets would be made of precious stones – a diamond for Saturn, a sapphire for Jupiter, a pearl for the moon, etc. – and the beverages contained in each planetary sphere would be fed through invisible pipes to seven taps around the rim of the bowl. Mercury would provide brandy, Venus mead, Mars a strong vermouth, and so on. The project was too costly and was never realised. Johannes Kepler, Prodromus Dissertationum Cosmographicarum, Continens Mysterium Cosmographicum de Admirabiii Proportione Orbium Coeslestium, Tubingen, G. Gruppenbach, 1596."

3. Kepler's first law of ellipse and second law of areas (modern representation with greatly exaggerated eccentricity).

4. The Eccentric Orbit of Mars. In the first edition of The New Astronomy Kepler suggested a new way of describing the solar system based on the observations of Tycho Brahe.

"n the first edition of The New Astronomy Kepler suggested a new way of describing the solar system based on the observations of Tycho Brahe. A fervent believer in the heliocentric view of the cosmos, Kepler paid particular attention to the orbit of Mars, which he attempted to explain in terms of the various systems of the world available to him. He calculated its sidereal period (the time taken for the planet to complete a single orbit of the sun and return to its exact starting point) from the numerous tables of measurements drawn up by Tycho Brahe: the period was 687 days. He first drew a circle to represent the earth’s orbit (which it virtually is) and marked on it the position of the earth – the observation point — at two distinct dates. Then he drew a straight line across the earth’s ecliptic plane to indicate a sighting of Mars. Detailed analysis of Tycho Brahe’s records enabled him to find five pairs of sightings of the planet at intervals of exactly 687 days. Some points where the pairs of lines met all lay inside the circumference of a circle. So he searched for an explanation which would satisfy his quest for perfection: an ellipse has a geometric centre and two foci. His conclusion was unequivocal: “A perfect ellipse is the only possible shape.”

Johannes Kepler, Astronomia Nova, 1609.

Paris, BNF, Rare Books Archive, gV. 454, p. 149"

Background from {[https://plato.stanford.edu/entries/kepler/]}

"Stanford Encyclopedia of Philosophy

Johannes Kepler

First published Mon May 2, 2011; substantive revision Thu May 21, 2015

Johannes Kepler (1571–1630) is one of the most significant representatives of the so-called Scientific Revolution of the 16th and 17th centuries. Although he received only the basic training of a “magister” and was professionally oriented towards theology at the beginning of his career, he rapidly became known for his mathematical skills and theoretical creativity. As a convinced Copernican, Kepler was able to defend the new system on different fronts: against the old astronomers who still sustained the system of Ptolemy, against the Aristotelian natural philosophers, against the followers of the new “mixed system” of Tycho Brahe—whom Kepler succeeded as Imperial Mathematician in Prague—and even against the standard Copernican position according to which the new system was to be considered merely as a computational device and not necessarily a physical reality. Kepler's complete corpus can be hardly summarized as a “system” of ideas like scholastic philosophy or the new Cartesian systems which arose in the second half of the 17th century. Nevertheless, it is possible to identify two main tendencies, one linked to Platonism and giving priority to the role of geometry in the structure of the world, the other connected with the Aristotelian tradition and accentuating the role of experience and causality in epistemology. While he attained immortal fame in astronomy because of his three planetary laws, Kepler also made fundamental contributions in the fields of optics and mathematics. To the little-known facts regarding Kepler’s indefatigable scientific activity belong his efforts to develop different technical devices, for instance, a water pump, which he tried to patent and apply in different practical contexts (for the documents, see KGW 21.2.2., pp. 509–57 and 667–691). To his contemporaries he was also a famous mathematician and astrologer; for his own part, he wanted to be considered a philosopher who investigated the innermost structure of the cosmos scientifically.

• 1. Life and Works

• 2. Philosophy, theology, cosmology

• 3. The five regular solids

• 4. Epistemology and philosophy of sciences

o 4.1 Realism

o 4.2 Causality

o 4.3 Philosophy of mathematics

o 4.4 Empiricism

• 5. Copernicanism reformed and the three planetary laws

• 6. Optics and metaphysics of light

• 7. Harmony and Soul

• Bibliography

o A. Primary Sources

o B. Secondary Literature

• Academic Tools

• Other Internet Resources

• Related Entries

________________________________________

1. Life and Works

Johannes Kepler was born on December 27, 1571 in Weil der Stadt, a little town near Stuttgart in Württemberg in southwestern Germany. Unlike his father Heinrich, who was a soldier and mercenary, his mother Katharina was able to foster Kepler's intellectual interests. He was educated in Swabia; firstly, at the schools Leonberg (1576), Adelberg (1584) and Maulbronn (1586); later, thanks to support for a place in the famous Tübinger Stift, at the University of Tübingen. Here, Kepler became Magister Artium (1591) before he began his studies in the Theological Faculty. At Tübingen, where he received a solid education in languages and in science, he met Michael Maestlin, who introduced him to the new world system of Copernicus (see Mysterium Cosmographicum, trans. Duncan, p. 63, and KGW 20.1, VI, pp. 144–180).

Before concluding his theology studies at Tübingen, in March/April 1594 Kepler accepted an offer to teach mathematics as the successor to Georg Stadius at the Protestant school in Graz (in Styria, Austria). During this period (1594–1600), he composed many official calendars and prognostications and published his first significant work, the Mysterium Cosmographicum (= MC), which catapulted him to fame overnight. On April 27, 1597 Kepler married his first wife, Barbara Müller von Mühleck. As a consequence of the anti-Protestant atmosphere in Graz and thanks also to the positive impact of his MC on the scientific community, he abandoned Graz and moved to Prague in 1600, to work under the supervision of the great Danish astronomer Tycho Brahe (1546–1601). His first contact with Tycho was, however, extremely traumatic, particularly because of the Ursus affair (see below Section 4.1). After Tycho's unexpected death on October 1601 Kepler succeeded him as Imperial Mathematician. During his time in Prague, Kepler was particularly productive. He completed his most important optical works, Astronomiae pars optica (=APO) and Dioptrice (=D), published several treatises on astrology (De fundamentis astrologiae certioribus, Antwort auf Roeslini Diskurs; Tertius interveniens), discussed Galileo's telescopic discoveries (Dissertatio cum nuncio sidereo), and composed his most significant astronomical work, the Astronomia nova (=AN), which contains his first two laws of planetary motion.

On August 3, 1611 Kepler's wife, Barbara Müller, died. In 1612 he moved to Linz, in Upper Austria, and became a professor at the Landschaftsschule. There, he served as Mathematician of the Upper Austrian Estates from 1612 to 1628. In 1613, he married Susanne Reuttinger, with whom he had six children. In 1615, he completed the mathematical works Stereometria doliorum and Messekunst Archimedis. At the end of 1617 Kepler successfully defended his mother, who had been accused of witchcraft. In 1619 he published his principal philosophical work, the Harmonice mundi (=HM), and wrote, partially at the same time, the Epitome astronomiae copernicanae (=EAC). In 1624 Kepler continued his investigations on mathematics, publishing his work on logarithms (Chilias logarithmorum…).

In pursuit of an accurate printer for the Tabulae Rudolphinae, he moved to Ulm near the end of 1626 and remained there until the end of 1627. In July 1628 he went to Sagan to enter the service of Albrecht von Wallenstein (1583–1634). He died on November 15, 1630 at Regensburg, where he was to present his financial claims before the imperial authorities (for Kepler's life, Caspar's biography (1993) is still the best work. KGW 19 contains biographically relevant documents).

2. Philosophy, theology, cosmology

There is probably no such thing as “Kepler's philosophy” in any pure form. Nevertheless, many attempts to deal with the “philosophy of Kepler” have been made, all of which are very valuable in their own way. Some studies have concentrated on a particular text (see, for instance, Jardine 1988, for the Defense of Tycho against Ursus), or have followed some particular ideas of Kepler over a longer period of his life and scientific career (see, for instance, Martens 2000, on Kepler's theory of the archetypes). Others have tried to determine from a philosophical point of view his place in the development of the astronomical revolution from the 15th to the 17th centuries (Koyré 1957 and 1961) or in the more general context of the scientific movement of the 17th century (Hall, 1963 and especially Burtt, 1924). Still others have discussed a long list of philosophical principles operating in Kepler's scientific world, and have claimed to have found, by means of such an analysis, compelling evidence for the interaction between science, philosophy, and religion (Kozhamthadam, 1994). If, in the particular case of Kepler, philosophy is immediately related to astronomy, mathematics and, finally, “cosmology” (a notion which arises much later), the core of these speculations is to be sought in the spectrum of problems with which he dealt in his Mysterium Cosmographicum and Harmonice Mundi (on this topic, Field 1988 is one of the most representative works on Kepler). In addition, because of the particular circumstances of his life and his fascinating personality and genius, the literature on Kepler is extremely wide-ranging, covering a spectrum from literary pieces like Max Brod's Tycho Brahes Weg zu Gott (1915)—though still not free from mistakes concerning Kepler—to general introductions in the genre of historical novels, and even fictional stories and charlatanry on astrology or, running for some years now, portraying him as the assassin of Tycho Brahe. However, according to recent reports, it is still a matter of controversy whether Tycho was assassinated at all (see the report at phys.org).

Kepler mastered, like the best scientists, the most complicated technical issues, especially in astronomy, but he always emphasized his philosophical, even theological, approach to the questions he dealt with: God manifests himself not only in the words of the Scriptures but also in the wonderful arrangement of the universe and in its conformity with the human intellect. Thus, astronomy represents for Kepler, if done philosophically, the best path to God (see Hübner 1975; Methuen 1998 and 2009; Jardine 2009). As Kepler at the core of his greatest astronomical work confesses (AN, Part II, chap. 7, KGW 6, p. 108, Engl. trans., p. 183), at the beginning of his career he “was able to taste the sweetness of philosophy … with no special interest whatsoever in astronomy.” And, even in his later work, after having calculated many ephemerides and different astronomical data, Kepler writes in a letter of February 17, 1619 to V. Bianchi: “I also ask you, my friends, that you do not condemn me to the treadmill of mathematical calculations; allow me time for philosophical speculation, my only delight!” (KGW 17, let. N° 827, p. 327, lin. 249– 51).

Especially where Kepler deals with the geometrical structure of the cosmos, he always returns to his Platonic and Neoplatonic framework of thought. Thus, the polyhedral hypothesis (see Section 3 below) he postulated for the first time in his MC represents a kind of “formal cause” constituting the foundational structure of the universe. In addition, an “efficient cause,” which realizes this structure in the corporeal world, is also needed. This is, of course, God the Creator, who accomplished His work according to the model of the five regular polyhedra. Kepler reinterprets the traditional statements about the Creation as an image of the Creator giving to the ancient ideas a more systematic and a quantitative character. Even the doctrine of the Trinity could be geometrically represented, taking the center for the Father, the spherical surface for the Son, and the intermediate space, which is mathematically expressed in the regularity of the relationship between the point and the surface, for the Holy Spirit. In Kepler's model, we have to be able to reduce all appearances to straightness and curvature as providing the foundation for the geometrical structure of the world's creation. The very first category, through which God produced a fundamental similitude of the created World to Himself, is that of quantity (see MC, chapter 2, KGW pp. 23–26). Furthermore, quantity was also introduced into the human soul for the specific purpose that this fundamental symmetry could be apprehended and known scientifically.

This kind of speculation also belongs to the basic principles of Kepler's philosophical optics. In Chapter 1 of APO (“On the Nature of Light”), Kepler gives a new account of this “Trinitarian Cosmogony.” As he admits in a letter to Thomas Harriot (1560–1621), his approach here is more theological than optical (KGW 15, let. 394, p. 348, lin. 18). Similar to his speculations in MC, Kepler explains again the symmetry between God and the Creation, but now he goes slightly beyond the limits of a theologico-geometrical reflection. Firstly, he seems to assume that the bodies of the world were provided in the Creation with some powers which enable them to exceed their geometrical limits and to act on other bodies (magnetic power is a good example of this). Secondly, the principle of symmetry introduced into matter constitutes “the most excellent thing in the whole corporeal world, the matrix of the animate faculties, and the chain linking the corporeal and spiritual world” (APO, Engl. trans., p. 19). Thirdly, as expressed by Kepler in a wonderful, long Latin sentence, with multiple subordinations, this principle “has passed over into the same laws (in leges easdem) by which the world was to be furnished” (ibid., p. 20; the original passage is in KGW 2, p. 19: the marginal note in the edition is “lucis encomium”). Finally, these reflections are concluded with a remark, in which—as with Copernicus, Marsilio Ficino, and others—the central position of the Sun is legitimated because of its function in spreading light and, indirectly, life. Similar speculations are still present in EAC (KGW 7, pp. 47–48 and 267). It is also worth noting that these speculations are of vital importance to the special way in which Kepler conceived of astrology (see, for instance, De fundamentis astrologiae certioribus with Engl. trans. and commentary in Field 1984).

3. The five regular solids

Philosophical, geometrical and even theological speculations related to the five regular polyhedra, the cube or hexahedron, the tetrahedron, the octahedron, the icosahedron and the dodecahedron, were known at least from the time of the ancient Pythagoreans. Since Plato's Timaeus, these five geometrical solids played a leading role, and for the later tradition they became known as the “five Platonic solids”.

Plato establishes at the physical and chemical level a correspondence between them and the five elements—earth, water, air, fire and ether—and tries to provide this correspondence with geometrical grounds. A further source of historically decisive importance is the fact that the five regular polyhedra are treated in Euclid's Elements of Geometry, a work that for Kepler, especially in the Platonic approach of Proclus, has a central position. At the very beginning of HM Kepler complains about the fact that the modern philosophical and mathematical school of Peter Ramus (1515–1572) had not been able to understand the architectonic structure of the Elements, which are crowned with the treatment of the five regular polyhedra. In addition, a revival of Platonic philosophy was taking place in Kepler's time and inspired not only philosophers and mathematicians but also architects, artists and illustrators (see Field 1997).

Amidst this general interest in the regular polyhedra during the Renaissance, Kepler was specifically concerned with their application in the resolution of a cosmological problem, namely the reality of the Copernican system (see Section 5 below). To achieve this goal, he introduced his polyhedral hypothesis already in MC, where he looked for an “a priori” foundation of the Copernican system (see Aiton 1977, Di Liscia 2009). The background for such an approach seems to be that the “a posteriori” way, which according to Kepler was taken over by Copernicus himself, cannot lead to a necessary affirmation of the reality of the new world system, but only to a probable, and hence to an “instrumental”, representation of it as a computational device. This is the “Osiander” or “Wittenberg Interpretation” of Copernicus which Kepler directly attacked not only in his MC but also later in his AN (see Westman 1972 and 1975). In MC, he claimed to have found an answer to the following three main questions: 1) the number of the planets; 2) the size of the orbits, i.e., the distances; 3) the velocities of the planets in their orbits. By referring to the polyhedral hypothesis (see Figure 1), Kepler found a definitive and simple answer to the first question. By intercalating the polyhedra between the spheres which carry the planets, one must inevitably finish with the sphere of Saturn surrounding the cube—there are no more polyhedra to be intercalated and, as remains the standard case for Renaissance and modern astronomy, there are no more planets to be carried by the spheres. It is absolutely decisive for the consistency of the argument that the necessity of the hypothesis is guaranteed by the fact that it already exists as a mathematical demonstration (by Euclid, Elements XIII, prop. 18, schol.), according to which there are only five regular (“Platonic” for the tradition) polyhedra. For the second and third questions the answer is, of course, not as evident as for the first one. However, Kepler was able to show that distances which are derived from the geometrical model of the five regular bodies fit much better with the Copernican system than with that of Ptolemy. The answer to the third question needs, in addition, the introduction of a notion of power that emanates from the Sun and extends to the outer limit of the universe (see Stephenson 1987, pp. 9–20).

In HM, Kepler continues his investigations of the polyhedra at a cosmological as well as a mathematical level. In the second book, dealing with “congruence” (which here does not mean, as it does today, “the same size and shape”, but the property of figures filling the surface together with other regular polygons – on the plane—or, to build closed geometric solids – hence, in space), Kepler made new mathematical discoveries working with tessellations. He discovered two new solids, the so-called “small and great stellated dodecahedrons”.

The treatment of the regular polyhedra constitutes one of the two principal pillars of HM, Book 5 (chapters 1–2), where the third law is formulated (see Caspar in KGW 6, Nachbericht, p. 497, and Engl. trans. p. xxxiii). Following his approach from MC and complementing it with occasional references to his EAC, Kepler makes use again of the Platonic solids to determine the number of the planets and their distances from the Sun. Meanwhile, he has learned that the application of his old polyhedral hypothesis has limits. As he tells us in a footnote in the second edition of his MC from 1621, he was earlier convinced of the possibility of explaining the eccentricities of the planetary orbits by values derived “a priori” from this hypothesis (MC, Engl. trans., p. 189). Now, with access to the observational data of Tycho, Kepler had to exclude this explanation and look for another. And this is one of the most significant achievements of his basic harmonies (which in turn are derived from the regular polygons), constituting the second great pillar of Book 5.

4. Epistemology and philosophy of sciences

Almost all of Kepler's scientific investigations reflect a philosophical background, and many of his philosophical questions find their final answer, even if they are of scientific interest, in the realm of theology. From a very modern point of view, one could highlight Kepler's epistemological thought in terms of four different items: realism; causality; his philosophy of mathematics; and his—own particular—empiricism.

4.1 Realism

Realism is a constant and integral part of Kepler's thought, and one which appears in sophisticated form from the outset. The reason for this is that his realism always runs parallel to his defense of the Copernican worldview, which appeared from his first public pronouncements and publications.

Many of Kepler's thoughts about epistemology can be found in his Defense of Tycho against Ursus or Contra Ursum (=CU), a work which emerged from a polemical framework, the plagiarism conflict between Nicolaus Raimarus Ursus (1551–1600) and Tycho Brahe: causality and physicalization of astronomical theories, the concept and status of astronomical hypotheses, the polemic “realism-instrumentalism”, his criticism of skepticism in general, the epistemological role of history, etc. It is one of the most significant works ever written on this subject and is sometimes compared with Bacon's Novum organum and Descartes' Discourse on Method (Jardine 1988, p. 5; for an excellent new edition and complete study of this work see Jardine / Segonds 2008).

The focus of the epistemological issues could be ranked mutatis mutandi with modern discussion surrounding the scientific status of astronomical theories (however, as Jardine has pointed out, it would be sounder to read Kepler's CU more as a work against skepticism than in the context of the modern realism/instrumentalism polemic). For Pierre Duhem (1861–1916), for instance, the position of Andreas Osiander, which was adopted by Ursus and which was, according to Duhem, naively criticized by Kepler in his MC, represents the modern approach known as “instrumentalism”. According to this epistemological position, held by Duhem himself, scientific theories are not to be closely linked to the concepts of truth and falsehood. Hypotheses and scientific laws are nothing more than “instruments” for describing and predicting phenomena (seldom for explaining them). The aim of physical theories is not to offer a causal explanation or to study the causes of phenomena, but simply to represent them. In the best-case scenario, theories are able to order and classify what is decisive for their predictive capacity (Duhem 1908, 1914).

Contrary to Tycho and Kepler, Ursus held a fictionalist position in astronomy. Yet in the very beginning of his work De hypothesibus, Ursus makes a clear declaration about the nature of astronomical theories, which is very similar to the approach suggested by Osiander in his forward to Copernicus' De revolutionibus: a hypothesis is a “fictitious supposition”, introduced just for the sake of “saving the motions of the heavenly bodies” and to “calculate them” (trans. Jardine 1988, p. 41)

Following his approach in MC and anticipating the opening pages of his later AN (see particularly AN, II.21: “Why, and to what extent, may a false hypothesis yield the truth?” Engl. trans., pp. 294–301), Kepler addresses the question of Copernicanism and its reception by thinkers such as Osiander, who emphasized that the truth of astronomical hypotheses cannot necessarily be deduced from the correct prediction of astronomical facts. According to this interpretation, Copernican hypotheses are not necessarily true even if they are able to save the phenomena, otherwise one would commit a fallacia affirmationis consequentis. However, according to Kepler, “this happens only by chance and not always, but only when the error in the one proposition meets another proposition, whether true or false, appropriate for eliciting the truth” (trans. Jardine, p. 140). To be noted is that, as Jardine (2005, p. 137) has pointed out, the modern scientific realist departs from a real independent world, while Kepler's notion of truth presupposes that neither nature nor the human mind are independent of God's mind (Jardine 2005, p. 137).

4.2 Causality

The reality of astronomical hypotheses—and hence the superiority of the Copernican world system—implied a physicalization of astronomical theories and, in turn, an accentuation of causality. Despite Kepler's criticism of Aristotle, this aspect can actually be considered the realization in the field of astronomy of the old Aristotelian ideal of knowledge: “knowledge” means to grasp the causes of the phenomena.

Thus, on the one hand, “causality” is a notion implying the most general idea of “actual scientific knowledge” which guides and stimulates each investigation. In this sense, Kepler already embarked in his MC on a causal investigation by asking for the cause of the number, the sizes and the “motions” (= the speeds) of the heavenly spheres (see Section 3 above).

On the other hand, “causality” implies in Kepler, according to the Aristotelian conception of physical science, the concrete “physical cause”, the efficient cause which produces a motion or is responsible for keeping the body in motion. Original to Kepler, however, and typical of his approach is the resoluteness with which he was convinced that the problem of equipollence of the astronomical hypotheses can be resolved and the consequent introduction of the concept of causality into astronomy – traditionally a mathematical science. This approach is already present in his MC, where he, for instance, relates for the first time the distances of the planets to a power which emerges from the Sun and decreases in proportion to the distance of each planet, up to the sphere of the fixed stars (see Stephenson 1987, pp. 9–10).

One of Kepler's decisive innovations in his MC is that he replaced the “mean Sun” of Copernicus with the real Sun, which was no longer merely a geometrical point but a body capable of physically influencing the surrounding planets. In addition, in notes to the 1621 edition of MC Kepler strongly criticizes the notion of “soul” (anima) as a dynamical factor in planetary motion and proposes to substitute “force” (vis) for it (see KGW 8, p. 113, Engl. trans. p. 203, note 3).

One of the most important philosophical aspects of Kepler's Astronomia Nova from 1609 (=AN) is its methodological approach and its causal foundation (see Mittelstrass 1972). Kepler was sufficiently conscious of the change of perspective he was introducing into astronomy. Hence, he decided to announce this in the full title of the work: Astronomia Nova, Aitiologetos, seu physica coelestis, tradita commentariis de motibus stellae Martis. Ex observationibus G. V. Tychonis Brahe: New Astronomy Based upon Causes or Celestial Physics Treated by Means of Commentaries on the Motions of the Star Mars from the observations of Tycho Brahe … (trans. Donahue). In the introduction to AN Kepler insists on his radical change of view: his work is about physics, not pure kinematical or geometrical astronomy. “Physics”, as in the traditional, Aristotelian understanding of the discipline, deals with the causes of phenomena, and for Kepler that constitutes his ultimate approach to deciding between rival hypotheses (AN, Engl. trans., p. 48; see Krafft 1991). On the other hand, since his celestial physics uses not only geometrical axioms but also other, non-mathematical axioms, the knowledge obtained often has a kernel of guesswork.

In the third part of AN, chapters 22–40, Kepler deals with the path of the Earth and intends to offer a physical account of the Copernican theory. By so doing he includes the idea that a certain notion of power should be made responsible for the regulation of the differences in velocities of the planets, which in turn have to be established in relation to the planets' distances. Now, the Copernican planetary theory departs from the general principle that the Earth moves regularly on an eccentric circle. For Kepler, on the contrary, the planets are moved irregularly, and the slower they are moved, the greater their distance is from the center of power, the Sun. Addressing the physical aspects of his new astronomy, he deals in chapters 32–40, perhaps the most idiosyncratic of the work, with his notion of motive power. Here, he combines different approaches and sources, sometimes producing—for the purpose of simplifying the whole geometrical construction of geometrical astronomy by introducing a power causing motions—a new confusion at the dynamical level. To begin with, it is not always absolutely clear what kind of power Kepler has in mind. He inclines, above all, to the idea of a magnetic power residing in the Sun, but he also mentions light and, at least indirectly, gravity (which he does not bring into operation in the central chapters of the AN but which is to a certain extent implied in his explanations using the model of the balance and which he surely accepts as true for the Sun-Moon system, as he explains in the general introduction). Secondly, it is not always clear what this power is and how it acts, especially when he is speaking merely analogically, “as if” (particularly in the case of light). Essentially, Kepler breaks down the motions of the planets into two components. On the one hand, the planets move around the Sun—at this state of the discussion—circularly. On the other hand, they exhibit a libration on the Sun-planet vector. The rotation of the Sun is responsible for the motion of the planets. Irradiating from the rotating Sun is a power which spreads at the ecliptic plain. This power diminishes with distance to the source of the power, that is, to the Sun. A decisive work for Kepler's development in his physical astronomy is William Gilbert's (1544–1603) De magnete (London, 1600), a work which also intends to offer a new physics for the new Copernican cosmology and which surely influenced Kepler's thoughts about this power. One of the main problems was, of course, how to apply the general principles of magnetism to planetary motion, first to explain the difference in velocity on a circular path, and later to give an account of the motion on an ellipse. Kepler conceives of a model with parallel magnetic fibers which links the Sun with the planets in such a way that the rotation of the Sun causes the motion of the planets around it. The fibers are born in the planets parallel and perpendicular to the lines of apsides by a kind of “animal power”. The planets themselves are polarized, that is, with one pole they are attracted to the Sun, with the other pole they are pushed away from it. This explains very well the direction of planetary motion: the planets all move in one direction because the Sun rotates in that direction. Nevertheless, a further problem still seems to remain unresolved: according to Kepler's explication, the planets should move around the Sun as fast as the Sun itself rotates, which is not the case. This phenomenon can be explained by referring to a property of matter, which for Kepler has an axiomatic character: the inclinatio ad quietem, that is, the tendency to rest (see especially AN, chap. 39; KGW 3, p. 256). As a consequence, the planets are moved around the Sun slower than they would be if the power of the Sun were at work alone.

Kepler's causal approach is above all present in his Epitome, a voluminous work which exercised a considerable influence on the later development of astronomy. In the second part of Book 4, he deals with the motion of the world's parts. Not the two first laws but rather the third law, which he had recently announced in his HM, is Kepler's starting point; for this law, rather than a calculational device for the path of one planet, represents a general cosmological statement, and thus it is more convenient for his approach here. At the same time, it should be pointed out that the third law is not necessarily the best point of departure for a dynamical, causal approach to motion, as Kepler intends here; for, in comparison with the previous causal approaches, the question of the location of the cause of power responsible for the production of motion remains relevant. The spheres, which in the traditional view transported the planets, had been abolished since the time of Tycho. Furthermore, Kepler is clearly against the “moving intelligences” of the Aristotelian tradition. The fact that the orbits are elliptical and not circular, shows that the motions are not caused by a spiritual power but rather by a natural one, which is internal to the composition of matter. The planets themselves are provided with “inertia”, a property, as Kepler understood it, that inhibits motion and represents an impediment to it. The motive power (vix motrix) comes indeed from the Sun, which sends its rays of light and power in all directions. These rays are captured by the planets. Kepler, however, tries to explain this behavior of the planets less through astrology and much more through magnetism (a physical phenomenon which was by no means clearly understood in his time). Firstly, the Sun rotates and, by so doing, sets in motion the planets around it. Secondly, since the planets are poles of magnets and the Sun itself acts with magnetic power, the planets are, at different parts of their orbits, either attracted or repelled; in this way the elliptical path is causally produced. Kepler partially gives up the mechanical approach by postulating a soul in the Sun which is responsible for its regular motion of rotation, a motion on which, finally, the entire system depends. In fact, the planets are also supposed by Kepler to rotate and are therefore provided with “a sort of soul” or some such principle which produces the rotation.

In addition to astronomy and cosmology Kepler expanded his causal approach to include the fields of optics (see Section 6 below) and harmonics (Section 7 below).

4.3 Philosophy of mathematics

Beyond his own original talent, it is clear that Kepler was trained in mathematics from his earliest studies at Tübingen. At least officially, his positions at Graz, Prague, Linz, Ulm and Sagan can be characterized as the typical professional occupations of a mathematician in the broadest sense, i.e., including astrology and astronomy, theoretical mechanics and pneumatics, metrology, and every topic that could in some way be related to mathematics. Besides the field of astronomy and optics, where mathematics is ordinarily applied in different ways, Kepler produced original contributions to the theory of logarithms and above all within his favorite field, geometry (especially with his stereometrical investigations). Thus, on account of his natural predilection and talent and the importance of mathematics, particularly of geometry, for his thought, it is not surprising to find many different passages in his works where he articulated his philosophy of mathematics. However, Kepler's principal exposition on this topic is to be found in his HM, a work in which the first two books are purely mathematical in content. As he himself declares, in HM he played the role “not of a geometer in philosophy but of a philosopher in this part of geometry” (KGW 6, p. 20, Engl. trans., p. 14).

While in philosophical questions related to mathematics, Proclus and Plato were Kepler's most important inspirational sources, he did not always see Plato and Aristotle as completely opposed, for the latter—in Kepler's interpretation—also accepted “a certain existence of the mathematical entities” (KGW 14, let. N° 226, p. 265; see Peters, p. 130). To a great extent Kepler understood his mathematical investigations of HM as a continuation of Euclid's Elements, especially of the analysis of irrationalities in Book 10. The central notion that he works out here is that of “constructability”. According to Kepler, each branch of knowledge must, in principle, be reducible to geometry if it is to be accepted as knowledge in the strong sense (although, in the case of the physics, this condition is, as the AN emphasizes, only a necessary and not a sufficient condition). Thus, the new principles he was elaborating over the years in astrology were geometrical ones. A similar case occurs with the basic notions of harmony, which, after Kepler, could be reduced to geometry. Of course, not every geometrical statement is equally relevant and equally fundamental. For Kepler, the geometrical entities, principles and propositions which are especially fundamental are those that can be constructed in the classical sense, i.e., using only ruler (without measurement units) and compass. On this are based further notions according to different degrees of “knowability” (scibilitas), which begins with the circle and its diameter. Once again, Kepler understood this within the framework of his cosmological and theological philosophy: geometry, and especially geometrically constructible entities, have a higher meaning than other kinds of knowledge because God has used them to delineate and to create this perfect harmonic world. From this point of view, it is clear that Kepler defends a Platonist conception of mathematics, that he cannot assume the Aristotelian theory of abstraction and that he is not able to accept algebra, at least in the way he understood it. So, for instance, there are figures that cannot be constructed “geometrically”, although they are often assumed as safe geometrical knowledge. The best example of this is perhaps the heptagon. This figure cannot be described outside of the circle, and in the circle its sides have, of course, a determinate magnitude, but this is not knowable. Kepler himself says that this is important because here he finds the explanation for why God did not use such figures to structure the world. Consequently, he devotes many pages to discussing the issue (KGW 6, Prop. 45, pp. 47–56, see also KGW 9, p. 147). Certainly for a geometer like Kepler, approximations constitute – as mathematical theory—a painful and precarious way to progress. The philosophical background for his rejection of algebra seems to be, at least partially, Aristotelian in some of its basic suppositions: geometrical quantities are continuous quantities which therefore cannot be treated with numbers that are, in the inverse, discrete quantities. But the difference from the Aristotelian ideal of science remains an important one: for Aristotle, a crossover between arithmetic and geometry is allowed only in the case of the “middle sciences”, while for Kepler all knowledge must be reduced to its geometrical foundations.

4.4 Empiricism

A general presentation of Kepler's philosophical attitude and principles is not complete without reference to his link to the world of experience. For, despite his mainly theoretical approach in the natural sciences, Kepler often emphasized the significance of experience and, in general, of empirical data. In his correspondence there are many remarks about the significance of observation and experience, as for instance in a letter to Herwart von Hohenburg from 1598 (KGW 13, let. N° 91, lines 150–152) or from 1603 to Fabricius (KGW 14, let. N° 262, p. 191, lines 129–130), to mention only two of his most important correspondents. Looking for empirical support for the Copernican system, Kepler compares different astronomical tables in his MC, and in AN he makes extensive use of Tycho's observational treasure trove. In MC (chapter 18) he quotes a long passage from Rheticus for the sake of rhetorical support when, as was the case here, the data of the tables he used did not fit perfectly with the calculated values from the polyhedral hypothesis. In this passage, the reader learns that the great Copernicus, whose world system Kepler defends in MC, said one day to Rheticus that it made no sense to insist on absolute agreement with the data, because these themselves were surely not perfect. After all, it is questionable whether Kepler, using for instance the Prutenic Tables (1551) of Erasmus Reinhold (1511–1553), had access to complete and correct empirical information to confirm the Copernican hypothesis in grand style, as he claimed (for an analysis of Reihold's tables and their influence see Gingerich 1993, pp. 205–255).

The situation changed completely when Kepler came into contact at Prague with Tycho's observations (which, as Kepler often reports, were seldom at his disposal). However, a change of attitude is evident in AN, where he used Tycho's observations without restriction (which is something he makes clear in the work's title). In part 2 (chap. 7–21), he presents the “vicarious hypothesis”, which in the end he refutes. This hypothesis represents the best result which can be reached within the limits of traditional astronomy. This works with circular orbits and with the supposition that the motion of a planet appears regular from a point on the lines of apsides. Against the traditional method, here, Kepler does not cut the eccentricity into equal parts but leaves the partition open. To check his hypothesis, he needs observations of Mars in opposition, where Mars, the Earth, and the Sun are at midnight on the same line. From Tycho, he “inherited” ten such observations between the years 1580 and 1600, and to them he added another two for 1602 and 1604. In chapters 17–21, Kepler carries out an observational and computational check of his vicarious hypothesis. On the one hand, he points out that this hypothesis is good enough, since the variations of the calculated positions from the observed positions fall within the limits of acceptability (2 minutes of arc). In fact, Kepler presents this hypothesis as the best hypothesis which can be proposed within the framework of a “traditional astronomy”, as opposed to his new astronomy, which he will offer in the following parts of the work. On the other hand, this hypothesis can be falsified if one takes the observations of the latitudes into consideration. Further calculations with these observations produce a difference of eight minutes, something that cannot be assumed because the observations of Tycho are reliable enough. Kepler's famous sentence runs: “these eight minutes alone will have led the way to the reformation of all of astronomy” (AN, KGW 3, p. 286; Engl. trans., p. 286). There seems to be agreement that Kepler's AN contains the first explicit consideration of the problem of observational error (for this question see Hon 1987 and Field 2005).

Kepler also gave an important place to experience in the field of optics. As a matter of fact, he began his research on optics because of a disagreement between theory and observation, and he made use of scientific instruments he had designed himself (see, for instance, KGW 21.1, p. 244). Recent research on the problem of the camera obscura and the “images in the air” shows, however, the limits of a traditional approach to Kepler's optics following the main current of the history of physics. Rather, his notion of experimentum needs to be contextualized within the social practices and epistemological commitments of his time (see Dupré 2008).

Finally, it should be mentioned that a similar significance is assigned to experience and empirical data in Kepler's harmonic-musical and astrological theories, two fields which are subordinated to his greater cosmological project of HM. For astrology, he uses meteorological data, which he recorded for many years, as confirmation material. This material shows that the Earth, as a whole living being, reacts to the aspects which occur regularly in the heavens. In his musical theory Kepler was a modern thinker, especially because of the role he gave to experience. As has been noted (Walker, 1978, p. 48), Kepler made acoustic experiments with a monochord long before he wrote his HM. In a letter to Herwart von Hohenburg (KGW 15, ep. 424, p. 450), he describes how he checked the sound of a string at different lengths, establishing in which cases the ear judges the sound to be pleasurable. Kepler does not accept that this limitation is founded on arithmetical speculations, even if this was already assumed by Plato, whom he often follows, and by the Pythagoreans. On the basis of his experiments, Kepler found that there are other divisions of the string that the ear perceives as consonant, i.e., thirds and sixths.

If cosmology is the main framework of Kepler's interest, there is no doubt that, as Field has pointed out, he “felt the need to seek observational support for his model of the Universe” (Field 1988, p. 28; see also Field 1982).

5. Copernicanism reformed and the three planetary laws

Today Kepler is remembered in the history of sciences above all for his three planetary laws, which he produced in very specific contexts and at different times. While it is questionable whether he would have understood these scientific statements as “laws”—and it is even arguable that he used this term with a different meaning than we do today—it seems to be clear that all three laws (as a linguistic convention, we may continue to use the term) suppose some fundamentals of Kepler's philosophy: (a) realism, (b) causality, (c) the geometric structure of the cosmos. Besides this, it should be remarked that the common denominator of all three laws is Kepler's defense of the Copernican worldview, a cosmological system which he was not able to defend without reforming it radically. It is noteworthy that already at the very beginning of his career Kepler vehemently defended the reality of the Copernican worldview in a way that he characterized, taking over the terminology from the standard Aristotelian epistemology, as “a priori” (see above Section 3 above and Di Liscia 2009).

The first two laws were published initially in AN (1609), although it is known that Kepler had arrived at these results much earlier. His first law establishes that the orbit of a planet is an ellipse with the Sun in one of the foci (see Figure 2). According to the second law, the radius vector from the Sun to a planet P sweeps out equal areas, for instance SP1P2 and SP3P4,in equal times. The planet P is therefore faster at perihelion, where it is closer to the Sun, and slower at aphelion, where it is farther from the Sun. In accordance with his dynamical approach, Kepler first found the second law and, then, as a further result because of the effect produced by the supposed force, the elliptical path of the planets (for the two first planetary laws see especially Aiton 1973, 1975a, Davis 1992a-e, and 1998a; Donahue 1994; Wilson 1968 and 1972).

Perhaps the most significant impact of Kepler's two laws can be found by considering their cosmological consequences. The first law abolishes the old axiom of the circular orbits of the planets, an axiom which was still valid not only for pre-Copernican astronomy and cosmology but also for Copernicus himself, and for Tycho and Galileo. The second law breaks with another axiom of traditional astronomy, according to which the motion of the planets is uniform in swiftness. The Ptolemaic tradition in astronomy was, of course, aware of this difficulty and applied a particularly effective device for saving the “appearance” of acceleration: the equant. Copernicus, for his own part, insisted on the necessity of the axiom of uniform circular motion. Ptolemy's equant was understood by Copernicus as a technical device based on the violation of this axiom. Kepler, on the contrary, affirms the reality of changes in the velocities of the planetary motions and provides a physical account for them. After struggling strenuously with established ideas which were located not only in the tradition before him but also in his own thinking, Kepler abandoned the circular path of planetary motion and in this way initiated a more empirical approach to cosmology (though see Brackenridge 1982).

Kepler published the third law, the so-called “harmonic law”, for the first time in his Harmonice mundi (1619), i.e., ten years later. In his Epitome, he provided a more systematic approach to all three laws, their grounds and implications (see Davis 2003; Stephenson 1987). In Book 5, chapter 3, as point 8 of 13 (KGW 6, p. 302; Engl. trans., pp. 411–12), Kepler expresses, almost accidentally, his fundamental relationship connecting elapsed times with distances, which in modern notation could be expressed as:

(T1/T2)2 = (a1/a2)3

with T1 and T2 representing the periodic times of two planets and a1 and a2 the length of their semi-major axes. A further formulation of this relationship, which is often found in the literature, is: a3/T2 = K, which expresses with K that the relationship between the third power of the distances and the square of the times is a constant (however, see Davis 2005, pp. 171–172; for the third planetary law see especially Stephenson 1987). As a consequence of the third law, the time a planet takes to travel around the Sun will significantly increase the farther away it is or the longer the radius of its orbit. Thus, for instance, Saturn's sidereal period is almost 30 years, while Mercury needs fewer than 88 days to go around the Sun. For the history of cosmology, it is important to make clear that the third law fulfils Kepler's search for a systematic representation and defense of the Copernican worldview, in which planets are not absolutely independent of each other but integrated in a harmonic world system.

6. Optics and metaphysics of light

Kepler contributed to the special field of optics with two seminal works, the Ad Vitellionem paralipomena (=APO) and the Dioptrice (=DI), the latter motivated in large part by the publication in 1610 of Galileo's Sidereal Messenger (Sidereus Nuncius). In his Conversation with the Sidereal Messenger (Dissertatio cum Nuncio Sidereo … a Galillaeo Galilaeo, KGW 4, pp. 281–311), he supported the factual information given by Galileo, indicating at the same time the necessity of giving an account of the causes of the observed phenomena. The background for his investigation into optics was undoubtedly the different particular questions of astronomical optics (see Straker 1971). In this context he concentrated his efforts on an explanation of the phenomena of eclipses, of the apparent size of the Moon and of atmospheric refraction. Kepler investigated the theory of the camera obscura very early and recorded its general principles (see commentary by M. Hammer in KGW 2, pp. 400–1 and Straker 1981). In addition, he worked intensively on the theory of the telescope and invented the refracting astronomical or ‘Keplerian’ telescope, which involved a considerable improvement over the Galilean telescope (see especially DI, Problem 86, KGW 4, pp. 387–88). Besides these impressive contributions, Kepler expanded his research program to embrace mathematics as well as anatomy, discussing for instance conic sections and explaining the process of vision (see Crombie 1991 and especially Lindberg 1976b).

In Chapter 1 of APO (“On the Nature of Light”), Kepler expounds 38 propositions concerning different properties of light: light flows in all directions from every point of a body's surface; it has no matter, weight, or resistance. Following—but also inverting—the Aristotelian argument for the temporality of motion, he affirms that the motion of light takes place not in time but in an instant (in momento). Light is propagated by straight lines (rays), which are not light itself but its motion. It is important to note that although light travels from one body to another, it is not a body but a two-dimensional entity which tends to expand to a curved surface. The two-dimensionality of light is probably the main reason why it is incorporeal. Motion in general plays a significant role in Kepler's philosophy of light. For Straker, the supposed link between optics and physics (especially in Prop. 20, where the mechanical analyses are introduced) “reveals the full extent of his commitment to a mechanical physics of light” (Straker 1971, p. 509).

Two questions are intensively discussed by modern specialists. Firstly, to what extent is the attribution of a mechanistic approach to Kepler justified? Secondly, how should one determine his place in the history of sciences, especially in the field of optics: do the main lines of thought in Kepler's optics indicate a continuity or rather a rupture with tradition? There are well–grounded arguments for different positions on both questions. For Crombie (1967, 1991) and Straker, Kepler develops a mechanical approach, which can be particularly appreciated in his explanation of vision using the model of the camera obscura. Besides this, Straker stresses that Kepler's basic mechanicism is also powerfully assisted by his conception of light as a non-active, passive entity. In addition, the concept of motion and the explanations using the model of the balance are indicative of a commitment to mechanicism (Straker 1970, pp. 502–3). On the contrary, Lindberg (1976a), who supports the “continuity side” of the dispute, has quite convincingly showed that, for Kepler, light has a constructive and active function in the universe, not only in optics but also in astrology, astronomy, and natural philosophy (for Kepler's criticism of the medieval tradition see also Chen-Morris / Unguru, 2001).

7. Harmony and Soul

From a philosophical point of view, Kepler considered the HM to be his main work and the one he most cherished. Containing his third planetary law, this work represents definitively a seminal contribution to the history of astronomy. But he did not reduce his long prepared project to an astronomical investigation—his first thoughts on the notion of “harmony” arose already in 1599, although he did not publish his work until 1619—but instead extensively discussed its mathematical foundations and its philosophical implications, including astrology, natural philosophy and psychology. Thus, Kepler's third planetary law appears in a context which goes far beyond astronomy and to a great extent takes up again the perspective of his youthful MC.

According to Kepler, it is necessary to distinguish “sensible” from “pure” harmony. The first is to be found among natural, sensible entities, like sounds in music or rays of light; both could be in proportion to one another and hence in harmony. He resolves this matter by combining three of the Aristotelian categories: quantity, relation and, finally, quality. Through the function of the category of relation Kepler passes over to the active function of the mind (or soul). It turns out that two things can be characterized as harmonic if they can be compared according to the category of quantity. But the fact that at least two things are needed shows that the property of “being harmonic” is not a property of an isolated thing. Furthermore, the relationship between the things cannot be found in the things themselves either; rather, it is produced by the mind: “in general every relation is nothing without mind apart from the things which it relates, because they do not have the relation which they are said to have unless the presence of some mind is assumed, to relate one to another” (KGW 6, p. 212; Engl. trans., p. 291). This process takes place through the comparison of different sensible things with an archetype (archetypus) present in the mind.

The next central question directly concerns gnoseology, for Kepler gives a psychological account of the path followed by sensible things into the mind. He resumes the scholastic species theory: immaterial species radiate from the sensible things and affect the sense organs by acting firstly on the “forecourts” and then on the internal functions. They arrive at the imagination and from there go over to the sensus communis, so that, according to the traditional teaching, the sensible information received is now able to be processed and used in statements. From here onwards, the sensible things are “preserved in the memory, brought forth by recollection, [and] distinguished by the higher faculty of the soul” (Caspar 1993, p. 269, cf. KGW 6, p. 214, lines 18–23; Engl. trans., p. 293). While harmony arises as an activity of the soul/mind consisting in relating quantitatively, Kepler adds, taking over the Aristotelian doctrine of categories, that harmony is a “qualitative relation” as well, involving the “quality of shape, being formed from the regular figures”, which provides the grounds for comparison (KGW 6, p. 216, lines 37–41; Engl. trans., p. 296). If this is how “things”, i.e., sensible entities, find their way into the soul in order to be compared, it by no means represents—as Kepler admitted—a sufficient explanation of how non-sensible things, i.e., mathematical entities, find their way into the mind. How do they come into the soul? Kepler accepted Aristotle's criticism of Pythagorean philosophy concerning numbers: both Kepler and Aristotle are convinced that numbers constitute ontologically a lower class within mathematical entities (for Kepler, they are derived from geometrical entities). Nevertheless, Aristotle's philosophy is insufficient to grasp the essence of mathematics. By aligning himself with Proclus, from whom he quotes a long passage of his Commentary on Euclides, Kepler defends Plato's theory of anamnêsis against Aristotle's doctrine of the tabula rasa. His discussion lies at the origin of the classical debate between empiricism and rationalism which was to dominate the philosophical scene for generations to come. A connection with idealism is, of course, apparent (see, for instance, Caspar 1993, Engl. trans., p. 269), and it is a fact that Kepler was positively received within German Idealism of the 19th century. Historically, however, it seems to be more accurate to link his position with the philosophical tradition of St. Augustine.

Besides psychology and gnoseology, the other main spectrum of questions Kepler deals with in Book 4 is his theory of “aspects”, i.e., astrology (HM, IV, Cap. 4–7), a further field of application of his psychology and further evidence of the role of geometry in his philosophy. The “aspects”, i.e., the angles between the planets, Moon and Sun, are all he wishes to save from the old astrology, which he harshly criticizes; for the aspects are or can be reduced to geometrical structures, the archetypes, which can be recognized by the soul. According to Kepler, there is no “mechanical influence” of the heavens (stars and constellations are not relevant in his astrology) which exerts a determining effect on the Earth and on human life. Rather, both the Earth and human beings, ultimately, like all other living entities, are provided with a soul in which the geometrical archetypes are present. By the formation of an aspect in the heavens, symmetry arises and stimulates the soul of the Earth or of human beings. “The Earth,” Kepler writes, responds to “what the aspects whistle” (KGW 11.2, p. 48; for his astrology see especially Field 1984 and Rabin 1997, Boner 2005 and 2006).

Bibliography

A. Primary Sources

Complete Editions

• Joannis Kepleri Astronomi Opera omnia, ed. Ch. Frisch, vols. 1–8, 2; Frankfurt a.M. and Erlangen: Heyder & Zimmer, 1858–1872.

• Johannes Kepler Gesammelte Werke, herausgegeben im Auftrag der Deutschen Forschungsgemeinschaft und der Bayerischen Akademie der Wissenschaften, unter der Leitung von Walther von Dyck und Max Caspar, Bd. 1–21.2.2; München: C.H. Beck'sche Verlagsbuchandlung, 1937–2009 (apparently finished) [ = KGW].

Selected English Translations of Individual Works

• Mysterium cosmographicum: Trans. A. M. Duncan, The secret of the universe Translation by A.M. Duncan / Introduction and commentary by E. J. Aiton, New York: Abaris, 1981.

• Apologia Tychonis contra Ursum: Trans. N. Jardine, The Birth of History and Philosophy and Science: Kepler's Defense of Tycho against Ursus with Essays on its Provenance and Significance, Cambridge: Cambridge University Press, 1984 (with corrections 1988).

• Ad Vitellionem paralipomena. Astronomiae pars optica: Trans. W. H. Donahue, Johannes Kepler. Optics: Paralipomena to Witelo & Optical Part of Astronomy, trans. William H. Donahue, New Mexico: Green Lion Press, 2000.

• Dissertatio cum Nuncio Sidereo and Narratio de Observatis Jovis Satellitibus: Trans. E. Rosen, Kepler's Conversation with Galileo's Sideral Messenger, New York and London: Johnson Reprint Corporation, 1965.

• De fundamentis astrologiae certioribus: = On Giving Astrology Sounder Foundations: Trans. J.V. Field, “A Lutheran Astrologer: Johannes Kepler”, Archive for History of Exact Sciences, 31/3 (1984): 189–272.

• Astronomia nova: Trans. W. H. Donahue, New Astronomy, Cambridge, New York: University Press, 1992; and Selections from Kepler's Astronomia Nova: A Science Classics Module for Humanities Studies, trans. W. H. Donahue, Santa Fe, NM: Green Lion Press, 2005.

• Harmonices mundi libri V: Trans. E. J. Aiton, A. M. Duncan, J.V. Field, The Harmony of the World. Philadelphia: American Philosophical Society (Memoirs of the American Philosophical Society), 1997.

• Epitome astronomiae copernicanae: Partial trans. C. G. Wallis, Epitome of Copernican Astronomy: IV and V, Chicago, London: Encyclopaedia Britannica (Great Books of the Western World, Volume 16], 1952, pp. 839–1004.

• Strena seu de nive sexangula: Trans. C. Hardie, The Six-Cornered Snowflake, Oxford: Clarendon Press, 1966.

• Somnium seu de astronomia lunari: Trans. E. Rosen, Kepler's Somnium: The Dream, or Posthumous Work on Lunar Astronomy, Madison: University of Wisconsin Press, 1967.

• Letters (selection): Trans. C. Baumgardt; introd. A. Einstein: Johannes Kepler: Life and Letters, New York: Philosophical Library, 1951.

B. Secondary Literature

• Aiton, E.J., 1973, “Infinitesimals and the Area Law”, in F. Krafft, K. Meyer, and B. Sticker (eds.), 1973: Internationales Kepler-Symposium Weil der Stadt 1971, Hildesheim: Gerstenberg, pp. 285–305.

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• Barker, P., and B.R. Goldstein, 1994, “Distance and Velocity in Kepler‘s Astronomy”, Annals of Science, 51: 59–73.

• –––, 1998, “Realism and Instrumentalism in Sixteenth Century Astronomy: A Reappraisal”, Perspective on Sciences, 6/3: 232–258.

• –––, 2001, “Theological Foundations of Kepler's Astronomy”, Osiris, 16: 88–113.

• Bialas, V., 1990, “Keplers komplizierter Weg zur Wahrheit: von neuen Schwierigkeiten, die ‘Astronomia Nova’ zu lesen”, Berichte zur Wissensschaftsgeschichte, 13/3: 167–176.

• Beer, A., and P. Beer (eds.), 1975, Kepler Four Hundred Years, Oxford: Pergamon Press.

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• Boner, P.J., 2005, “Soul-searching with Kepler: An analysis of Anima, in his astrology”, Journal for the History of Astronomy 36/1: 7–20.

• –––, 2006, “Kepler's Living Cosmology: Bridging the Celestial and Terrestrial Realms”, Centaurus, 48: 32–39.

• –––, 2013, Kepler’s Cosmological Synthesis: Astrology, Mechanism and the Soul, Leiden: Brill.

• Brackenridge, J.B., 1982, “Kepler, elliptical orbits, and celestial circularity: a study in the persistence of metaphysical commitment. I”, Annals of Science 39/2: 117–143 and II in Annals of Science, 39/3: 265–295.

• Buchdahl, G., 1972, “Methodological aspects of Kepler's theory of refraction”, Studies in History and Philosophy of Science, 3: 265–298.

• Burtt, E.A., 1924, The Metaphysical Foundations of Modern Physical Science, New York: Kegan Paul; 2nd rev. ed., 1954, Garden City, NY: Doubleday, reprinted 2003, New York: Dover Publications.

• Caspar, M., 1993, Johannes Kepler, New York: Dover Publications.

• Chen-Morris, R.D., and Unguru, Sabetai, 2001, “Kepler's Critique of the Medieval Perspectivist Tradition”, in Simon & Débarbat 2001, pp. 83–92.

• Crombie, J.A., 1967, “The Mechanistic Hypothesis, and the Study of Vision: Some Optical Ideas as a Background to the Invention of the Microscope”, in S. Bradbury and G.L.E. Turner (eds.), Historical Aspects of Microscopy, Cambridge: Heffer (for the Royal Microscopical Society), pp. 3–112.

• –––, 1991, “Expectation, modelling and assent in the history of optics. II, Kepler and Descartes”, Studies in History and Philosophy of Science, 22/1: 89–115.

• –––, Cohen, B., 1985, The Birth of a New Physics, New York and London: W.W. Norton.

• Davis, A.E.L., 1975, “Systems of Conics in Kepler's Work”, Vistas in Astronomy, 18: 673–685.

• –––, 1992a, “Kepler's resolution of individual planetary motion”, Centaurus, 35/2: 97–102.

• –––, 1992b, “Kepler's ‘distance law’ – myth not reality”, Centaurus, 35/2: 103–120.

• –––, 1992c, “Grading the eggs (Kepler's sizing-procedure for the planetary orbit)”, Centaurus, 35/2: 121–142.

• –––, 1992d, “Kepler's road to Damascus”, Centaurus, 35/2: 143–164.

• –––, 1992e, “Kepler's physical framework for planetary motion”, Centaurus, 35/2: 165–191.

• –––, 1998a, “Kepler's unintentional ellipse—a celestial detective story”, The Mathematical Gazette, 82/493: 37–43.

• –––, 1998b, “Kepler, the ultimate Aristotelian”, Acta historiae rerum naturalium necnon technicarum, 2: 65–73.

• –––, 2003, “The Mathematics of the Area Law: Kepler's successful proof in Epitome Astronomiae Copernicanae, (1621)”, Archive for History of Exact Sciences, 57/5: 355–393.

• –––, 2005, “Kepler's Angular Measure of Uniformity: how it provided a potential proof of his Third Law”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.), pp. 157–173.

• –––, 2010, “Kepler's Astronomia Nova: a geometrical success story,” in A. Hadravová, T.J. Mahoney and P. Hadrava (eds.), Kepler's Heritage in the Space Age, (Acta historiae rerum naturalium necnon technicarum, Volume 10), Prague: National Technical Museum.

• Di Liscia, Daniel A., 2009, “Kepler's A Priori Copernicanism in his Mysterium Cosmographicum,” in M.A. Granada and E. Mehl (eds.), pp. 283–317.

• Donahue, William, 1994, “Kepler's invention of the second planetary law”, British Journal for the History of Science, 27/92: 89–102.

• Dreyer, J.L.E., 1963, Tycho Brahe, A Picture of Scientific Life and Work in the Sixteenth Century, New York: Dover Publications.

• Duhem, P., 1908, SOZEIN TA, FAINOMENA: Essai sur la notion de théorie physique de Platon à Galilée, Paris: A. Hermann.

• –––, 1914, La théorie physique, son objet et sa structure, Paris: Chevalier et Rivière.

• Dupré, Sven, 2008, “Inside the Camera Obscura: Kepler's Experiment and Theory of Optical Imagery”, Early Science and Medicine, 13: 219–244.

• Field, J.V., 1982, “Kepler's Cosmological Theories: their agreement with observation”, Quarterly Journal of the Royal Astronomical Society, 23: 556–568.

• –––, 1983, “Cosmology in the work of Kepler and Galileo”, P. Galluzzi (ed.) Novità celesti e crisi del sapere, [Supplement volume to Annali del Istituto e Museo di Storia della Scienza di Firenze], Florence: Giunti Barbèra, pp. 207–215

• –––, 1984, “A Lutheran Astrologer: Johannes Kepler”, Archive for History of Exact Science, 31: 189–272.

• –––, 1988, Kepler's Geometrical Cosmology, London and Chicago: Athlone and University of Chicago Press.

• –––, 1997, “Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler”, Archive for History of Exact Sciences, 50/3–4: 241–289.

• –––, 1998, “Kepler's mathematization of cosmology”, Acta historiae rerum naturalium necnon technicarum, 2: 27–48.

• –––, 2005, “Tycho Brahe, Johannes Kepler and the concept of error”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.), pp. 143–155.

• –––, 2009, “Kepler's Harmony of the World”, in Richard L. Kremer and Jarosław Włodarczyk (eds.), Johannes Kepler: From Tübingen to Zagan (Proceedings of a conference held in Zielona Gora in June 2008), Studia Copernicana, 42: 11–28.

• Folta, J. (ed.), 1998, Mysterium cosmographicum 1596–1996, Prague [Acta Historiae rerum naturalium necnon technicarum, New Series, Vol. 2].

• Gingerich, Owen, 1993, The Eye of Heaven. Ptolemy, Copernicus, Kepler, New York: American Institute of Physics.

• Granada, M.A., 2008, “Kepler and Bruno on the Infinity of the Universe and of Solar Systems”, Journal for the History of Astronomy, 39: 1–27.

• Granada, M.A., and E. Mehl (eds.), 2009, Nouveau ciel, nouvelle terre / La révolution copernicienne en Allemagne 1530–1630, Paris: Belles Lettres.

• Hall, Rupert, 1963, From Galileo to Newton 1630–1720, New York: Collins.

• Holton, G., 1956, “Johannes Kepler's universe: its physics and metaphysics”, American Journal of Physics, 24: 340–351.

• Hübner, J., 1975, Die Theologie Johannes Keplers zwischen Orthodoxie und Naturwissenschaft, Tübingen: J.C.B. Mohr.

• Hon, G., 1987, “On Kepler's awareness of the problem of experimental error”, Annals of Science, 44/6, 545–591.

• Itokazu, A.G., 2009, “On the Equivalence of the Hypotheses in Part 1 of Johannes Kepler's New Astronomy”, Journal for the History of Astronomy, 50: 173–190.

• Jardine, N., 1988, The Birth of History and Philosophy and Science: Kepler's Defense of Tycho against Ursus with Essays on its Provenance and Significance, Cambridge: Cambridge University Press, 1984 (with corrections 1988).

• –––, 2005, “The Many Significances of Kepler's Contra Ursum”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.), pp. 129–142.

• –––, 2009, “Kepler, God, and the Virtues of Copernican Hypotheses”, in M. A. Granada and E. Mehl (eds.), pp. 269–277.

• Jardine, N., and A. Segonds, 1999, “Kepler as Reader and Translator of Aristotle”, in C. Blackwell and S. Kusukawa, eds., Philosophy in the Sixteenth and Seventeenth Centuries: Conversations with Aristotle, Aldershot: Ashgate, pp. 206–233.

• Jardine, N., and A. Segonds, 2001, “A Challenge to the Reader: Ramus on Astrologia, without Hypotheses”, in M. Feingold, J.S. Freedman, and W. Rother (eds.), The Influence of Petrus Ramus. Studies in Sixteenth and Seventeenth Century Philosophy and Sciences, Basel: Schwabe, pp. 248–266.

• Jardine, N., and A. Segonds, 2008, La Guerre des Astronomes: La querelle au sujet de l'origine du système géo-hèliocentrique à la fin du XVIe siècle (Science et humanisme, 9–10), Paris: Belles Lettres, 2 volumes.

• Koyré, A., 1957, From the Closed World to the Infinite Universe, Baltimore, MD: Johns Hopkins Press.

• –––, 1961, La révolution astronomique. Copernic – Kepler – Borelli, [Histoire de la Pensée III. École Pratique des Hautes Études. Sorbonne], Paris: Hermann.

• Kozhamthadam, Job, 1994, The Discovery of Kepler's Laws., The Interaction of Science, Philosophy and Religion, Notre Dame and London: University of Notre Dame Press.

• Krafft, F., 1991, “The New Celestial Physics of Johannes Kepler”, in S. Unguru (ed.), pp. 185–227.

• Launert, D., 1999, Nicolaus Reimers (Raimarus Ursus). Günstling Rantzaus – Brahes Feind Leben und Werk, Munich (Dr. Rauner Verlag) [Algoritmus N° 29].

• Lindberg, D. C., 1976a, “The Genesis of Kepler's Theory of Light: Light Metaphysics from Plotinus to Kepler”, Osiris (NS), 2: 5–42.

• –––, 1976b, Theories of Vision from Al-Kindi to Kepler, Chicago and London: University of Chicago Press.

• –––, 1991, “Kepler and the incorporeality of light”, in S. Unguru (ed.), pp. 229–250.

• Martens, R., 1998, “Kepler's use of Archetypes in his Defense against Aristotelian Scepticism”, in J. Folta (ed.), Mysterium cosmographicum 1596–1996, Prague 1998 [Acta historiae rerum naturalium necnon technicarum 2], pp. 74–89.

• –––, 2000, Kepler's Philosophy and the New Astronomy, Princeton, NJ: Princeton University Press.

• Methuen, C., 1998, Kepler's, Tübingen: Stimulus to a Theological Mathematics (St. Andrews Studies in Reformation History), Aldershot: Ashgate.

• –––, 2009, “De la sola scriptura, à l'Astronomia nova: Principe d'autorité, principe d'accommodation et réforme de l'astronomie dans l'oeuvre de Jean Kepler”, in M.A. Granada and E. Mehl (eds.), pp. 319–334.

• Mittelstrass, J., 1972, “Methodological Aspects of Keplerian Astronomy”, Studies in History and Philosophy of Science, 3: 213–32.

• Kepler, Harmonik, and Th. Peters, 1941–42, “Über Näherungskonstruktionen und Mechanismen im ersten Buch der Harmonik Keplers und seine Forderung nach Beschränkung der Konstruktionsmittel allein auf Zirkel und Lineal”, Deutsche Mathematik, 6/1: 118–132.

• Rabin, S., 1997, “Kepler's Attitude toward Pico and the Anti-astrology Polemic”, Renaissance Quatterly, 50: 750–770.

• Rosen, E., 1986, Three Imperial Mathematicians: Kepler Trapped Between Tycho Brahe and Ursus, New York: Abaris Books.

• Schofield, C.J., 1981, Tychonic and Semi-Tychonic World Systems, New York: Arno Presse.

• Simon, G., 1979, Kepler astronome astrologue, Paris: Gallimard.

• Simon, G., and S. Débarbat (eds.), 2001, Optics and Astronomy. Proceedings of the XXth International Congress of History of Science (Liège, 20–26 July 1997), Turnhout: Brepols.

• Stephenson, B., 1987, Kepler's Physical Astronomy [Studies in the History of Mathematics and Physical Sciences 13] New York, Berlin: Springer Verlag.

• –––, 1994, The Music of the Heavens: Kepler's Harmonic Astronomy, Princeton, NJ: Princeton University Press.

• Straker, S.M., 1970, Kepler‘s Optics: A Study in the Development of Seventeenth-Century Natural Philosophy, Ph.D Dissertation, Indiana University.

• –––, 1981, “Kepler, Tycho, and the ‘Optical Part of Astronomy’: the genesis of Kepler's theory of pinhole images”, Archive for History of Exact Sciences, 24/4: 267–293.

• Unguru, S. (ed.), 1991, Physics, Cosmology and Astronomy, 1300–1700: Tension and Accommodation (Boston Studies in the Philosophy of Science 126), Dordrecht, Boston and London: Kluwer Academinc Publishers.

• Voelkel, James R., 1999, Johannes, Kepler and the New Astronomy, New York and Oxford: Oxford University Press.

• –––, 2001, The Composition of Kepler's, Astronomia Nova, Princeton, NJ: Princeton University Press.

• Walker, Daniel P., 1967, “Kepler's Celestial Music”, Journal of the Warburg and Courtauld Institutes, 30: 228–250.

• –––, 1978, Studies in Musical Science in the Late Renaissance, London: Warburg Institute.

• Whiteside, D.T., 1974, “Keplerian Planetary Eggs, Laid and Unlaid, 1600–1605”, Journal for the History of Astronomy, 5: 1–21.

• Wilson, C., 1968, “Kepler's Derivation of the Elliptical Path”, Isis: 59: 5–25; reprinted in Wilson 1989.

• –––, 1972, “How did Kepler Discover His First Two Laws?”, Scientific American, 226: 93–106; reprinted in Wilson 1989.

• –––, 1974, “Newton and Some Philosophers on Kepler's Laws”, Journal for the History of Ideas, 35: 231–58.

• –––, 1978, “Horrocks, Harmonies, and the Exactitude of Kepler's Third Law”, Science and History: Studies in Honor of Edward Rosen, [Studia Copernicana 16] Wrocław (Ossolineum), pp. 235–59.

• –––, 1989, Astronomy from Kepler to Newton: Historical Essays, Aldershot: Variorum Reprints.

• Westman, R. S., 1972, “Kepler's theory of hypothesis and the ‘Realist Dilemma’”, Studies in History and Philosophy of Science, 3: 233–264.

• –––, 1975, “The Melanchthon Circle, Rheticus, and the Wittenberg Interpretation of the Copernican Theory”, Isis, 66: 164–193.

• Zik, Yaakov, 2006, “Mathematical Instruments: The Technological Infrastructure of Seventeenth-Century Science”, in M. Folkerts (ed.), Astronomy as a Model for the Sciences in Early Modern Times, Augsburg: Rauner, pp. 355–368."

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Episode 22: The Kepler Problem - The Mechanical Universe

Episode 22. The Kepler Problem: The deduction of Kepler's laws from Newton's universal law of gravitation is one of the crowning achievements of Western thou...

Episode 22: The Kepler Problem - The Mechanical Universe

Episode 22. The Kepler Problem: The deduction of Kepler's laws from Newton's universal law of gravitation is one of the crowning achievements of Western thought.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor."

https://www.youtube.com/watch?v=0tSmY4fn5xY

Images:

1. Kepler's 3 Laws of Planetary Motion

2. The Platonic Solids – Kepler believed these shapes determined how far each of the six known planets lay from the sun.

3. Kepler’s Second Law: a planet orbiting the sun sweeps out equal areas in equal times. In this image, the planet takes the same time to move from A to B as it takes to move from C to D.

4. In 1604, Kepler discovered the inverse square law of light intensity. If you double your distance from the sun, the amount of light reaching you is lowered by a factor of four. If you triple your distance from the sun, the amount of light you get is reduced by a factor of nine.

Background from {[https://www.famousscientists.org/johannes-kepler/]}

"Johannes Kepler

Lived 1571 to 1630.

Johannes Kepler played a key role in the profound changes in human thinking during the scientific revolution. In Kepler’s lifetime:

religion clashed with religion

religion clashed with science

the old ideas of Ptolemy and Aristotle clashed with the new discoveries of Copernicus and Galileo

the superstition of astrology clashed with the science of astronomy

Kepler reflected the times he lived in. Seen through modern eyes, he had somewhat contradictory ideas. He was:

an unorthodox Protestant

a follower of Copernicus and Galileo

a brilliant mathematician and scientist who discovered that the solar system’s planets follow elliptical paths, not circular paths

an astrologer, whose horoscopes were sought out as among the best available

Such contradictions were not unusual during the scientific revolution. Isaac Newton, who lived in a later time than Kepler, (1643 to 1727) did not work in the way a modern scientist would. Also a Protestant with unorthodox views, Newton spent more time investigating the true meaning of the Bible’s words and on the pseudoscience of alchemy than he did on mathematics or physics!

Johannes Kepler’s Early Life and Education

Johannes Kepler was born on December 27, 1571, in the town of Weil der Stadt, which then lay in the Holy Roman Empire, and is now in Germany.

His mother, Katharina Guldenmann, was a herbalist who helped run an inn owned by her father. When Johannes was about five his father Heinrich was killed fighting as a mercenary in Holland.

Physically Damaged, Intellectually Strong

The young Johannes Kepler was prone to ill-health. His hands were crippled and his eyesight permanently impaired by smallpox. Despite these difficulties, guests at his grandfather’s inn were astonished by his ability to solve any problem they brought him involving numbers.

His herbalist mother, Katharina, tried to convey her love of the natural world to her son. She made a point of taking him out at night to show him interesting things in the heavens, including a comet and a lunar eclipse.

In doing so, she set her son on a course that would transform our understanding of the solar system and the universe.

School

Kepler was formally schooled in Latin, the language of academics, lawyers and churchmen throughout Europe. Hoping to become a Protestant minister he attended the Protestant Seminary of Maulbronn.

After successfully completing his studies at Maulbronn he moved to the University of Tübingen. There, although he took courses in Theology, Greek, Hebrew, and Philosophy, it was in Mathematics that he stood out.

He was one of the few students deemed intellectually and mathematically capable of being taught the work of Nicolaus Copernicus. Kepler decided that Copernicus’s heliocentric hypothesis, that the sun lay at the center of the solar system, was correct.

In 1594, aged 23, Kepler became a lecturer in astronomy and mathematics at the Protestant School in the city of Graz, Austria.

Kepler Discovers the Truth about Planets’ Orbits

Kepler thought Copernicus’s heliocentric view of the solar system was right.

His own belief, one which would transform science, was that the sun exerted a force on the planets orbiting it.

In 1596, at age 25, he published a book – Mystery of the Cosmos. His book explained why, logically, the sun lay at the center of the solar system.

Kepler noted that Mercury and Venus always seem to be close to the sun, unlike Mars, Jupiter and Saturn. This is because Mercury and Venus’s orbits are closer to the sun than Earth’s. Kepler said that if the sun and all the planets orbited Earth, there is no reason why Mercury and Venus should always be near the sun.

Kepler Sees the Hand of God

Including Earth, there were six known planets. (Uranus, Neptune and Pluto had not been discovered in Kepler’s era.)

Looking for evidence of ‘God the mathematician,’ Kepler was able to justify six planets and their distances from the sun in terms of the five Platonic solids. These are the five highly symmetrical, regular, 3D solids whose perfect symmetry allows them to be used as dice.

spheres spaced by the five Platonic Solids.

Kepler Seeks Better Data

Thrilled that he had found evidence of God’s hand in the solar system’s design, Kepler now sought better data.

He hoped that more accurate astronomical observations would prove his theory.

By great good fortune, at exactly the same time as Kepler sought better data, one of Europe’s foremost astronomers, Tycho Brahe, was hoping to recruit an assistant to carry out astronomy calculations.

The two started working together, but it was not a match made in heaven! Tycho was notoriously quarrelsome – he had worn a metal nose ever since he lost his real nose in a duel fought over the merits of a particular mathematical formula!

Tycho Brahe’s Data

Early in the year 1600, Kepler moved from Graz to the town of Benatek, which today is in the Czech Republic. There Tycho, who was the imperial mathematician to Holy Roman Emperor Rudolph II, had an observatory.

Tycho gave Kepler access to some of his Mars observations, but not all. The pair fell out and Kepler spent some time in Prague, then Graz, before they resolved their differences and got together again.

During 1601, Kepler carried out calculations of planet movements for Tycho. In October 1601, Tycho died. Kepler replaced him as the imperial mathematician.

Kepler now had unrestricted access to all of Tycho’s astronomical data. With an incredible amount of hard work he began to deduce the laws that govern the movements of the planets. The first law he found is now called his second law!

Kepler’s Second Law – The Law of Areas

Kepler’s Second Law: a planet orbiting the sun sweeps out equal areas in equal times. In this image, the planet takes the same time to move from A to B as it takes to move from C to D. The green and blue shaded areas are equal. The planet speeds up when it is closer to the sun.

From Tycho’s very accurate observations, Kepler discovered that Mars does not move in a perfect circle around the sun.

This was scandalous! Everyone ‘knew’ that heavenly bodies were perfect, traveling along a path whose shape was perfect – the circle.

It took a long time before people came to terms with Kepler’s shocking discovery and started to believe it. All the more so because the planets’ orbits are actually very close to circular.

It is a tribute to both the precision of Tycho’s observations (made, remember without a telescope) and Kepler’s calculating prowess that Kepler was ever able to discover the truth.

Kepler found Mars was moving in an oval-like orbit and that sometimes it was closer to the sun than at others. When close to the sun, it moved faster than when it was farther away.

In 1602, Kepler deduced what came to be called his second law: a line drawn from planet to sun sweeps out equal areas in equal times.

Kepler’s First Law – The Law of Orbits

Kepler tried to figure out the mathematical shape of Mars’s orbit. After about 40 misses, in 1605, he got it right. Mars follows an elliptical path around the sun.

And now he formulated what would become Kepler’s first law: planets orbit the sun in ellipses, with the sun at one focus.

You can draw an ellipse using pencil, paper, string and pushpins. The closer the pins are together, the closer your ellipse will be to circular. If points A and B coincide, you will draw a circle. A and B are the ellipse’s focuses.

Kepler’s Third Law – The Law of Periods

Kepler never gave up his idea that regular polygons determine the orbits of the planets. As a fortunate result of this wrong thinking, he continued calculating and theorizing.

In 1618, his continuing research led to his third law of planetary motion:

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Restated crudely, this law means that if we square the time it takes a planet to complete one orbit around the sun, we’ll find it’s proportional to the planet’s distance from the sun cubed.

Even more crudely: the farther a planet is from the sun, the slower it moves along its orbital path.

Summing up Kepler’s Laws and their Significance

In a 17 year period, when he was aged between 30 and 47, Kepler took Copernicus’s mathematically flawed heliocentric solar system by the scruff of the neck and put it on a rock-solid mathematical footing.

During his own lifetime, the huge significance of his work won little or no recognition.

Later, however, his work provided the platform for Isaac Newton to discover the law of universal gravitation. When Newton (perhaps with double meaning) said that if he had seen further, it was by standing on the shoulders of giants, there can be no doubt that Johannes Kepler was one of the giants.

More of Kepler’s Achievements

The Tides

Although he greatly admired Galileo’s work, Kepler disagreed with him about the cause of tides on Earth. Galileo believed they were caused by the earth spinning. Kepler, correctly, identified that they were caused by the moon.

Kepler wrote about a ‘magnetic force’ that today we call the force of gravity.

The moon… is a mass, akin to the mass of the earth, attracts the waters by a magnetic force, not because they are liquid, but because they possess earthy substance, and so share in the movements of a heavy body.

Optics

Kepler made important discoveries in optics.

Light Intensity

In 1604, Kepler discovered the inverse square law of light intensity.

If you double your distance from the sun, the amount of light reaching you is lowered by a factor of four. If you triple your distance from the sun, the amount of light you get is reduced by a factor of nine.

The consequence of this law is, for example, that because Jupiter is five times farther than Earth is from the sun, each square meter of Jupiter receives only one-twenty-fifth of the sun’s energy compared with a square meter on Earth.

When Newton, guided by Kepler’s Third Law, discovered the law of universal gravitation, he found that gravity also follows an inverse square law. So, if you triple your distance from the sun, the force of the sun’s gravity you feel is divided by nine.

Everything is Upside Down

It was Kepler who discovered that the lenses in our eyes invert images. This means images on our retinas are upside down. Kepler rightly concluded that the inverted image is corrected by our brains.

Designing a Better Telescope

In 1610, Kepler was excited by news from Italy, where Galileo had discovered four moons orbiting Jupiter.

In 1611, Kepler turned his attention to Galileo’s telescope design, improving it by using two convex lenses. This allowed higher magnifications. Kepler’s design is now the standard design for refracting telescopes.

Kepler’s ‘Last Theorem’

Fermat’s Last Theorem, sometimes called Fermat’s Conjecture, took more than 300 years to prove.

Kepler left the world a puzzle that resisted all attempts at formal proof for 400 years: the Kepler Conjecture.

Kepler’s Conjecture looks at how you can pack a bunch of equally sized spheres into the smallest possible space; he says there are two most efficient ways: cubic close packing and hexagonal close packing.

Kepler was right. A formal proof of his conjecture was published in 2015 by Thomas Hales and coworkers.

It is impossible to pack spheres more efficiently than the arrangements shown in the image: cubic close packing and hexagonal close packing. Image original by Cdang, then modified.

It turns out that Kepler’s ideas about packing are useful in modern chemistry, where atoms in metals are arranged in close packed arrangements: for example copper’s atoms are cubic close packed, while magnesium’s atoms are hexagonal close packed.

Logarithms

In 1616, Kepler became aware of John Napier’s invention of logarithms.

Using logarithms, any multiplication can be turned into an addition; any division can be turned into a subtraction.

For example, adding 3.5627685 to 3.7736402 by hand is much easier than multiplying 3654 by 5938. Try it and see! And in Kepler’s day, everything had to be calculated by hand! Consulting a table of logarithms allows you to convert the result of the addition or subtraction into the result of the equivalent multiplication or division.

Napier’s logarithms allowed mathematicians to greatly simplify their calculations. Kepler was overjoyed when he learned of them. And no wonder: he carried out a huge number of calculations in his work, and logarithms speeded everything up enormously.

But Kepler went one step further.

Mathematicians had not been able to understand how logarithms worked. They just saw that they seemed to give the right answers. Kepler realized that if logarithms had no firm mathematical basis, his calculations might be discredited someday.

He overcame this hurdle in the way any genius would – he himself proved how logarithms transform multiplications and divisions into additions and subtractions.

Kepler’s proof allowed all mathematicians to use logarithms with no misgivings.

The End

In Kepler’s time, Central Europe was convulsed with tensions between Catholicism and Protestantism – resulting in the Thirty Years’ War.

Symbolic of the paranoia of these times, Kepler’s nature-loving herbalist mother was arrested in Württemberg in 1620, charged with witchcraft. She was held in prison for over a year, and the tortures she faced if she did not confess to witchcraft were described to her in detail. Kepler himself traveled to Württemberg and successfully defended his mother, finally winning her release.

Kepler moved around a lot during the last few years of his life. His job as a mathematician disappeared and, in an effort to make ends meet, he ended up casting horoscopes for a military commander – General Wallenstein.

Johannes Kepler fell ill and died at age 58, on November 15, 1630 in the German city of Regensburg. He was survived by a son and a daughter from his first marriage, to Barbara Müller, who died rather young. He was also survived by his second wife, Susanna Reuttinger, and two sons and a daughter from that marriage.

Today Kepler’s grave is lost. The graveyard he was buried in was destroyed during religious battles a few years after he was buried.

Although Kepler cast horoscopes, his feelings about what was important in his work are summed up in these words:

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Episode 22. The Kepler Problem: The deduction of Kepler's laws from Newton's universal law of gravitation is one of the crowning achievements of Western thought.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor."

https://www.youtube.com/watch?v=0tSmY4fn5xY

Images:

1. Kepler's 3 Laws of Planetary Motion

2. The Platonic Solids – Kepler believed these shapes determined how far each of the six known planets lay from the sun.

3. Kepler’s Second Law: a planet orbiting the sun sweeps out equal areas in equal times. In this image, the planet takes the same time to move from A to B as it takes to move from C to D.

4. In 1604, Kepler discovered the inverse square law of light intensity. If you double your distance from the sun, the amount of light reaching you is lowered by a factor of four. If you triple your distance from the sun, the amount of light you get is reduced by a factor of nine.

Background from {[https://www.famousscientists.org/johannes-kepler/]}

"Johannes Kepler

Lived 1571 to 1630.

Johannes Kepler played a key role in the profound changes in human thinking during the scientific revolution. In Kepler’s lifetime:

religion clashed with religion

religion clashed with science

the old ideas of Ptolemy and Aristotle clashed with the new discoveries of Copernicus and Galileo

the superstition of astrology clashed with the science of astronomy

Kepler reflected the times he lived in. Seen through modern eyes, he had somewhat contradictory ideas. He was:

an unorthodox Protestant

a follower of Copernicus and Galileo

a brilliant mathematician and scientist who discovered that the solar system’s planets follow elliptical paths, not circular paths

an astrologer, whose horoscopes were sought out as among the best available

Such contradictions were not unusual during the scientific revolution. Isaac Newton, who lived in a later time than Kepler, (1643 to 1727) did not work in the way a modern scientist would. Also a Protestant with unorthodox views, Newton spent more time investigating the true meaning of the Bible’s words and on the pseudoscience of alchemy than he did on mathematics or physics!

Johannes Kepler’s Early Life and Education

Johannes Kepler was born on December 27, 1571, in the town of Weil der Stadt, which then lay in the Holy Roman Empire, and is now in Germany.

His mother, Katharina Guldenmann, was a herbalist who helped run an inn owned by her father. When Johannes was about five his father Heinrich was killed fighting as a mercenary in Holland.

Physically Damaged, Intellectually Strong

The young Johannes Kepler was prone to ill-health. His hands were crippled and his eyesight permanently impaired by smallpox. Despite these difficulties, guests at his grandfather’s inn were astonished by his ability to solve any problem they brought him involving numbers.

His herbalist mother, Katharina, tried to convey her love of the natural world to her son. She made a point of taking him out at night to show him interesting things in the heavens, including a comet and a lunar eclipse.

In doing so, she set her son on a course that would transform our understanding of the solar system and the universe.

School

Kepler was formally schooled in Latin, the language of academics, lawyers and churchmen throughout Europe. Hoping to become a Protestant minister he attended the Protestant Seminary of Maulbronn.

After successfully completing his studies at Maulbronn he moved to the University of Tübingen. There, although he took courses in Theology, Greek, Hebrew, and Philosophy, it was in Mathematics that he stood out.

He was one of the few students deemed intellectually and mathematically capable of being taught the work of Nicolaus Copernicus. Kepler decided that Copernicus’s heliocentric hypothesis, that the sun lay at the center of the solar system, was correct.

In 1594, aged 23, Kepler became a lecturer in astronomy and mathematics at the Protestant School in the city of Graz, Austria.

Kepler Discovers the Truth about Planets’ Orbits

Kepler thought Copernicus’s heliocentric view of the solar system was right.

His own belief, one which would transform science, was that the sun exerted a force on the planets orbiting it.

In 1596, at age 25, he published a book – Mystery of the Cosmos. His book explained why, logically, the sun lay at the center of the solar system.

Kepler noted that Mercury and Venus always seem to be close to the sun, unlike Mars, Jupiter and Saturn. This is because Mercury and Venus’s orbits are closer to the sun than Earth’s. Kepler said that if the sun and all the planets orbited Earth, there is no reason why Mercury and Venus should always be near the sun.

Kepler Sees the Hand of God

Including Earth, there were six known planets. (Uranus, Neptune and Pluto had not been discovered in Kepler’s era.)

Looking for evidence of ‘God the mathematician,’ Kepler was able to justify six planets and their distances from the sun in terms of the five Platonic solids. These are the five highly symmetrical, regular, 3D solids whose perfect symmetry allows them to be used as dice.

spheres spaced by the five Platonic Solids.

Kepler Seeks Better Data

Thrilled that he had found evidence of God’s hand in the solar system’s design, Kepler now sought better data.

He hoped that more accurate astronomical observations would prove his theory.

By great good fortune, at exactly the same time as Kepler sought better data, one of Europe’s foremost astronomers, Tycho Brahe, was hoping to recruit an assistant to carry out astronomy calculations.

The two started working together, but it was not a match made in heaven! Tycho was notoriously quarrelsome – he had worn a metal nose ever since he lost his real nose in a duel fought over the merits of a particular mathematical formula!

Tycho Brahe’s Data

Early in the year 1600, Kepler moved from Graz to the town of Benatek, which today is in the Czech Republic. There Tycho, who was the imperial mathematician to Holy Roman Emperor Rudolph II, had an observatory.

Tycho gave Kepler access to some of his Mars observations, but not all. The pair fell out and Kepler spent some time in Prague, then Graz, before they resolved their differences and got together again.

During 1601, Kepler carried out calculations of planet movements for Tycho. In October 1601, Tycho died. Kepler replaced him as the imperial mathematician.

Kepler now had unrestricted access to all of Tycho’s astronomical data. With an incredible amount of hard work he began to deduce the laws that govern the movements of the planets. The first law he found is now called his second law!

Kepler’s Second Law – The Law of Areas

Kepler’s Second Law: a planet orbiting the sun sweeps out equal areas in equal times. In this image, the planet takes the same time to move from A to B as it takes to move from C to D. The green and blue shaded areas are equal. The planet speeds up when it is closer to the sun.

From Tycho’s very accurate observations, Kepler discovered that Mars does not move in a perfect circle around the sun.

This was scandalous! Everyone ‘knew’ that heavenly bodies were perfect, traveling along a path whose shape was perfect – the circle.

It took a long time before people came to terms with Kepler’s shocking discovery and started to believe it. All the more so because the planets’ orbits are actually very close to circular.

It is a tribute to both the precision of Tycho’s observations (made, remember without a telescope) and Kepler’s calculating prowess that Kepler was ever able to discover the truth.

Kepler found Mars was moving in an oval-like orbit and that sometimes it was closer to the sun than at others. When close to the sun, it moved faster than when it was farther away.

In 1602, Kepler deduced what came to be called his second law: a line drawn from planet to sun sweeps out equal areas in equal times.

Kepler’s First Law – The Law of Orbits

Kepler tried to figure out the mathematical shape of Mars’s orbit. After about 40 misses, in 1605, he got it right. Mars follows an elliptical path around the sun.

And now he formulated what would become Kepler’s first law: planets orbit the sun in ellipses, with the sun at one focus.

You can draw an ellipse using pencil, paper, string and pushpins. The closer the pins are together, the closer your ellipse will be to circular. If points A and B coincide, you will draw a circle. A and B are the ellipse’s focuses.

Kepler’s Third Law – The Law of Periods

Kepler never gave up his idea that regular polygons determine the orbits of the planets. As a fortunate result of this wrong thinking, he continued calculating and theorizing.

In 1618, his continuing research led to his third law of planetary motion:

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Restated crudely, this law means that if we square the time it takes a planet to complete one orbit around the sun, we’ll find it’s proportional to the planet’s distance from the sun cubed.

Even more crudely: the farther a planet is from the sun, the slower it moves along its orbital path.

Summing up Kepler’s Laws and their Significance

In a 17 year period, when he was aged between 30 and 47, Kepler took Copernicus’s mathematically flawed heliocentric solar system by the scruff of the neck and put it on a rock-solid mathematical footing.

During his own lifetime, the huge significance of his work won little or no recognition.

Later, however, his work provided the platform for Isaac Newton to discover the law of universal gravitation. When Newton (perhaps with double meaning) said that if he had seen further, it was by standing on the shoulders of giants, there can be no doubt that Johannes Kepler was one of the giants.

More of Kepler’s Achievements

The Tides

Although he greatly admired Galileo’s work, Kepler disagreed with him about the cause of tides on Earth. Galileo believed they were caused by the earth spinning. Kepler, correctly, identified that they were caused by the moon.

Kepler wrote about a ‘magnetic force’ that today we call the force of gravity.

The moon… is a mass, akin to the mass of the earth, attracts the waters by a magnetic force, not because they are liquid, but because they possess earthy substance, and so share in the movements of a heavy body.

Optics

Kepler made important discoveries in optics.

Light Intensity

In 1604, Kepler discovered the inverse square law of light intensity.

If you double your distance from the sun, the amount of light reaching you is lowered by a factor of four. If you triple your distance from the sun, the amount of light you get is reduced by a factor of nine.

The consequence of this law is, for example, that because Jupiter is five times farther than Earth is from the sun, each square meter of Jupiter receives only one-twenty-fifth of the sun’s energy compared with a square meter on Earth.

When Newton, guided by Kepler’s Third Law, discovered the law of universal gravitation, he found that gravity also follows an inverse square law. So, if you triple your distance from the sun, the force of the sun’s gravity you feel is divided by nine.

Everything is Upside Down

It was Kepler who discovered that the lenses in our eyes invert images. This means images on our retinas are upside down. Kepler rightly concluded that the inverted image is corrected by our brains.

Designing a Better Telescope

In 1610, Kepler was excited by news from Italy, where Galileo had discovered four moons orbiting Jupiter.

In 1611, Kepler turned his attention to Galileo’s telescope design, improving it by using two convex lenses. This allowed higher magnifications. Kepler’s design is now the standard design for refracting telescopes.

Kepler’s ‘Last Theorem’

Fermat’s Last Theorem, sometimes called Fermat’s Conjecture, took more than 300 years to prove.

Kepler left the world a puzzle that resisted all attempts at formal proof for 400 years: the Kepler Conjecture.

Kepler’s Conjecture looks at how you can pack a bunch of equally sized spheres into the smallest possible space; he says there are two most efficient ways: cubic close packing and hexagonal close packing.

Kepler was right. A formal proof of his conjecture was published in 2015 by Thomas Hales and coworkers.

It is impossible to pack spheres more efficiently than the arrangements shown in the image: cubic close packing and hexagonal close packing. Image original by Cdang, then modified.

It turns out that Kepler’s ideas about packing are useful in modern chemistry, where atoms in metals are arranged in close packed arrangements: for example copper’s atoms are cubic close packed, while magnesium’s atoms are hexagonal close packed.

Logarithms

In 1616, Kepler became aware of John Napier’s invention of logarithms.

Using logarithms, any multiplication can be turned into an addition; any division can be turned into a subtraction.

For example, adding 3.5627685 to 3.7736402 by hand is much easier than multiplying 3654 by 5938. Try it and see! And in Kepler’s day, everything had to be calculated by hand! Consulting a table of logarithms allows you to convert the result of the addition or subtraction into the result of the equivalent multiplication or division.

Napier’s logarithms allowed mathematicians to greatly simplify their calculations. Kepler was overjoyed when he learned of them. And no wonder: he carried out a huge number of calculations in his work, and logarithms speeded everything up enormously.

But Kepler went one step further.

Mathematicians had not been able to understand how logarithms worked. They just saw that they seemed to give the right answers. Kepler realized that if logarithms had no firm mathematical basis, his calculations might be discredited someday.

He overcame this hurdle in the way any genius would – he himself proved how logarithms transform multiplications and divisions into additions and subtractions.

Kepler’s proof allowed all mathematicians to use logarithms with no misgivings.

The End

In Kepler’s time, Central Europe was convulsed with tensions between Catholicism and Protestantism – resulting in the Thirty Years’ War.

Symbolic of the paranoia of these times, Kepler’s nature-loving herbalist mother was arrested in Württemberg in 1620, charged with witchcraft. She was held in prison for over a year, and the tortures she faced if she did not confess to witchcraft were described to her in detail. Kepler himself traveled to Württemberg and successfully defended his mother, finally winning her release.

Kepler moved around a lot during the last few years of his life. His job as a mathematician disappeared and, in an effort to make ends meet, he ended up casting horoscopes for a military commander – General Wallenstein.

Johannes Kepler fell ill and died at age 58, on November 15, 1630 in the German city of Regensburg. He was survived by a son and a daughter from his first marriage, to Barbara Müller, who died rather young. He was also survived by his second wife, Susanna Reuttinger, and two sons and a daughter from that marriage.

Today Kepler’s grave is lost. The graveyard he was buried in was destroyed during religious battles a few years after he was buried.

Although Kepler cast horoscopes, his feelings about what was important in his work are summed up in these words:

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Tycho Brahe, Johannes Kepler and Planetary Motion

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun. 1. The orbit of a planet is an ell...

Tycho Brahe, Johannes Kepler and Planetary Motion

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are almost circles, and careful observation and calculation is required in order to establish that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Johannes Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, also have elliptical orbits.

Kepler's work improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation.

Kepler's laws are part of the foundation of modern astronomy and physics.

Tycho Brahe (14 December 1546 – 24 October 1601), born Tycho Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations. He was born in Scania, then part of Denmark, now part of modern-day Sweden. Tycho was well known in his lifetime as an astronomer, astrologer and alchemist, and has been described more recently as "the first competent mind in modern astronomy to feel ardently the passion for exact empirical facts.

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer. A key figure in the 17th century scientific revolution, he is best known for his laws of planetary motion, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation."

https://www.youtube.com/watch?v=x3ALuycrCwI

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In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are almost circles, and careful observation and calculation is required in order to establish that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Johannes Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, also have elliptical orbits.

Kepler's work improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation.

Kepler's laws are part of the foundation of modern astronomy and physics.

Tycho Brahe (14 December 1546 – 24 October 1601), born Tycho Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations. He was born in Scania, then part of Denmark, now part of modern-day Sweden. Tycho was well known in his lifetime as an astronomer, astrologer and alchemist, and has been described more recently as "the first competent mind in modern astronomy to feel ardently the passion for exact empirical facts.

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer. A key figure in the 17th century scientific revolution, he is best known for his laws of planetary motion, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation."

https://www.youtube.com/watch?v=x3ALuycrCwI

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