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The fundamental concepts and principles of mechanics, or dynamics, were established in the 17th century. Copernicus gave the notion of reference system in 1543, and Galileo stated the principle of inertia in 1638 in his important work Discorsi e dimostrazioni mathematiche intorno a due nove scienze alla meccanica ed i movimenti locali.1 A particle on which no force is exerted has a constant velocity. Linear uniform motion is a state relative to the observer, and not a process. It is the variation of the velocity that is a process resulting from an external action. Many scientists participated in this evolution, such as Tycho Brahe, Kepler, Descartes, and Christiaan Huygens, to name a few.

The speculation that there could be extraterrestrial life began soon after Copernicus demoted the Earth to the status of just one of the planets. Galileo’s 1610 telescopic observation of the lunar surface led Kepler to suggest that the circular features seen on the Moon might be artificial constructions by lunar inhabitants. John Wilkins (1614–1672), Master of Trinity College, Cambridge and Bishop of Chester, wrote in his 1638 book The Discovery of a World in the Moon that the Moon may be inhabited by intelligent beings, and speculated on future journeys to the Moon and encounters with a lunar civilization. William Herschel was a firm believer in life on the Moon. In 1780, he wrote “… the almost absolute certainty, of her (the Moon) being inhabited” and observed features on the Moon were the “works of the lunarians and may be called their towns.” The idea that the Moon is populated by life remained popular well into the nineteenth century. An 1835 news report in the New York Sun, wrongly attributed to observations by John Herschel, reported the observation of forests, fields, beaches, and even winged creatures on the Moon. The Scottish astronomer Thomas Dick (1774–1857), in his 1838 book Celestial Scenery, estimated that there were over 22 trillion inhabitants on planets of the Solar System.

In this text we present a collection of various inventions in Geometry, from Euclid to Grothendieck and Thom, passing by Kepler, Euler, Riemann and Poincare, and we indicate for all of them a deep link with the organization and functions of the brains of animals, especially humans. In particular, we present works conducted with Tamar Flash and Alain Berthoz, and many other collaborators, using a variety of geometries, all included in the affine geometry, which are combined to prepare, plan and control the execution of human movements, for drawing or for walking. We also present a recent model, elaborated with David Rudrauf, Kenneth Williford, and other collaborators, showing that the three-dimensional projective geometry underlines a space for consciousness. Additionally, we introduce a new kind of geometry developed with Alain Berthoz, and many collaborators, which is able to organize different action spaces (at least five); this geometry corresponds to a 2-category of fibered categories of geometrical spaces over several networks in the brain, seen as different sites for different topos. The unifying theme of these works is adaptation.

Kepler showed that the planets orbit in ellipses around the Sun at one focus. In this chapter, an account of his life and work is given, and the groundwork is laid to enable the reader to understand the geometry of ellipses. Kepler became Tycho’s assistant, and inherited his data, which he spent many years analysing to develop an answer to the crisis in cosmology caused by Tycho’s critique of the theories of Ptolemy and Copernicus. He was the first astronomer to think in terms of physical causes of orbits as well as to describe them. Equations are derived for an ellipse in polar coordinates centred on one focus.

The seventeenth-century scientists began to develop mechanics as a branch of natural philosophy. In contradistinction to Aristotle, these philosophers were strongly driven by the quantitative measurement of phenomena. They based their theories on measurements. We first look at Christiaan Huyghens, who gave us centripetal force and the first accurate clock. Then, we move to Isaac Newton, who built on the work of Tycho, Kepler, Galileo, Huyghens and Isaac Barrow to produce a tour de force that gave a systematic account of mechanics using the new arts of differential and integral calculus that he did so much to develop. Newton’s theory of the Solar System is described in great detail, focusing on planetary orbits. The mathematics is made as easy to follow as possible.

NumPy arrays are the workhorses of numerics in Python, extending it by remarkable numerical capabilities. For example, they can be used just like simple variables to evaluate an arithmetic expression for many different values without programming a loop. In the first section, we combine the power of NumPy and Astropy and compute the positions of objects on the celestial sphere. Moreover, we introduce Matplotlib to produce plots from array data. Further applications are shown in the context of Kepler’s laws and tidal forces, for example, printing formatted tables and plotting vector maps.

Before getting into the life of Johannes Kepler and the significance of SN 1604, some background information will help the reader understand the society Johannes lived in.

In contrast to the Newtonian force of gravity of a spherically symmetric body (“cause”), the Kepler orbits (“effects”) are not spherically symmetric. Does this contradict and thus disprove the metaphysical rule “equat causa effectum” (the effect equals its cause)? I will show this contradiction to be resolved through taking into account the initial conditions. Spherically symmetric sets of Kepler orbits are constructed by means of the conserved quantities energy, angular momentum, and Laplace–Runge–Lenz vector. The quantum analogs (Bohr orbitals) are considered as well. The fact, that the Kepler hodographs and orbits are plane, is shown by means of qualitative and intuitive arguing. The old rule “equat causa effectum” thus leads to new insights due to another point of view. This should be exploited in teaching physics, too.

Johann Elert Bode (1747–1826) was a German astronomer who lived about two centuries after Kepler. Bode attempted anew to answer the same question that initially concerned Kepler before his discoveries of the three planetary laws, namely “why are the planets spaced around the Sun as they are?” Kepler’s first answer was his failed constructions of inscribed and circumscribed circles around regular polygons, followed by his second answer of the constructions of inscribed and circumscribed spheres around Platonic solids.

From the earliest times astronomical information was used for calendars, time, climate, farming, seasons, and phenomena predictions. The Sun, Moon, five planets, and star patterns were known. Treatises and tables were used to calculate positions of planets. There are records of astronomical data in different forms from different countries. Ptolemy created accurate geometrical models for compiling positions of planets. The Chinese had records of celestial phenomena. The Mayan had their codices containing almanacs. The Islamic cultures had astronomical zijes of astronomical tables. In the 15th and 16th centuries there were a number of almanacs printed with the printing presses. In 1628 Kepler produced the Rudolphine Tables, the first real improvement since Ptolemy. With the printing press many almanacs were published with astronomical data and other information. The Royal Observatory at Greenwich was founded in 1675. The Connaisance des Temps was first published in 1678. In 1700 the Calendar Edict introduced the Gregorian Calendar to Germany and required a stamp tax on calendars, which supported the astronomers of the Berlin Observatory. Navigators could determine latitude by angular measurements of Polaris or the Sun. Different navigational instruments were developed for measuring celestial angles with increasing precision.

We have examined the motivations which induced Johannes Kepler to travel the way he did, both physically and spiritually, finally consummated in the legacy he left behind in his major works. Kepler himself was convinced that he had deciphered God’s blueprint for the harmony of his creation. In this respect, Kepler stood at the summit of all the combined efforts of those that preceded him in the history of science. And to this day, he has not been surpassed by any other.

In a similar (mathematical) fashion, Kepler’s 1st law, namely the possibility of an elliptical path for a body under the influence of gravity, is shown compatible with Newtonian Mechanics. Kepler’s 2nd law of differentiated speeds along an elliptical path is also readily derived from Newtonian Mechanics.

In his inaugural speech for the chair of astronomy of Gresham College in 1657, the young Christopher Wren expressed confidence that, thanks to “the industry of some writers of our age,” the foundation of Copernican astronomy, laid by Galileo and Kepler, would soon be completed and perfected. A few years later, in 1670, another Gresham professor delivered a lecture which attempted to provide an unquestionable proof of Copernican astronomy. Although unsuccessful in this respect, Hooke’s Attempt to Prove the Motion of the Earth by Observations, published in 1674, did present three new principles of the Copernican “system of the world.” The first principle introduced the idea “that all celestial bodies whatsoever, have an attraction or gravitating power towards their own centres.” This attraction is not limited to the constituents of each body, rather it extends “to all the other celestial bodies that are within the sphere of their activity.” Terrestrial gravity is part of a wider physical system, since the sun, the moon and all the other planets “have an influence” over the earth, as this latter does on them.

Significant progress, discoveries, and major breakthroughs, in chemistry, biology, geology, sociology, psychology, economics, molecular biology, medicine, and other scientific disciplines, all came after Newton; after physics was discovered; after the scientific method was adopted; after mathematics was endowed a central role in science by Kepler, Galileo, and Newton; and after Descartes emphasized the mechanical and causality aspects of nature. It is certainly reasonable to state that the decisive success of physics in the 17th century gave inspiration and confidence to later pioneers in the sciences; the chemists, the biologists, the geologists, the psychologist, the economists, and the others, that they should indeed investigate and theorize, that there was surely something essential out there to be discovered, and that the hidden secrets of nature can be uncovered, just as it occurred in physics. It is in that sense that the work of Kepler, Galileo, and Newton, as well as Copernicus, Tycho Brahe, and Descartes, consequently and practically directly, promoted, inspired, and led to the progress of scientific discovery in general, well beyond physics.

The order of things, as outlined in Chap. 2, seemed to have been one impetus for Kepler’s quest. However, this went hand in hand with his attempt to discover what might be lying beneath it, that is, what was really causing it—in other words, the world behind the world as we perceive it through our limited senses.

We start in medieval times, briefly considering the prevailing world view and emphasising their version of Aristotelian physics. We then look at the motivations and causes for the Renaissance, focusing on the rebirth of astronomy. The remainder of the chapter discusses in more detail the contributions of four seminal figures on the path to modern science: Copernicus, Tycho, Kepler, and Galileo.

In this chapter we are going to start analysing the simplest model of one-dimensional particle motion, which corresponds to the motion under the influence of a one-dimensional potential and no friction forces. This model is of special interest as it can be analytically solved in many cases. Motion in more than one dimension, most of the times, has no analytical solution and must be solved numerically. There are some exceptions however, and here we are going to analyse the two-dimensional motion under a central potential, and in particular we are going to analyse Kepler’s potential. The analytic solution for the motion under the influence of Kepler’s potential is derived explicitly by solving the corresponding differential equations of motion. Within the broad class of central potentials we shall also analyse potentials of the type \(kr^n\) for integer powers of n. As we shall observe, due to their particular functional dependence, we shall be able to treat central potentials in most aspects as if they were one-dimensional. We are also going to justify mathematically that the two-body problem with an internal central potential can be studied as a one dimensional problem.

Thousands of transiting exoplanets have been discovered to date, thanks in great part to the Kepler space mission. As in all populations, and certainly in the case of exoplanets, one finds unique objects with distinct characteristics. Here we will describe the properties and behaviour of a small group of ‘disintegrating’ exoplanets discovered over the last few years (KIC 12557548b, K2-22b, and others). They evaporate, lose mass unraveling their naked cores, produce spectacular dusty comet-like tails, and feature highly variable asymmetric transits. Apart from these exoplanets, there is observational evidence for even smaller ‘exo-’objects orbiting other stars: exoasteroids and exocomets. Most probably, such objects are also behind the mystery of Boyajian’s star. Ongoing and upcoming space missions such as TESS and PLATO will hopefully discover more objects of this kind, and a new era of the exploration of small extrasolar systems bodies will be upon us.

Kepler certainly understood at the time of his formulation of the third law that planets are not just points of light; rather, as Galileo’s telescopic observations of the moon suggested, they are real physical entities just like our planet Earth, and perhaps they are made of sand, rocks, or ice, with hills and valleys, as the moon seemed to be.

The prospects for the further development of stellar astrophysics remain golden. Much can still be done based on the Kepler data, the TESS mission is providing data over most of the sky, at a 10 min cadence in the extended phase, PLATO will provide data comparable with those of Kepler for relatively near-by stars, ground-based observations yield data of even higher quality for selected stars, and Gaia is revolutionizing the determination of global stellar properties. On the modelling side ever more realistic simulations are becoming possible of dynamic phenomena in stellar interiors, and new techniques, known as artificial intelligence or deep learning, are helping the analysis of the huge amounts of data that are becoming available. Much of this effort is aimed at applications of stellar physics in other areas of astrophysics, such as exoplanets or Galactic archaeology. In the excitement of these developments, in the spirit of Michael we should not forget the basic goal of improving our physical understanding of stars, combining human intelligence and intuition with mathematical analysis and modelling, observations, and data interpretation. This should be kept in mind in our work and, even more importantly, in the education of coming generations of stellar astrophysicists.

If there are globes in the Heaven similar to our Earth, do we vie with them over who occupies the better portion of the Universe? For if their globes are nobler, we are not the noblest of rational creatures. Then how can all things be for mans’ sake? How can we be the masters of God’s handiwork? … This system of planets, on one of which we human dwell, is located in the very bosom of the World, around the heart of the Universe, that is, the Sun… We humans live on the globe which by right belongs to the primary rational creature, the noblest of the corporeal creatures.

Following the discovery of his first and second laws and the publication of Astronomia Nova in 1609, Kepler spent the next ten years in futile effort searching for any further regularities or patterns in Tycho’s data using a variety of imaginative but fruitless approaches.

Johannes Kepler is one of the most atypical figures in the history of science. In fact, there is no other character who better embodies the passage from Renaissance magic to the dawn of modern science. Passionate about astrology and mysticism, prisoner of the religious and baroque culture of the time, he nevertheless managed to turn his gaze to the heavens, and what he ultimately saw paved the way for modern developments in science. The depositary of Tycho Brahe's extraordinarily accurate measurements of the position of the stars and the movement of the planets, Kepler discovered the elliptical orbits and a cosmos governed not by Aristotelian perfection, but by the laws of physics and geometry. The Copernican vision paradoxically found confirmation in the magical atmosphere of Rudolph II’s Prague, which teemed with astrologers, fortune tellers and alchemists.

The final version of Kepler’s horsocope for his new master contained a serious and very specific warning for the beginning of the year 1634. It was set to occur in or near a castle in the extreme west of what is today the Czech Republic.

Almost exactly 1500 years ago, the Roman statesman and philosopher Boethius wrote The Consolation of Philosophy, a conversation with Lady Philosophy in which he proclaimed the superiority of a life of the mind over wealth and fame. What does Urania, the Muse of Astronomy, have to offer? Here we argue that astronomy offers us a cosmological worldview within which we can frame our daily lives, providing a cosmic perspective sorely needed in the modern world. Boethius himself exhibited such a perspective as he sat in prison during the last days of his life, as did Giordano Bruno and many others, both in the realm of natural philosophy and natural science. Poets, philosophers, and scientists fostered it as the scientific revolution took root in the decades and centuries following Galileo, Kepler, and Newton, even as John Donne feared in the wake of the Copernican theory “’Tis all in pieces, all coherence gone.” In the twentieth century, astronomers Harlow Shapley and Carl Sagan actively promoted a much more robust cosmic perspective, based on modern science and featuring many worlds, extraterrestrial intelligence, and an expansive image of cosmic evolution. The cosmic perspective, similar to but more robust than what Sagan called “the cosmic connection,” is widespread today and is growing even among theologians, philosophers, and a public otherwise too far removed from science. One of the “consolations of astronomy” is that it allows us to envision our place in the universe of space and time, offering an inspiring vision of both life and death.

Johannes Kepler (1571–1630) was a German mathematician, astronomer, astrologer, and a key figure in the seventeenth century Scientific Revolution.

The fascinating story of Tycho, Kepler, and Newton, illustrates humanity’s profound challenge in fully understanding and describing the Solar System in terms of the correct heliocentric model, and ultimately in terms of physical laws of motion, and the gravitational forces applied to the planets.

In 1600 Tycho Brahe invited Kepler to Prague. After Tycho’s death in 1601 Kepler was appointed his successor as imperial mathematician at the royal court. Now, being in the possession of Tycho’s entire astronomical data and with a strong focus on Mars, Kepler would make his most significant discoveries which would prove to be true and lasting.

Rotation plays an important role in the evolution of stars. For decades rotation-age relationships have been used as surface rotation can be measured rather easily. Here we show the result of the measurement of rotation periods for a large number of Kepler solar-like stars along with their level of magnetic activity. We also study solar-like stars for which oscillations have not been detected using 1 month of Kepler data in order to understand what can be the reason behind these non-detections. Surprisingly the conclusion of our analysis is that magnetic activity is not the main reason for not detecting acoustic modes in those stars.

In 1596 at the age of 26 Kepler published his first major astronomical work, ‘Mysterium Cosmographicum’ (‘The Cosmographic Mystery’).

The Roman Numeral system was in use in Europe for nearly 1800 years, far longer than the current Hindu-Arabic system has been in existence.

The motivation for Kepler’s second law was a noticeable and decisive variation in Tycho’s data on Mars regarding calculated speed. Apparently nobody before Kepler ever inquired about the speed of the planets; instead the focus was merely on location and timing.

Chapter 1 is a historical essay that provides the context and sets the stage for all that follows. It recalls humanity’s efforts to understand the planets, moons, comets, and asteroids and how they move, from the time of the Greeks, to Isaac Newton and the Scientific Revolution, to the discovery of the vast, expanding universe of galaxies in the twentieth century. In describing the decisive contributions of Kepler, Newton, Hubble, and those of other pioneering scientists, the chapter considers some of the elementary mathematical elements involved by introducing the ellipse, parabola, and hyperbola as well as very basic concepts and principles of the physics of motion.

Building on the foundations in helioseismology and asteroseismology by pioneers such as Michael Thompson, observational progress in asteroseismology of solar-like oscillators has exploded over the past decade thanks to space-based missions such as CoRoT and Kepler/K2. Over the coming decade, this revolution is set to continue through facilities that will provide a several orders of magnitude increase of known solar-like oscillators, providing unprecedented possibilities to study stellar physics, exoplanets, and galactic stellar populations. In this contribution I will provide a brief history of the detection of solar-like oscillations, followed by a review of the first results by the TESS mission. Finally, I will discuss the prospects and challenges for asteroseismology of solar-like oscillators in the coming decade and beyond using space-based telescopes such WFIRST and PLATO, as well as ground-based radial velocity observations using dedicated networks such as SONG, photometric transient surveys, and 8-m class telescopes.

Why would Greek geometers study the conic sections? We have no direct answer to that question in any ancient text that has survived. It was not, generally, for the scientific applications of conic sections. The earliest practical applications we know of are in sundials, with medieval Arabic scientists. Only in the early modern period, with Galileo (1564–1642), was the parabolic path of bodies in motion recognized. And only with Kepler (1571–1630) were the orbits of planets seen to be ellipses. An exception to this picture is a short work, On Burning Mirrors, whose author, Diocles, was active about 200 BC. It was, it seems, the first to give the focus of a parabola, with the property that a surface formed by revolving a parabola about its axis would reflect the sun’s rays to the focus of that parabola. But it did not spark work in applications; there is no surviving mention of On Burning Mirrors until the sixth century AD, by Eutocius of Askalon, and, again, in Islamic work of the eleventh century.