Johannes Kepler (1571–1630) is one of the most significant representatives of the so-called Scientific Revolution of the 16 th and 17 th 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. Johannes Kepler was born on December 27, 1571 to Heinrich Kepler and Katharina Guldenmann in the Stuttgart region of Germany. His family was believed to be very wealthy but by the time Kepler was born, the wealth in the family had declined drastically.
Johannes Kepler (1571–1630) is one of the most significantrepresentatives of the so-called Scientific Revolution of the16th and 17th centuries. Although he receivedonly the basic training of a “magister” and wasprofessionally oriented towards theology at the beginning of hiscareer, he rapidly became known for his mathematical skills andtheoretical creativity. As a convinced Copernican, Kepler was able todefend the new system on different fronts: against the old astronomerswho still sustained the system of Ptolemy, against the Aristoteliannatural philosophers, against the followers of the new “mixedsystem” of Tycho Brahe—whom Kepler succeeded as ImperialMathematician in Prague—and even against the standard Copernicanposition according to which the new system was to be considered merelyas a computational device and not necessarily a physicalreality. Kepler's complete corpus can be hardly summarized as a“system” of ideas like scholastic philosophy or the newCartesian systems which arose in the second half of the 17thcentury. Nevertheless, it is possible to identify two main tendencies,one linked to Platonism and giving priority to the role of geometry inthe structure of the world, the other connected with the Aristoteliantradition and accentuating the role of experience and causality inepistemology. While he attained immortal fame in astronomy because ofhis three planetary laws, Kepler also made fundamental contributionsin the fields of optics and mathematics. To the little-known factsregarding Kepler’s indefatigable scientific activity belong hisefforts to develop different technical devices, for instance, a waterpump, which he tried to patent and apply in different practicalcontexts (for the documents, see KGW 21.2.2., pp. 509–57 and667–691). To his contemporaries he was also a famousmathematician and astrologer; for his own part, he wanted to beconsidered a philosopher who investigated the innermost structure ofthe cosmos scientifically.
- 4. Epistemology and philosophy of sciences
1. Life and Works
Johannes Kepler was born on December 27, 1571 in Weil der Stadt, alittle town near Stuttgart in Württemberg in southwestern Germany.Unlike his father Heinrich, who was a soldier and mercenary, his motherKatharina was able to foster Kepler's intellectual interests. Hewas educated in Swabia; firstly, at the schools Leonberg (1576),Adelberg (1584) and Maulbronn (1586); later, thanks to support for aplace in the famous Tübinger Stift, at theUniversity of Tübingen. Here, Kepler became MagisterArtium (1591) before he began his studies in the TheologicalFaculty. At Tübingen, where he received a solid education inlanguages and in science, he met Michael Maestlin, who introduced himto the new world system of Copernicus (see MysteriumCosmographicum, trans. Duncan, p. 63, and KGW 20.1, VI, pp.144–180).
Before concluding his theology studies at Tübingen, inMarch/April 1594 Kepler accepted an offer to teach mathematics as thesuccessor to Georg Stadius at the Protestant school in Graz (inStyria, Austria). During this period (1594–1600), he composedmany official calendars and prognostications and published his firstsignificant work, theMysterium Cosmographicum (= MC), which catapulted him to fameovernight. On April 27, 1597 Kepler married his first wife, BarbaraMüller von Mühleck. As a consequence of the anti-Protestantatmosphere in Graz and thanks also to the positive impact of his MC onthe scientific community, he abandoned Graz and moved to Prague in1600, to work under the supervision of the great Danish astronomerTycho 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 October1601 Kepler succeeded him as Imperial Mathematician. During his timein Prague, Kepler was particularly productive. He completed his mostimportant optical works, Astronomiae parsoptica(=APO) and Dioptrice (=D), published several treatises onastrology (De fundamentis astrologiaecertioribus, Antwort auf Roeslini Diskurs; Tertiusinterveniens), discussed Galileo's telescopic discoveries(Dissertatio cum nuncio sidereo), and composed his mostsignificant astronomical work, theAstronomia nova (=AN), which contains his first two laws ofplanetary motion.
On August 3, 1611 Kepler's wife, Barbara Müller, died.In 1612 he moved to Linz, in Upper Austria, and became a professor atthe Landschaftsschule. There, he served as Mathematician of the UpperAustrian Estates from 1612 to 1628. In 1613, he married SusanneReuttinger, with whom he had six children. In 1615, he completed themathematical works Stereometria doliorum and MessekunstArchimedis. At the end of 1617 Kepler successfully defended hismother, who had been accused of witchcraft. In 1619 he published hisprincipal philosophical work, the Harmonice mundi (=HM), andwrote, partially at the same time, the Epitome astronomiaecopernicanae (=EAC). In 1624 Kepler continued his investigationson mathematics, publishing his work on logarithms (Chiliaslogarithmorum…).
In pursuit of an accurate printer for the TabulaeRudolphinae, he moved to Ulm near the end of 1626 and remainedthere until the end of 1627. In July 1628 he went to Sagan to enter theservice of Albrecht von Wallenstein (1583–1634). He died on November15, 1630 at Regensburg, where he was to present his financial claimsbefore the imperial authorities (for Kepler's life,Caspar's biography (1993) is still the best work. KGW 19contains biographically relevant documents).
2. Philosophy, theology, cosmology
There is probably no such thing as “Kepler'sphilosophy” in any pure form. Nevertheless, many attempts to dealwith the “philosophy of Kepler” have been made, all ofwhich are very valuable in their own way. Some studies haveconcentrated on a particular text (see, for instance, Jardine 1988, forthe Defense of Tycho against Ursus), or have followed someparticular ideas of Kepler over a longer period of his life andscientific career (see, for instance, Martens 2000, on Kepler'stheory of the archetypes). Others have tried to determine from aphilosophical point of view his place in the development of theastronomical revolution from the 15th to the 17thcenturies (Koyré 1957 and 1961) or in the more general contextof the scientific movement of the 17th century (Hall, 1963and especially Burtt, 1924). Still others have discussed a long list ofphilosophical 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 ofKepler, philosophy is immediately related to astronomy, mathematicsand, finally, “cosmology” (a notion which arises muchlater), the core of these speculations is to be sought in the spectrumof problems with which he dealt in his MysteriumCosmographicum and Harmonice Mundi (on this topic, Field1988 is one of the most representative works on Kepler). In addition,because of the particular circumstances of his life and his fascinatingpersonality and genius, the literature on Kepler is extremelywide-ranging, covering a spectrum from literary pieces like MaxBrod's Tycho Brahes Weg zu Gott (1915)—thoughstill not free from mistakes concerning Kepler—to generalintroductions in the genre of historical novels, and even fictionalstories 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 complicatedtechnical issues, especially in astronomy, but he always emphasizedhis philosophical, even theological, approach to the questions hedealt with: God manifests himself not only in the words of theScriptures but also in the wonderful arrangement of the universe andin its conformity with the human intellect. Thus, astronomy representsfor Kepler, if done philosophically, the best path to God (seeHübner 1975; Methuen 1998 and 2009; Jardine 2009). As Kepler atthe core of his greatest astronomical work confesses (AN, Part II,chap. 7, KGW 6, p. 108, Engl. trans., p. 183), at the beginning ofhis 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 anddifferent astronomical data, Kepler writes in a letter of February17, 1619 to V. Bianchi: “I also ask you, my friends, that you donot condemn me to the treadmill of mathematical calculations; allow metime 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 thecosmos, he always returns to his Platonic and Neoplatonic framework ofthought. Thus, the polyhedral hypothesis (see Section 3 below) hepostulated for the first time in his MC represents a kind of“formal cause” constituting the foundational structure ofthe universe. In addition, an “efficient cause,” whichrealizes this structure in the corporeal world, is also needed. Thisis, of course, God the Creator, who accomplished His work according tothe model of the five regular polyhedra. Kepler reinterprets thetraditional statements about the Creation as an image of the Creatorgiving to the ancient ideas a more systematic and a quantitativecharacter. Even the doctrine of the Trinity could be geometricallyrepresented, taking the center for the Father, the spherical surfacefor the Son, and the intermediate space, which is mathematicallyexpressed in the regularity of the relationship between the point andthe surface, for the Holy Spirit. In Kepler's model, we have to beable to reduce all appearances to straightness and curvature asproviding the foundation for the geometrical structure of theworld's creation. The very first category, through which Godproduced a fundamental similitude of the created World to Himself, isthat of quantity (see MC, chapter 2, KGW pp. 23–26). Furthermore,quantity was also introduced into the human soul for the specificpurpose that this fundamental symmetry could be apprehended and knownscientifically.
This kind of speculation also belongs to the basic principles ofKepler's philosophical optics. In Chapter 1 of APO (“Onthe Nature of Light”), Kepler gives a new account of this“Trinitarian Cosmogony.” As he admits in a letter toThomas Harriot (1560–1621), his approach here is more theological thanoptical (KGW 15, let. 394, p. 348, lin. 18). Similar to hisspeculations in MC, Kepler explains again the symmetry between God andthe Creation, but now he goes slightly beyond the limits of atheologico-geometrical reflection. Firstly, he seems to assume that thebodies of the world were provided in the Creation with some powerswhich enable them to exceed their geometrical limits and to act onother bodies (magnetic power is a good example of this). Secondly, theprinciple of symmetry introduced into matter constitutes “themost excellent thing in the whole corporeal world, the matrix of theanimate faculties, and the chain linking the corporeal and spiritualworld” (APO, Engl. trans., p. 19). Thirdly, as expressed byKepler in a wonderful, long Latin sentence, with multiplesubordinations, this principle “has passed over into the samelaws (in leges easdem) by which the world was to befurnished” (ibid., p. 20; the original passage is in KGW 2, p.19: the marginal note in the edition is “lucisencomium”). Finally, these reflections are concluded with aremark, in which—as with Copernicus, Marsilio Ficino, andothers—the central position of the Sun is legitimated becauseof its function in spreading light and, indirectly, life. Similarspeculations are still present in EAC (KGW 7, pp. 47–48 and 267). It isalso worth noting that these speculations are of vital importance tothe special way in which Kepler conceived of astrology (see, forinstance, De fundamentis astrologiae certioribus with Engl.trans. and commentary in Field 1984).
3. The five regular solids
Philosophical, geometrical and even theological speculations relatedto the five regular polyhedra, the cube or hexahedron, thetetrahedron, the octahedron, the icosahedron and the dodecahedron,were known at least from the time of the ancient Pythagoreans. SincePlato's Timaeus, these five geometrical solids played aleading role, and for the later tradition they became known as the“five Platonic solids”.
Figure 1: Table 3 in Mysterium Cosmographicum, with Kepler'smodel illustrating the intercalation of the five regular solidsbetween the imaginary spheres of the planets (cf. KGW 1, pp. 26–27).
Plato establishes at the physical and chemical level acorrespondence between them and the five elements—earth, water,air, fire and ether—and tries to provide this correspondencewith geometrical grounds. A further source of historically decisiveimportance is the fact that the five regular polyhedra are treated inEuclid's Elements of Geometry, a work that for Kepler,especially in the Platonic approach of Proclus, has a centralposition. At the very beginning of HM Kepler complains about the factthat the modern philosophical and mathematical school of Peter Ramus(1515–1572) had not been able to understand the architectonicstructure of the Elements, which are crowned with thetreatment of the five regular polyhedra. In addition, a revival ofPlatonic philosophy was taking place in Kepler's time and inspired notonly philosophers and mathematicians but also architects, artists andillustrators (see Field 1997).
Amidst this general interest in the regular polyhedra during theRenaissance, Kepler was specifically concerned with their applicationin the resolution of a cosmological problem, namely the reality of theCopernican system (see Section 5 below). To achieve this goal, heintroduced his polyhedral hypothesis already in MC, where he lookedfor an “a priori” foundation of the Copernican system (seeAiton 1977, Di Liscia 2009). The background for such an approach seemsto be that the “a posteriori” way, which according toKepler was taken over by Copernicus himself, cannot lead to anecessary affirmation of the reality of the new world system, but onlyto a probable, and hence to an “instrumental”,representation of it as a computational device. This is the“Osiander” or “Wittenberg Interpretation” ofCopernicus which Kepler directly attacked not only in his MC but alsolater in his AN (see Westman 1972 and 1975). In MC, he claimed to havefound an answer to the following three main questions: 1) the numberof the planets; 2) the size of the orbits, i.e., the distances; 3) thevelocities of the planets in their orbits. By referring to thepolyhedral hypothesis (see Figure 1), Kepler found a definitive andsimple answer to the first question. By intercalating the polyhedrabetween the spheres which carry the planets, one must inevitablyfinish with the sphere of Saturn surrounding the cube—there areno more polyhedra to be intercalated and, as remains the standard casefor Renaissance and modern astronomy, there are no more planets to becarried by the spheres. It is absolutely decisive for the consistencyof the argument that the necessity of the hypothesis is guaranteed bythe fact that it already exists as a mathematical demonstration (byEuclid, Elements XIII, prop. 18, schol.), according to whichthere are only five regular (“Platonic” for the tradition)polyhedra. For the second and third questions the answer is, ofcourse, not as evident as for the first one. However, Kepler was ableto show that distances which are derived from the geometrical model ofthe five regular bodies fit much better with the Copernican systemthan with that of Ptolemy. The answer to the third question needs, inaddition, the introduction of a notion of power that emanates from theSun and extends to the outer limit of the universe (see Stephenson1987, pp. 9–20).
In HM, Kepler continues his investigations of the polyhedra at acosmological as well as a mathematical level. In the second book,dealing with “congruence” (which here does not mean, as itdoes today, “the same size and shape”, but the property offigures filling the surface together with other regular polygons– on the plane—or, to build closed geometric solids– hence, in space), Kepler made new mathematical discoveriesworking 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 twoprincipal pillars of HM, Book 5 (chapters 1–2), where the third law isformulated (see Caspar in KGW 6, Nachbericht, p. 497, and Engl. trans.p. xxxiii). Following his approach from MC and complementing it withoccasional references to his EAC, Kepler makes use again of thePlatonic solids to determine the number of the planets and theirdistances from the Sun. Meanwhile, he has learned that the applicationof his old polyhedral hypothesis has limits. As he tells us in afootnote in the second edition of his MC from 1621, he was earlierconvinced of the possibility of explaining the eccentricities of theplanetary orbits by values derived “a priori” from thishypothesis (MC, Engl. trans., p. 189). Now, with access to theobservational data of Tycho, Kepler had to exclude this explanation andlook for another. And this is one of the most significant achievementsof his basic harmonies (which in turn are derived from the regularpolygons), constituting the second great pillar of Book 5.
4. Epistemology and philosophy of sciences
Almost all of Kepler's scientific investigations reflect aphilosophical background, and many of his philosophical questions findtheir final answer, even if they are of scientific interest, in therealm of theology. From a very modern point of view, one couldhighlight Kepler's epistemological thought in terms of fourdifferent items: realism; causality; his philosophy of mathematics; andhis—own particular—empiricism.
Realism is a constant and integral part of Kepler's thought,and one which appears in sophisticated form from the outset. The reasonfor this is that his realism always runs parallel to his defense of theCopernican worldview, which appeared from his first publicpronouncements and publications.
Many of Kepler's thoughts about epistemology can be found inhis Defense of Tycho against Ursus or Contra Ursum(=CU), a work which emerged from a polemical framework, the plagiarismconflict between Nicolaus Raimarus Ursus (1551–1600) and Tycho Brahe:causality and physicalization of astronomical theories, the concept andstatus of astronomical hypotheses, the polemic“realism-instrumentalism”, his criticism of skepticism ingeneral, the epistemological role of history, etc. It is one of themost significant works ever written on this subject and is sometimescompared with Bacon's Novum organum andDescartes' Discourse on Method (Jardine 1988, p. 5; foran excellent new edition and complete study of this work see Jardine /Segonds 2008).
The focus of the epistemological issues could be ranked mutatismutandi with modern discussion surrounding the scientific statusof astronomical theories (however, as Jardine has pointed out, it wouldbe sounder to read Kepler's CU more as a work against skepticismthan in the context of the modern realism/instrumentalism polemic). ForPierre Duhem (1861–1916), for instance, the position of AndreasOsiander, which was adopted by Ursus and which was, according to Duhem,naively criticized by Kepler in his MC, represents the modern approachknown as “instrumentalism”. According to thisepistemological position, held by Duhem himself, scientific theoriesare 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 tooffer a causal explanation or to study the causes of phenomena, butsimply to represent them. In the best-case scenario, theories are ableto order and classify what is decisive for their predictive capacity(Duhem 1908, 1914).
Contrary to Tycho and Kepler, Ursus held a fictionalist position inastronomy. Yet in the very beginning of his work Dehypothesibus, Ursus makes a clear declaration about the nature ofastronomical theories, which is very similar to the approach suggestedby Osiander in his forward to Copernicus' Derevolutionibus: a hypothesis is a “fictitioussupposition”, introduced just for the sake of “saving themotions of the heavenly bodies” and to “calculatethem” (trans. Jardine 1988, p. 41)
Following his approach in MC and anticipating the opening pages ofhis later AN (see particularly AN, II.21: “Why, and to whatextent, may a false hypothesis yield the truth?” Engl. trans.,pp. 294–301), Kepler addresses the question of Copernicanism and itsreception by thinkers such as Osiander, who emphasized that the truthof astronomical hypotheses cannot necessarily be deduced from thecorrect prediction of astronomical facts. According to this interpretation,Copernican hypotheses are not necessarily true even if they are able tosave the phenomena, otherwise one would commit a fallaciaaffirmationis consequentis. However, according to Kepler,“this happens only by chance and not always, but only when theerror in the one proposition meets another proposition, whether true orfalse, 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 naturenor the human mind are independent of God's mind (Jardine 2005,p. 137).
The reality of astronomical hypotheses—and hence thesuperiority of the Copernican world system—implied aphysicalization of astronomical theories and, in turn, an accentuationof causality. Despite Kepler's criticism of Aristotle, thisaspect can actually be considered the realization in the field ofastronomy of the old Aristotelian ideal of knowledge:“knowledge” means to grasp the causes of thephenomena.
Thus, on the one hand, “causality” is a notion implyingthe most general idea of “actual scientific knowledge”which guides and stimulates each investigation. In this sense, Kepleralready embarked in his MC on a causal investigation by asking for thecause 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, theconcrete “physical cause”, the efficient cause whichproduces a motion or is responsible for keeping the body in motion.Original to Kepler, however, and typical of his approach is theresoluteness with which he was convinced that the problem ofequipollence of the astronomical hypotheses can be resolved and theconsequent introduction of the concept of causality into astronomy– traditionally a mathematical science. This approach is alreadypresent in his MC, where he, for instance, relates for the first timethe distances of the planets to a powerwhich emerges fromthe Sun and decreases in proportion to the distance of eachplanet, 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 hereplaced the “mean Sun” of Copernicus with the real Sun,which was no longer merely a geometrical point but a body capable ofphysically influencing the surrounding planets. In addition, in notesto the 1621 edition of MC Kepler strongly criticizes the notion of“soul” (anima) as a dynamical factor in planetarymotion 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'sAstronomia Nova from 1609 (=AN) is its methodologicalapproach and its causal foundation (see Mittelstrass 1972). Kepler wassufficiently conscious of the change of perspective he was introducinginto astronomy. Hence, he decided to announce this in the full titleof 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 orCelestial Physics Treated by Means of Commentaries on the Motions ofthe Star Mars from the observations of Tycho Brahe …(trans. Donahue). In the introduction to AN Kepler insists on hisradical change of view: his work is about physics, not purekinematical or geometrical astronomy. “Physics”, as in thetraditional, Aristotelian understanding of the discipline, deals withthe causes of phenomena, and for Kepler that constitutes his ultimateapproach to deciding between rival hypotheses (AN, Engl. trans.,p. 48; see Krafft 1991). On the other hand, since his celestialphysics uses not only geometrical axioms but also other,non-mathematical axioms, the knowledge obtained often has a kernel ofguesswork.
In the third part of AN, chapters 22–40, Kepler deals with the pathof the Earth and intends to offer a physical account of the Copernicantheory. By so doing he includes the idea that a certain notion of powershould be made responsible for the regulation of the differences invelocities of the planets, which in turn have to be established inrelation to the planets' distances. Now, the Copernicanplanetary theory departs from the general principle that the Earthmoves regularly on an eccentric circle. For Kepler, on the contrary,the planets are moved irregularly, and the slower they are moved, thegreater their distance is from the center of power, the Sun. Addressingthe physical aspects of his new astronomy, he deals in chapters 32–40,perhaps the most idiosyncratic of the work, with his notion of motivepower. Here, he combines different approaches and sources, sometimesproducing—for the purpose of simplifying the whole geometricalconstruction of geometrical astronomy by introducing a power causingmotions—a new confusion at the dynamical level. To begin with,it is not always absolutely clear what kind of power Kepler has inmind. He inclines, above all, to the idea of a magnetic power residingin the Sun, but he also mentions light and, at least indirectly,gravity (which he does not bring into operation in the central chaptersof the AN but which is to a certain extent implied in his explanationsusing the model of the balance and which he surely accepts as true forthe 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, “asif” (particularly in the case of light). Essentially, Keplerbreaks down the motions of the planets into two components. On the onehand, the planets move around the Sun—at this state of thediscussion—circularly. On the other hand, they exhibit alibration on the Sun-planet vector. The rotation of the Sun isresponsible for the motion of the planets. Irradiating from therotating Sun is a power which spreads at the ecliptic plain. This powerdiminishes with distance to the source of the power, that is, to theSun. A decisive work for Kepler's development in his physicalastronomy is William Gilbert's (1544–1603) De magnete(London, 1600), a work which also intends to offer a new physics forthe new Copernican cosmology and which surely influencedKepler's thoughts about this power. One of the main problems was,of course, how to apply the general principles of magnetism toplanetary motion, first to explain the difference in velocity on acircular path, and later to give an account of the motion on anellipse. Kepler conceives of a model with parallel magnetic fiberswhich links the Sun with the planets in such a way that the rotation ofthe Sun causes the motion of the planets around it. The fibers are bornin the planets parallel and perpendicular to the lines of apsides by akind of “animal power”. The planets themselves arepolarized, that is, with one pole they are attracted to the Sun, withthe other pole they are pushed away from it. This explains very wellthe direction of planetary motion: the planets all move in onedirection because the Sun rotates in that direction.Nevertheless, a further problem still seems to remain unresolved:according to Kepler's explication, the planets should movearound the Sun as fast as the Sun itself rotates, which is not thecase. This phenomenon can be explained by referring to a property ofmatter, which for Kepler has an axiomatic character: the inclinatioad quietem, that is, the tendency to rest (see especially AN,chap. 39; KGW 3, p. 256). As a consequence, the planets are movedaround the Sun slower than they would be if the power of the Sun wereat work alone.
Kepler's causal approach is above all present in hisEpitome, a voluminous work which exercised a considerableinfluence on the later development of astronomy. In the second part ofBook 4, he deals with the motion of the world's parts. Not thetwo first laws but rather the third law, which he had recentlyannounced 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 moreconvenient for his approach here. At the same time, it should bepointed out that the third law is not necessarily the best point ofdeparture for a dynamical, causal approach to motion, as Kepler intendshere; for, in comparison with the previous causal approaches, thequestion of the location of the cause of power responsible for theproduction of motion remains relevant. The spheres, which in thetraditional view transported the planets, had been abolished since thetime of Tycho. Furthermore, Kepler is clearly against the“moving intelligences” of the Aristotelian tradition. Thefact that the orbits are elliptical and not circular, shows that themotions are not caused by a spiritual power but rather by a naturalone, which is internal to the composition of matter. The planetsthemselves are provided with “inertia”, aproperty, as Kepler understood it, that inhibits motion and representsan impediment to it. The motive power (vix motrix) comesindeed from the Sun, which sends its rays of light and power in alldirections. These rays are captured by the planets. Kepler, however,tries to explain this behavior of the planets less through astrologyand much more through magnetism (a physical phenomenon which was by nomeans clearly understood in his time). Firstly, the Sun rotates and, byso doing, sets in motion the planets around it. Secondly, since theplanets are poles of magnets and the Sun itself acts with magneticpower, the planets are, at different parts of their orbits, eitherattracted or repelled; in this way the elliptical path is causallyproduced. Kepler partially gives up the mechanical approach bypostulating a soul in the Sun which is responsible for its regularmotion of rotation, a motion on which, finally, the entire systemdepends. In fact, the planets are also supposed by Kepler to rotate andare therefore provided with “a sort of soul” or some suchprinciple which produces the rotation.
In addition to astronomy and cosmology Kepler expanded his causalapproach to include the fields of optics (see Section 6 below) andharmonics (Section 7 below).
4.3 Philosophy of mathematics
Beyond his own original talent, it is clear that Kepler was trained inmathematics from his earliest studies at Tübingen. At leastofficially, his positions at Graz, Prague, Linz, Ulm and Sagan can becharacterized as the typical professional occupations of amathematician in the broadest sense, i.e., including astrology andastronomy, theoretical mechanics and pneumatics, metrology, and everytopic that could in some way be related to mathematics. Besides thefield of astronomy and optics, where mathematics is ordinarily appliedin different ways, Kepler produced original contributions to thetheory of logarithms and above all within his favorite field, geometry(especially with his stereometrical investigations). Thus, on accountof his natural predilection and talent and the importance ofmathematics, particularly of geometry, for his thought, it is notsurprising to find many different passages in his works where hearticulated his philosophy of mathematics. However, Kepler's principalexposition on this topic is to be found in his HM, a work in which thefirst two books are purely mathematical in content. As he himselfdeclares, in HM he played the role “not of a geometer inphilosophy but of a philosopher in this part of geometry” (KGW6, p. 20, Engl. trans., p. 14).
While in philosophical questions related to mathematics, Proclus andPlato were Kepler's most important inspirational sources, he did notalways see Plato and Aristotle as completely opposed, for thelatter—in Kepler's interpretation—also accepted “acertain existence of the mathematical entities” (KGW 14,let. N° 226, p. 265; see Peters, p. 130). To a great extent Keplerunderstood his mathematical investigations of HM as a continuation ofEuclid's Elements, especially of the analysis ofirrationalities in Book 10. The central notion that he works out hereis that of “constructability”. According to Kepler, eachbranch of knowledge must, in principle, be reducible to geometry if itis to be accepted as knowledge in the strong sense (although, in thecase of the physics, this condition is, as the AN emphasizes, only anecessary and not a sufficient condition). Thus, the new principleshe was elaborating over the years in astrology were geometricalones. A similar case occurs with the basic notions of harmony, which,after Kepler, could be reduced to geometry. Of course, not everygeometrical statement is equally relevant and equally fundamental. ForKepler, the geometrical entities, principles and propositions whichare especially fundamental are those that can be constructed in theclassical sense, i.e., using only ruler (without measurement units)and compass. On this are based further notions according to differentdegrees of “knowability” (scibilitas), whichbegins with the circle and its diameter. Once again, Kepler understoodthis within the framework of his cosmological and theologicalphilosophy: geometry, and especially geometrically constructibleentities, have a higher meaning than other kinds of knowledge becauseGod has used them to delineate and to create this perfect harmonicworld. From this point of view, it is clear that Kepler defends aPlatonist conception of mathematics, that he cannot assume theAristotelian theory of abstraction and that he is not able to acceptalgebra, 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 safegeometrical knowledge. The best example of this is perhaps theheptagon. This figure cannot be described outside of the circle, andin the circle its sides have, of course, a determinate magnitude, butthis is not knowable. Kepler himself says that this is importantbecause here he finds the explanation for why God did not use suchfigures to structure the world. Consequently, he devotes many pages todiscussing the issue (KGW 6, Prop. 45, pp. 47–56, see also KGW9, p. 147). Certainly for a geometer like Kepler, approximationsconstitute – as mathematical theory—a painful andprecarious way to progress. The philosophical background for hisrejection of algebra seems to be, at least partially, Aristotelian insome of its basic suppositions: geometrical quantities are continuousquantities which therefore cannot be treated with numbers that are, inthe inverse, discrete quantities. But the difference from theAristotelian ideal of science remains an important one: for Aristotle,a crossover between arithmetic and geometry is allowed only in thecase of the “middle sciences”, while for Kepler allknowledge must be reduced to its geometrical foundations.
A general presentation of Kepler's philosophical attitude andprinciples is not complete without reference to his link to the worldof experience. For, despite his mainly theoretical approach in thenatural sciences, Kepler often emphasized the significance ofexperience and, in general, of empirical data. In his correspondencethere are many remarks about the significance of observation andexperience, as for instance in a letter to Herwart von Hohenburg from1598 (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 twoof his most important correspondents. Looking for empirical support forthe Copernican system, Kepler compares different astronomical tables inhis MC, and in AN he makes extensive use of Tycho'sobservational treasure trove. In MC (chapter 18) he quotes a longpassage from Rheticus for the sake of rhetorical support when, as wasthe case here, the data of the tables he used did not fit perfectlywith the calculated values from the polyhedral hypothesis. In thispassage, the reader learns that the great Copernicus, whose worldsystem Kepler defends in MC, said one day to Rheticus that it made nosense to insist on absolute agreement with the data, because thesethemselves were surely not perfect. After all, it is questionablewhether Kepler, using for instance the Prutenic Tables (1551)of Erasmus Reinhold (1511–1553), had access to complete and correctempirical information to confirm the Copernican hypothesis in grandstyle, as he claimed (for an analysis of Reihold's tables andtheir influence see Gingerich 1993, pp. 205–255).
The situation changed completely when Kepler came into contact atPrague with Tycho's observations (which, as Kepler oftenreports, were seldom at his disposal). However, a change of attitude isevident in AN, where he used Tycho's observations withoutrestriction (which is something he makes clear in the work'stitle). In part 2 (chap. 7–21), he presents the “vicarioushypothesis”, which in the end he refutes. This hypothesisrepresents the best result which can be reached within the limits oftraditional astronomy. This works with circular orbits and with thesupposition that the motion of a planet appears regular from a point onthe lines of apsides. Against the traditional method, here, Kepler doesnot cut the eccentricity into equal parts but leaves the partitionopen. To check his hypothesis, he needs observations of Mars inopposition, where Mars, the Earth, and the Sun are at midnight on thesame line. From Tycho, he “inherited” ten suchobservations between the years 1580 and 1600, and to them he addedanother two for 1602 and 1604. In chapters 17–21, Kepler carries out anobservational and computational check of his vicarious hypothesis. Onthe one hand, he points out that this hypothesis is good enough, sincethe variations of the calculated positions from the observed positionsfall within the limits of acceptability (2 minutes of arc). In fact,Kepler presents this hypothesis as the best hypothesis which can beproposed within the framework of a “traditionalastronomy”, as opposed to his new astronomy, which he will offerin the following parts of the work. On the other hand, this hypothesiscan be falsified if one takes the observations of the latitudes intoconsideration. Further calculations with these observations produce adifference of eight minutes, something that cannot be assumed becausethe observations of Tycho are reliable enough. Kepler's famoussentence runs: “these eight minutes alone will have led the wayto the reformation of all of astronomy” (AN, KGW 3, p. 286; Engl.trans., p. 286). There seems to be agreement that Kepler's ANcontains the first explicit consideration of the problem ofobservational error (for this question see Hon 1987 and Field2005).
Kepler also gave an important place to experience in the field ofoptics. As a matter of fact, he began his research on optics because ofa disagreement between theory and observation, and he made use ofscientific instruments he had designed himself (see, for instance, KGW21.1, p. 244). Recent research on the problem of the cameraobscura and the “images in the air” shows, however,the limits of a traditional approach to Kepler's opticsfollowing the main current of the history of physics. Rather, hisnotion of experimentum needs to be contextualized within thesocial practices and epistemological commitments of his time (seeDupré 2008).
Finally, it should be mentioned that a similar significance isassigned to experience and empirical data in Kepler'sharmonic-musical and astrological theories, two fields which aresubordinated to his greater cosmological project of HM. For astrology,he uses meteorological data, which he recorded for many years, asconfirmation material. This material shows that the Earth, as a wholeliving being, reacts to the aspects which occur regularly in theheavens. In his musical theory Kepler was a modern thinker, especiallybecause of the role he gave to experience. As has been noted (Walker,1978, p. 48), Kepler made acoustic experiments with a monochord longbefore 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 atdifferent lengths, establishing in which cases the ear judges the soundto be pleasurable. Kepler does not accept that this limitation isfounded on arithmetical speculations, even if this was already assumedby Plato, whom he often follows, and by the Pythagoreans. On the basisof his experiments, Kepler found that there are other divisions of thestring that the ear perceives as consonant, i.e., thirds andsixths.
If cosmology is the main framework of Kepler's interest,there is no doubt that, as Field has pointed out, he “felt theneed 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 forhis three planetary laws, which he produced in very specific contextsand at different times. While it is questionable whether he would haveunderstood these scientific statements as “laws”—andit is even arguable that he used this term with a different meaningthan we do today—it seems to be clear that all three laws (as alinguistic convention, we may continue to use the term) suppose somefundamentals of Kepler's philosophy: (a) realism, (b) causality, (c)the geometric structure of the cosmos. Besides this, it should beremarked that the common denominator of all three laws is Kepler'sdefense of the Copernican worldview, a cosmological system which hewas not able to defend without reforming it radically. It isnoteworthy that already at the very beginning of his career Keplervehemently defended the reality of the Copernican worldview in a waythat he characterized, taking over the terminology from the standardAristotelian epistemology, as “a priori” (see aboveSection 3 above and Di Liscia 2009).
Figure 2. Kepler's first law of ellipse andsecond law of areas (modern representation withgreatly exaggerated eccentricity).
The first two laws were published initially in AN (1609), although itis known that Kepler had arrived at these results much earlier. Hisfirst law establishes that the orbit of a planet is an ellipse withthe Sun in one of the foci (see Figure 2). According to the secondlaw, the radius vector from the Sun to a planet P sweeps out equalareas, for instance SP1P2and SP3P4,in equal times. Theplanet P is therefore faster at perihelion, where it iscloser to the Sun, and slower at aphelion, where it is farther fromthe Sun. In accordance with his dynamical approach, Kepler firstfound the second law and, then, as a further result because of theeffect produced by the supposed force, the elliptical path of theplanets (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 canbe found by considering their cosmological consequences. The first lawabolishes the old axiom of the circular orbits of the planets, an axiomwhich was still valid not only for pre-Copernican astronomy andcosmology 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 thisdifficulty and applied a particularly effective device for saving the“appearance” of acceleration: the equant. Copernicus, forhis own part, insisted on the necessity of the axiom of uniformcircular motion. Ptolemy's equant was understood by Copernicusas a technical device based on the violation of this axiom. Kepler, onthe contrary, affirms the reality of changes in the velocities of theplanetary motions and provides a physical account for them. Afterstruggling strenuously with established ideas which were located notonly in the tradition before him but also in his own thinking, Keplerabandoned the circular path of planetary motion and in this wayinitiated a more empirical approach to cosmology (though seeBrackenridge 1982).
Kepler published the third law, the so-called “harmoniclaw”, for the first time in his Harmonice mundi (1619),i.e., ten years later. In his Epitome, he provided a moresystematic approach to all three laws, their grounds and implications(see Davis 2003; Stephenson 1987). In Book 5, chapter 3, as point 8 of13 (KGW 6, p. 302; Engl. trans., pp. 411–12), Kepler expresses, almostaccidentally, his fundamental relationship connecting elapsed timeswith distances, which in modern notation could be expressedas:
(T1/T2)2 = (a1/a2)3
with T1 and T2 representingthe periodic times of two planets and a1and a2 the length of their semi-major axes. Afurther formulation of this relationship, which is often found in theliterature, is: a3/T2= K, which expresses with K that the relationshipbetween the third power of the distances and the square of the timesis a constant (however, see Davis 2005, pp. 171–172; for the thirdplanetary law see especially Stephenson 1987). As a consequence of thethird law, the time a planet takes to travel around the Sun willsignificantly increase the farther away it is or the longer the radiusof its orbit. Thus, for instance, Saturn's sidereal period isalmost 30 years, while Mercury needs fewer than 88 days to go aroundthe Sun. For the history of cosmology, it is important to make clearthat the third law fulfils Kepler's search for a systematicrepresentation and defense of the Copernican worldview, in whichplanets are not absolutely independent of each other but integrated ina harmonic world system.
6. Optics and metaphysics of light
Kepler contributed to the special field of optics with two seminalworks, the AdVitellionem paralipomena (=APO) and theDioptrice (=DI), the latter motivated in large part by thepublication in 1610 of Galileo's Sidereal Messenger(Sidereus Nuncius). In his Conversation with the SiderealMessenger (Dissertatio cum Nuncio Sidereo … aGalillaeo Galilaeo, KGW 4, pp. 281–311), he supported the factualinformation given by Galileo, indicating at the same time the necessityof giving an account of the causes of the observed phenomena. Thebackground for his investigation into optics was undoubtedly thedifferent particular questions of astronomical optics (see Straker1971). In this context he concentrated his efforts on an explanation ofthe phenomena of eclipses, of the apparent size of the Moon and ofatmospheric refraction. Kepler investigated the theory of thecamera obscura very early and recorded its general principles(see commentary by M. Hammer in KGW 2, pp. 400–1 and Straker 1981). Inaddition, he worked intensively on the theory of the telescope andinvented the refracting astronomical or ‘Keplerian’telescope, which involved a considerable improvement over the Galileantelescope (see especially DI, Problem 86, KGW 4, pp. 387–88). Besidesthese impressive contributions, Kepler expanded his research program toembrace mathematics as well as anatomy, discussing for instance conicsections and explaining the process of vision (see Crombie 1991 andespecially Lindberg 1976b).
In Chapter 1 of APO (“On the Nature of Light”), Keplerexpounds 38 propositions concerning different properties of light:light flows in all directions from every point of a body'ssurface; it has no matter, weight, or resistance. Following—butalso inverting—the Aristotelian argument for the temporality ofmotion, he affirms that the motion of light takes place not in time butin an instant (in momento). Light is propagated by straightlines (rays), which are not light itself but its motion. It isimportant 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 toa curved surface. The two-dimensionality of light is probably the mainreason why it is incorporeal. Motion in general plays a significantrole in Kepler's philosophy of light. For Straker, the supposedlink between optics and physics (especially in Prop. 20, where themechanical analyses are introduced) “reveals the full extent ofhis 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 toKepler justified? Secondly, how should one determine his place in thehistory of sciences, especially in the field of optics: do the mainlines of thought in Kepler's optics indicate acontinuity or rather a rupture with tradition? Thereare well–grounded arguments for different positions on bothquestions. For Crombie (1967, 1991) and Straker, Kepler develops amechanical approach, which can be particularly appreciated in hisexplanation of vision using the model of the cameraobscura. Besides this, Straker stresses that Kepler's basicmechanicism is also powerfully assisted by his conception of light asa non-active, passive entity. In addition, the concept of motion andthe explanations using the model of the balance are indicative of acommitment to mechanicism (Straker 1970, pp. 502–3). On thecontrary, Lindberg (1976a), who supports the “continuityside” of the dispute, has quite convincingly showed that, forKepler, light has a constructive and active function in the universe,not only in optics but also in astrology, astronomy, and naturalphilosophy (for Kepler's criticism of the medieval tradition see alsoChen-Morris / Unguru, 2001).
7. Harmony and Soul
From a philosophical point of view, Kepler considered the HM to behis main work and the one he most cherished. Containing his thirdplanetary law, this work represents definitively a seminal contributionto the history of astronomy. But he did not reduce his long preparedproject to an astronomical investigation—his first thoughts onthe notion of “harmony” arose already in 1599, although hedid not publish his work until 1619—but instead extensivelydiscussed its mathematical foundations and its philosophicalimplications, including astrology, natural philosophy and psychology.Thus, Kepler's third planetary law appears in a context whichgoes far beyond astronomy and to a great extent takes up again theperspective of his youthful MC.
According to Kepler, it is necessary to distinguish“sensible” from “pure” harmony. The first isto be found among natural, sensible entities, like sounds in music orrays of light; both could be in proportion to one another and hence inharmony. He resolves this matter by combining three of the Aristoteliancategories: quantity, relation and, finally, quality. Through thefunction of the category of relation Kepler passes over to the activefunction of the mind (or soul). It turns out that two things can becharacterized as harmonic if they can be compared according to thecategory of quantity. But the fact that at least two things are neededshows that the property of “being harmonic” is not aproperty of an isolated thing. Furthermore, the relationship betweenthe things cannot be found in the things themselves either; rather, itis produced by the mind: “in general every relation is nothingwithout mind apart from the things which it relates, because they donot have the relation which they are said to have unless the presenceof some mind is assumed, to relate one to another” (KGW 6, p.212; Engl. trans., p. 291). This process takes place through thecomparison of different sensible things with an archetype(archetypus) present in the mind.
The next central question directly concerns gnoseology, for Keplergives a psychological account of the path followed by sensible thingsinto the mind. He resumes the scholastic species theory: immaterialspecies radiate from the sensible things and affect the sense organsby acting firstly on the “forecourts” and then on theinternal functions. They arrive at the imagination and from there goover to the sensus communis, so that, according to thetraditional teaching, the sensible information received is now able tobe processed and used in statements. From here onwards, the sensiblethings are “preserved in the memory, brought forth byrecollection, [and] distinguished by the higher faculty of thesoul” (Caspar 1993, p. 269, cf. KGW 6, p. 214, lines 18–23;Engl. trans., p. 293). While harmony arises as an activity of thesoul/mind consisting in relating quantitatively, Kepler adds, takingover 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 becompared, it by no means represents—as Kepler admitted—asufficient explanation of how non-sensible things, i.e., mathematicalentities, find their way into the mind. How do they come into thesoul? Kepler accepted Aristotle's criticism of Pythagoreanphilosophy concerning numbers: both Kepler and Aristotle are convincedthat numbers constitute ontologically a lower class withinmathematical entities (for Kepler, they are derived from geometricalentities). Nevertheless, Aristotle's philosophy is insufficientto grasp the essence of mathematics. By aligning himself with Proclus,from whom he quotes a long passage of his Commentary onEuclides, Kepler defends Plato's theoryof anamnêsis against Aristotle's doctrine ofthe tabula rasa. His discussion lies at the origin of theclassical debate between empiricism and rationalism which was todominate the philosophical scene for generations to come. A connectionwith idealism is, of course, apparent (see, for instance, Caspar 1993,Engl. trans., p. 269), and it is a fact that Kepler was positivelyreceived within German Idealism of the 19thcentury. Historically, however, it seems to be more accurate to linkhis position with the philosophical tradition of St. Augustine.
Besides psychology and gnoseology, the other main spectrum ofquestions Kepler deals with in Book 4 is his theory of“aspects”, i.e., astrology (HM, IV, Cap. 4–7), a furtherfield of application of his psychology and further evidence of the roleof geometry in his philosophy. The “aspects”, i.e., theangles between the planets, Moon and Sun, are all he wishes to savefrom the old astrology, which he harshly criticizes; for the aspectsare or can be reduced to geometrical structures, the archetypes, whichcan be recognized by the soul. According to Kepler, there is no“mechanical influence” of the heavens (stars andconstellations are not relevant in his astrology) which exerts adetermining effect on the Earth and on human life. Rather, both theEarth and human beings, ultimately, like all other living entities, areprovided with a soul in which the geometrical archetypes are present.By the formation of an aspect in the heavens, symmetry arises andstimulates the soul of the Earth or of human beings. “TheEarth,” Kepler writes, responds to “what the aspectswhistle” (KGW 11.2, p. 48; for his astrology see especially Field1984 and Rabin 1997, Boner 2005 and 2006).
A. Primary Sources
- 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 imAuftrag der Deutschen Forschungsgemeinschaft und derBayerischen Akademie der Wissenschaften, unter der Leitung vonWalther 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 Defenseof Tycho against Ursus with Essays on its Provenance andSignificance, Cambridge: Cambridge University Press, 1984 (withcorrections 1988).
- Ad Vitellionem paralipomena. Astronomiae parsoptica: 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 deObservatis Jovis Satellitibus: Trans. E. Rosen,Kepler's Conversation with Galileo's SideralMessenger, New York and London: Johnson Reprint Corporation,1965.
- De fundamentis astrologiae certioribus: = OnGiving Astrology Sounder Foundations: Trans. J.V. Field, “ALutheran Astrologer: Johannes Kepler”, Archive for History ofExact Sciences, 31/3 (1984): 189–272.
- Astronomia nova: Trans. W. H. Donahue, NewAstronomy, Cambridge, New York: University Press, 1992; andSelections from Kepler's Astronomia Nova: A ScienceClassics 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 PhilosophicalSociety), 1997.
- Epitome astronomiae copernicanae: Partial trans. C.G. Wallis, Epitome of Copernican Astronomy: IV and V,Chicago, London: Encyclopaedia Britannica (Great Books of the WesternWorld, 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 LunarAstronomy, 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 AreaLaw”, in F. Krafft, K. Meyer, and B. Sticker (eds.), 1973:Internationales Kepler-Symposium Weil der Stadt 1971,Hildesheim: Gerstenberg, pp. 285–305.
- –––, 1975a, “How Kepler discovered theelliptical orbit”, Mathematical Gazette, 59/410:250–260.
- –––, 1975b, “Johannes Kepler and theastronomy without hypotheses”, Japanese Studies in theHistory of Science, 14: 49–71.
- –––, 1977, “Kepler and theMysterium Cosmographicum”, Sudhoffs Archiv,61/2: 173–194.
- –––, 1978, “Kepler's path tothe construction and rejection of his first oval orbit for Mars”,Annals of Science, 35/2: 173–190.
- Barker, P., and B.R. Goldstein, 1994, “Distance andVelocity in Kepler‘s Astronomy”, Annals of Science,51: 59–73.
- –––, 1998, “Realism andInstrumentalism in Sixteenth Century Astronomy: A Reappraisal”,Perspective on Sciences, 6/3: 232–258.
- –––, 2001, “Theological Foundationsof Kepler's Astronomy”, Osiris, 16:88–113.
- Bialas, V., 1990, “Keplers komplizierter Weg zur Wahrheit:von neuen Schwierigkeiten, die ‘Astronomia Nova’ zulesen”, Berichte zur Wissensschaftsgeschichte, 13/3:167–176.
- Beer, A., and P. Beer (eds.), 1975, Kepler Four HundredYears, Oxford: Pergamon Press.
- Bennett, B, 1998, “Kepler's Response to theMystery: 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: ErwinRauner Verlag.
- Boner, P.J., 2005, “Soul-searching with Kepler: Ananalysis of Anima, in his astrology”, Journal for theHistory of Astronomy36/1: 7–20.
- –––, 2006, “Kepler's LivingCosmology: 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, ellipticalorbits, and celestial circularity: a study in the persistence ofmetaphysical commitment. I”, Annals of Science39/2: 117–143 and II in Annals of Science,39/3: 265–295.
- Buchdahl, G., 1972, “Methodological aspects ofKepler's theory of refraction”, Studies in History andPhilosophy of Science, 3: 265–298.
- Burtt, E.A., 1924, The Metaphysical Foundations ofModern Physical Science, New York: Kegan Paul; 2nd rev. ed.,1954, Garden City, NY: Doubleday, reprinted 2003, New York: DoverPublications.
- Caspar, M., 1993, Johannes Kepler, NewYork: Dover Publications.
- Chen-Morris, R.D., and Unguru, Sabetai, 2001, “Kepler'sCritique of the Medieval Perspectivist Tradition”, in Simon& Débarbat 2001, pp. 83–92.
- Crombie, J.A., 1967, “The MechanisticHypothesis, and the Study of Vision: Some Optical Ideas as aBackground to the Invention of the Microscope”, in S. Bradbury andG.L.E. Turner (eds.), Historical Aspects of Microscopy,Cambridge: Heffer (for the Royal Microscopical Society), pp. 3–112.
- –––, 1991, “Expectation, modellingand 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 aNew Physics, New York and London: W.W. Norton.
- Davis, A.E.L., 1975, “Systems of Conics in Kepler'sWork”, Vistas in Astronomy, 18:673–685.
- –––, 1992a, “Kepler's resolution ofindividual planetary motion”, Centaurus,35/2: 97–102.
- –––, 1992b, “Kepler's ‘distancelaw’ – 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 toDamascus”, Centaurus, 35/2: 143–164.
- –––, 1992e, “Kepler's physicalframework for planetary motion”, Centaurus,35/2: 165–191.
- –––, 1998a, “Kepler's unintentionalellipse—a celestial detective story”, TheMathematical Gazette, 82/493: 37–43.
- –––, 1998b, “Kepler, the ultimateAristotelian”, Acta historiae rerum naturalium necnontechnicarum, 2: 65–73.
- –––, 2003, “The Mathematics of theArea Law: Kepler's successful proof in Epitome AstronomiaeCopernicanae, (1621)”, Archive for History of ExactSciences, 57/5: 355–393.
- –––, 2005, “Kepler's AngularMeasure of Uniformity: how it provided a potential proof of his ThirdLaw”, in F. Boockmann, D.A. Di Liscia, and H. Kothmann (eds.),pp. 157–173.
- –––, 2010, “Kepler's Astronomia Nova: ageometrical success story,” in A. Hadravová, T.J. Mahoneyand P. Hadrava (eds.), Kepler's Heritage in the Space Age,(Acta historiae rerum naturalium necnon technicarum, Volume10), Prague: National Technical Museum.
- Di Liscia, Daniel A., 2009, “Kepler's APriori Copernicanism in his MysteriumCosmographicum,” in M.A. Granada and E. Mehl (eds.),pp. 283–317.
- Donahue, William, 1994, “Kepler's invention of thesecond planetary law”, British Journal for the History ofScience, 27/92: 89–102.
- Dreyer, J.L.E., 1963, Tycho Brahe, A Pictureof Scientific Life and Work in the Sixteenth Century, New York:Dover Publications.
- Duhem, P., 1908, SOZEIN TA, FAINOMENA: Essai sur lanotion de théorie physique de Platon àGalilée, Paris: A. Hermann.
- –––, 1914, La théorie physique,son objet et sa structure, Paris: Chevalier etRivière.
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Other Internet Resources
Johannes Kepler Astrology Images
- Kepler Biography, MacTutor, University of St. Andrews.
- Biography of Johannes Kepler, The Galileo Project, Rice University.
- Kepler's Planetary Laws, by A.E.L. Davis (Imperial College London), in Historical Topics at MacTutor, University of St. Andrews.
- Planetary Motion Tackled Kinematically,by A.E.L. Davis (Imperial College London), at MacTutor, University of St. Andrews.
Aristotle Copernicus, Nicolaus cosmology: and theology Galileo Galilei Pythagoreanism
Support from the Université de Versailles-Saint-Quentin enYvelines (ESR Moyen âge/Temps modernes, Prof. Emmanuel Bury),where the author was a Prof. Invité from February 1, 2010 toMay 30, 2010, made it possible for him to complete this entry. Theauthor also wishes to thank David T. McAuliffe, and his colleaguePatrick J. Boner for their suggestions as to how to improve the textlinguistically. The editors of the SEP wish to thank Sheila Rabin andJill Kraye, respectively, for their outstanding efforts in refereeingand editing this work.
Johannes Kepler Horoscope
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Daniel A. Di Liscia<[email protected]>