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The life and activity of Kepler coincided with an important period in the history of astronomy, when every decade new instruments were being invented, bringing about many great discoveries. At this time many scientists and artisans participated in the construction of astronomical instruments. The contribution made by them is fairly well known, but Kepler's work in this field has been little investigated. However, in this domain he has done no less than others and in some respects surpassed them and even forestalled future developments. In Kepler's time great advances were made in the construction of astronomical instruments for measurements of angles. A large triquetrum proved to be well adapted for Kepler's purpose, though not exceeding in precision Tycho Brahe's instruments. A device constructed by Kepler for the observation of the Moon deserves attention. An invaluable contribution to observational astronomy was made by Kepler when he designed his telescope. The scheme proposed by him, consisting of a biconvex object lens and a biconvex eyepiece, determined for centuries the progress of instruments for observations and angular measurements. Establishing, on the basis of Copernicus' work, the true structure of the solar system, Kepler opened a new page in the field of practical modelling of armillary spheres, planetaria and telluria. Thanks to his initiative many such instruments were made. Kepler himself proposed in 1596 and 1624–1625 two original models of planetaria, and his investigations were also used in the design of celestial globes.

Kepler's manuscript heritage is one of the most remarkable collections of documents on the history of science, and is the pride of the Archives of the USSR Academy of Sciences. It comprises an almost complete collection of his manuscripts, as well as vast quantities of notes, preliminary materials, and manuscripts by other scientists. It includes Kepler's correspondence with many scientists and other eminent contemporaries. Other documents have been added later, including some belonging to former owners of the collection. There are also several family papers, treasured through two and a half centuries by successive generations of Kepler's descendants. The Archives also contain other material relevant to the study of Kepler's work, the history of the manuscript collection, and the publication of his works, particularly the Opera Omnia edited by Professor Ch. Frisch in 1858–1871. The present note gives the facts on the history of the Keplerian manuscript heritage and of its scientific use. The most important steps in the work of description and study of these papers, particularly that carried out in the Archives of the Academy are described.

Galileo and Kepler both continued the work begun by Copernicus, Galileo through his astronomical observations and the popularization of Copernicus' ideas, and Kepler through the development of these ideas. Their connections were quite close in 1610 when Kepler helped Galileo in his struggle for “Nuncius Sidereus”, and in 1618 (“the case of three comets”) when Galileo and Kepler took up different positions. Tycho Brahe's observations of the apparition of previous comets demonstrated the absence of noticeable parallaxes and also the cosmical origin of comets. Orazio Grassi, Professor of the “Collegium Romanum”, came to the same conclusions. Scipio Chiaramonti from Cesena wrote against Tycho Brahe and Grassi. Kepler supported Tycho; his attacks on Chiaramonti were right in essence but very disagreeable in form, a fact which Kepler later regretted. Galileo and his pupil Mario Guiducci were opponents of Tycho Brahe and of the Collegium Romanum. They defended the earthly origin of comets. Galileo's theory was wrong but his defence in the famous “Saggiatore” was remarkable for the general statement which stressed that the study of Nature was more valuable than the bookish wisdom of authorities.

A recently found printed funeral sermon for the alchemist and physician Martin Ruland II (1569–1611) is accompanied by an elegy by Kepler to his deceased friend Martin. This entirely unknown and unnoticed poem of Kepler's led to a closer study of his alchemical-chemical interests. The main sources for this research were Kepler's correspondence with his “alchemical friends”, especially Martin Ruland II and Joachim Tanckius. Also by considering his published scientific works it is possible to see that, although Kepler did not himself experiment in the laboratory, he was excellently informed about the experiments of the contemporary alchemists (e.g. Tycho Brahe at the Court of Rudolf II in Prague). As a strong opponent to the magical, cabbalistic currents in alchemy —e.g. Robert Fludd—Kepler's alchemical knowledge glimmers through his ideas about natural philosophy.

Kepler's theology stands between the various fronts in a time of religious turmoil. Educated in the Lutheran faith, throughout his life Kepler felt himself a member of the Lutheran Church. However, he refuted the teaching of the universal presence of the Body of Christ which is at the core of the orthodox Lutheran teaching. On this point he agrees with Calvinism, though he could not accept the core of the latter's theology, the doctrine of predestination. He searched for religious unity, but again he could not accept the hierarchical structure of the Roman Church. Thus he devoted himself to the “Book of Nature”, looking on his work in this field as the service of a priest. In this way natural science gained for him the character of a theology of nature. Natural science serves for the praise of the Creator—and indirectly for the salvation of souls. Theology now has to develop a new language to make its teaching more understandable in our time.

This paper, published by courtesy of the Royal Astronomical Society, is a revised version of “Heavenly harmony and earthly harmonics”, delivered as the 1971 Harold Jeffreys lecture of the Royal Astronomical Society, on 10 December 1971 at the meeting which marked the fourth centenary of the birth of Johannes Kepler in December 1571. Two separate topics are discussed. The first is the numerical relationship between the orbital distances of the planets, the “heavenly harmony” which Kepler found so fascinating. The second topic is the study of the Earth's gravitational field and shape by analysis of satellite orbits, and in particular the determination of high-order harmonics in the geopotential by analysis of harmonious, or resonant, orbits which have orbital periods such that the satellite follows the same set of ground tracks over the Earth day after day.

The following text consists of some extracts from a lecture on Kepler's life given by Professor Sutter at the Akademischer Festakt arranged by the Province of Styria and its capital Graz. The full lecture is published under the title Johannes Keplers Lebensweg by Verlag J. A. Kienreich, Graz, 1972 (47 pp.).

In Kepler's day, explanation of planetary orbits meant either: showing that they followed from the basic properties of matter, or: analysing them into uniform circular elements. These explanatory “schemas” were independent of each other; each investigator used one or the other but not both. Neither, by itself, could have led to the discovery of Kepler's laws. Kepler succeeded because of his unique capacity to work with both mathematical and physical schemas simultaneously. To reach his goal he had to work out for himself a new and essentially modern concept of scientific method. It seems likely that if he had not succeeded in his task, none of his contemporaries or successors for at least a generation would have done so.

Every generation writes the history of science anew, selecting with partiality the topics especially close to its own mentality, interests and views, as if projecting the difficulties and problems of the science of its period into the past. On the pages of ancient treatises, through a phraseology and archaic argumentation that seem strange to us, we unexpectedly find the outlines of familiar ideas, detect the echo of doubts worrying us, take notice of the course of reasoning which is similar to ours, and then the authors of these old volumes that seemed something like bronze statues suddenly come to life, descend from their pedestals and their ideas become close and comprehensible. In our report we want to outline the parallels between the circle of ideas that were interesting for Johann Kepler and nourished his scientific genius, and the views which considerably determine the outlook of modern physicists. In spite of the erroneous but nevertheless widely spread opinion Kepler enters the history of science not only as the discoverer of three famous laws of planetary motions. His scientific legacy is extremely rich but very complicated and contradictory. The main motive of the whole of Kepler's activity, i.e. the search for the “world harmony”, has recently gained special significance. Here in this short note we want to emphasize this point. As is known, logically irreproachable reasoning never leads to any discovery that overthrows our notions referring to the structure of the world. Only by overthrowing the circle of the customary ideas can we enlarge our knowledge of nature. In our days such a disrupture of the well-established canons was due after Niels Bohr's “mad idea”. Many topics which are quite clear and understandable for us were, in the distant epoch when Kepler lived and worked, mad ideas. Mad was the very thought of the possibility of mathematical description of nature, of the necessity of comparison of theory with experiment, of physical causes of motion, of the unity of the world, i.e. of the universal laws which govern different events both on Earth and in the sky, of the rejection of the circular motions and the substitution of them by the elliptical ones. Yet the most “mad” among Keplerian ideas was the idea of the “world harmony”. Changing and developing it brought Kepler to his most important discoveries. In it we see the forerunner of Einstein's “cosmic religious feeling”, that is the feeling of the universal objective order which underlies different phenomena. In it we trace the deepest analogue to modern views on the role of the symmetry principles in the system of the laws of Nature. Both Kepler and modern physicists impart symmetry with a two-fold function. It is symmetry that plays a part in a principle which enables them to fill the gap in reasoning and to predict and describe a phenomenon even before it is properly understood. At the same time symmetry comes forward as the final aim of the search for new laws of nature. The difference is that a modern physicist uses the symmetry principle together with other methods of knowledge and has no intention to consider them as absolute, while Kepler saw in them the highest truth, “the manifestation of the wisdom of the creation”. Nevertheless, disrespectful references to Kepler, which sometimes can be heard, as astrologer and even as a mediocre scientist who was able to achieve his outstanding results only through hard work and industry, are thoroughly erroneous. Kepler and the modern physicist have much in common in their approach to the investigation of the regularities underlying the structure of the world. In fact, the modern physicist trying to embrace a vast amount of experimental data in the compact form of a conservation law, to discover the underlying symmetry and to unite numerous conservation laws into one universal law, solves the problem which like the well-known two-body problem can be called a “Keplerian problem”.

BETWEEN August and October 1971 many conferences in honor of Kepler were held across Europe? Several of these symposia (reviewed in this paper) took place in a number of cities where Kepler lived and worked and where many others have since labored to uncover the significance of his writings: in Weil der Stadt, Kepler's birthplace; in Linz, where he was district mathematician from 1612 to 1626; in Leningrad, where Kepler's manuscripts were brought in the eighteenth century by Catherine the Great; in London, at a meeting of the British Society for the History of Science held at the Warburg Institute; and in Zag~in, where Kepler spent the last two years of his life in great loneliness before his death in Regensburg. Special displays of Kepleriana could be seen not only at the various symposia, but also in Prague, where Kepler succeeded Tycho Brahe as Court Astronomer to the Emperor Rudolph II and where he composed his great Astronomiae pars opticae (1604) and the Astronomia Nova (1609); and in Tiibingen, the site of Kepler's university training, his work with Michael M/istlin and his adoption of the Copernican theory. A series of lectures on Kepler's life were given at Graz, location of the young astronomer's

Kepler's Six-cornered Snowflake (1611) is the first work on the problem of crystal structure. Despite its modest size it is remarkably rich in ideas. One of Kepler's major discoveries is the geometry of the packing of spheres (as is well known, the principle of closest packing is basic to modern structural crystallography). Kepler dealt with the densest cubical packing (F), and described two less dense packings for spheres—the hexagonal (H) and the simple cubic (P). He was not aware of the densest hexagonal packing. Starting from the spherical packings, Kepler drew conclusions about the parallelohedra—the convex polyhedra which can fill space in a regular manner. In this he anticipates the conclusions of R. J. Hauy (1784) and E. S. Fedorov (1885). Kepler discovered three of Fedorov's four perfect parallelohedra—the cube, the hexagonal prism, with a pinacoid, and the rhombic dodecahedron. He only omitted one, a particular combination of cube and octahedron. Kepler's work contains indirect pointers to the law of constant angles for a six-sided snow crystal. Thus one can see Kepler as a forerunner of the discoverers of the law (N. Steno, 1669; M. W. Lomonosov, 1749; Rome de l'Isle, 1783). We note particularly Kepler's ideas on the dependence of all natural shapes on the formative power of the Earth; in this we see him as one of the first predecessors of Pierre Curie and his universal principle of symmetry (1894).

The name of Kepler heads the list of those associated with the development of a whole new period in mathematics—the period of variable quantities. At the very beginning of the seventeenth century his investigations of the laws of planetary motion led him into the difficult and important area of integral calculation. Though he was not able to perform integrations as such, he treated the new concept in terms of the area bounded by a curve, and developed procedures that come near to the modern methods of numerical integration. Nor was this all; he developed these methods and used them in his Nova Stereometria to determine the volume of various quite complicated solids of revolution. The questions of differential calculation were not ignored either: in the same book Kepler deals with extrema and the isoperimeter problem. In his work on optics, Kepler was the first to introduce into geometry the principle of continuity and the concept of a point at infinity—an important step towards the creation of projective geometry. His work on the theory of polygons and polyhedra was also noteworthy, particularly his discovery of the regular star-shaped polyhedra and of the space-filling property of these and other regular convex solids. The very scope of the calculations which he performed in his many years of search for the planetary laws is sufficient to earn Kepler his place as the greatest master of calculation of his times. While carrying out this arduous task he turned his creative ability to the perfection of methods and aids that would speed up the work. He was among the first to work out an original theory of logarithm construction, and he prepared logarithmic tables not unlike those of Napier, though not identical with them. He was indeed the first to use logarithmic calculation in important astronomical researches. His tables outlived those of Burgi and Napier, and were reissued in 1700. In the 12,700 pages of Kepler manuscript material preserved in the archives of the USSR Academy of Sciences it can be seen that Kepler was not only helper but very probably initiator in the invention of the first calculating machine by his friend, the Tubingen Professor W. Schickard. In the same manuscript collection his German text on the logarithmic calculation, his “Weinvisierbuchlein”, and other writings all show his substantial contribution to the development of German-language scientific literature. Kepler's views on the nature of mathematical concepts are reflected in his manuscript (also in the Academy's collection) “De quantitatibus Libelli”. Parts of this also supplement our knowledge of Kepler's contribution to the development of infinitesimal methods and other problems.