The practical effect of his clearing of the “second inequality” was to refer the orbit of Mars directly to the sun, and he found that the area between successive distances of Mars from the sun (instead of the sum of the distances) was strictly proportional to the time taken, in short, equal areas were described in equal times (2nd Law) when referred to the sun in the focus of the ellipse (1st Law).
He announced that (1) The planet describes an ellipse, the sun being in one focus; and (2) The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time. These are Kepler’s first and second Laws though not discovered in that order, and it was at once clear that Ptolemy’s “bisection of the excentricity” simply amounted to the fact that the centre of an ellipse bisects the distance between the foci, the sun being in one focus and the angular velocity being uniform about the empty focus. For so many centuries had the fetish of circular motion postponed discovery. It was natural that Kepler should assume that his laws would apply equally to all the planets, but the proof of this, as well as the reason underlying the laws, was only given by Newton, who approached the subject from a totally different standpoint.
This commentary on Mars was published in 1609, the year of the invention of the telescope, and Kepler petitioned the Emperor for further funds to enable him to complete the study of the other planets, but once more there was delay; in 1612 Rudolph died, and his brother Matthias who succeeded him, cared very little for astronomy or even astrology, though Kepler was reappointed to his post of Imperial Mathematician. He left Prague to take up a permanent professorship at the University of Linz. His own account of the circumstances is gloomy enough. He says, “In the first place I could get no money from the Court, and my wife, who had for a long time been suffering from low spirits and despondency, was taken violently ill towards the end of 1610, with the Hungarian fever, epilepsy and phrenitis. She was scarcely convalescent when all my three children were at once attacked with smallpox. Leopold with his army occupied the town beyond the river just as I lost the dearest of my sons, him whose nativity you will find in my book on the new star. The town on this side of the river where I lived was harassed by the Bohemian troops, whose new levies were insubordinate and insolent; to complete the whole, the Austrian army brought the plague with them into the city. I went into Austria and endeavoured to procure the situation which I now hold. Returning in June, I found my wife in a decline from her grief at the death of her son, and on the eve of an infectious fever, and I lost her also within eleven days of my return. Then came fresh annoyance, of course, and her fortune was to be divided with my step-sisters. The Emperor Rudolph would not agree to my departure; vain hopes were given me of being paid from Saxony; my time and money were wasted together, till on the death of the Emperor in 1612, I was named again by his successor, and suffered to depart to Linz.”
Being thus left a widower with a ten-year-old daughter Susanna, and a boy Louis of half her age, he looked for a second wife to take charge of them. He has given an account of eleven ladies whose suitability he considered. The first, an intimate friend of his first wife, ultimately declined; one was too old, another an invalid, another too proud of her birth and quarterings, another could do nothing useful, and so on. Number eight kept him guessing for three months, until he tired of her constant indecision, and confided his disappointment to number nine, who was not impressed. Number ten, introduced by a friend, Kepler found exceedingly ugly and enormously fat, and number eleven apparently too young. Kepler then reconsidered one of the earlier ones, disregarding the advice of his friends who objected to her lowly station. She was the orphan daughter of a cabinetmaker, educated for twelve years by favour of the Lady of Stahrenburg, and Kepler writes of her: “Her person and manners are suitable to mine; no pride, no extravagance; she can bear to work; she has a tolerable knowledge of how to manage a family; middle-aged and of a disposition and capability to acquire what she still wants”.
Wine from the Austrian vineyards was plentiful and cheap at the time of the marriage, and Kepler bought a few casks for his household. When the seller came to ascertain the quantity, Kepler noticed that no proper allowance was made for the bulging parts, and the upshot of his objections was that he wrote a book on a new method of gauging—one of the earliest specimens of modern analysis, extending the properties of plane figures to segments of cones and cylinders as being “incorporated circles”. He was summoned before the Diet at Ratisbon to give his opinion on the Gregorian Reform of the Calendar, and soon afterwards was excommunicated, having fallen foul of the Roman Catholic party at Linz just as he had previously at Gratz, the reason apparently being that he desired to think for himself. Meanwhile his salary was not paid any more regularly than before, and he was forced to supplement it by publishing what he called a “vile prophesying almanac which is scarcely more respectable than begging unless it be because it saves the Emperor’s credit, who abandons me entirely, and with all his frequent and recent orders in council, would suffer me to perish with hunger”.
In 1617 he was invited to Italy to succeed Magini as Professor of Mathematics at Bologna. Galileo urged him to accept the post, but he excused himself on the ground that he was a German and brought up among Germans with such liberty of speech as he thought might get him into trouble in Italy. In 1619 Matthias died and was succeeded by Ferdinand III, who again retained Kepler in his post. In the same year Kepler reprinted his “Mysterium Cosmographicum,” and also published his “Harmonics” in five books dedicated to James I of England. “The first geometrical, on the origin and demonstration of the laws of the figures which produce harmonious proportions; the second, architectonical, on figurate geometry and the congruence of plane and solid regular figures; the third, properly Harmonic, on the derivation of musical proportions from figures, and on the nature and distinction of things relating to song, in opposition to the old theories; the fourth, metaphysical, psychological, and astrological, on the mental essence of Harmonics, and of their kinds in the world, especially on the harmony of rays emanating on the earth from the heavenly bodies, and on their effect in nature and on the sublunary and human soul; the fifth, astronomical and metaphysical, on the very exquisite Harmonics of the celestial motions and the origin of the excentricities in harmonious proportions.” The extravagance of his fancies does not appear until the fourth book, in which he reiterates the statement that he was forced to adopt his astrological opinions from direct and positive observation. He despises “The common herd of prophesiers who describe the operations of the stars as if they were a sort of deities, the lords of heaven and earth, and producing everything at their pleasure. They never trouble themselves to consider what means the stars have of working any effects among us on the earth whilst they remain in the sky and send down nothing to us which is obvious to the senses, except rays of light.” His own notion is “Like one who listens to a sweet melodious song, and by the gladness of his countenance, by his voice, and by the beating of his hand or foot attuned to the music, gives token that he perceives and approves the harmony: just so does sublunary nature, with the notable and evident emotion of the bowels of the earth, bear like witness to the same feelings, especially at those times when the rays of the planets form harmonious configurations on the earth,” and again “The earth is not an animal like a dog, ready at every nod; but more like a bull or an elephant, slow to become angry, and so much the more furious when incensed.” He seems to have believed the earth to be actually a living animal, as witness the following: “If anyone who has climbed the peaks of the highest mountains, throw a stone down their very deep clefts, a sound is heard from them; or if he throw it into one of the mountain lakes, which beyond doubt are bottomless, a storm will immediately arise, just as when you thrust a straw into the ear or nose of a ticklish animal, it shakes its head, or runs shudderingly away. What so like breathing, especially of those fish who draw water into their mouths and spout it out again through their gills, as that wonderful tide! For although it is so regulated according to the course of the moon, that, in the preface to my ‘Commentaries on Mars,’ I have mentioned it as probable that the waters are attracted by the moon, as iron by the loadstone, yet if anyone uphold that the earth regulates its breathing according to the motion of the sun and moon, as animals have daily and nightly alternations of sleep and waking, I shall not think his philosophy unworthy of being listened to; especially if any flexible parts should be discovered in the depths of the earth, to supply the functions of lungs or gills.”
In the same book Kepler enlarges again on his views in reference to the basis of astrology as concerned with nativities and the importance of planetary conjunctions. He gives particulars of his own nativity. “Jupiter nearest the nonagesimal had passed by four degrees the trine of Saturn; the Sun and Venus in conjunction were moving from the latter towards the former, nearly in sextiles with both: they were also removing from quadratures with Mars, to which Mercury was closely approaching: the moon drew near to the trine of the same planet, close to the Bull’s Eye even in latitude. The 25th degree of Gemini was rising, and the 22nd of Aquarius culminating. That there was this triple configuration on that day—namely the sextile of Saturn and the Sun, the sextile of Mars and Jupiter, and the quadrature of Mercury and Mars, is proved by the change of weather; for after a frost of some days, that very day became warmer, there was a thaw and a fall of rain.” This alleged “proof” is interesting as it relies on the same principle which was held to justify the correction of an uncertain birth-time, by reference to illnesses, etc., met with later. Kepler however goes on to say, “If I am to speak of the results of my studies, what, I pray, can I find in the sky, even remotely alluding to it? The learned confess that several not despicable branches of philosophy have been newly extricated or amended or brought to perfection by me: but here my constellations were, not Mercury from the East in the angle of the seventh, and in quadratures with Mars, but Copernicus, but Tycho Brahe, without whose books of observations everything now set by me in the clearest light must have remained buried in darkness; not Saturn predominating Mercury, but my lords the Emperors Rudolph and Matthias, not Capricorn the house of Saturn but Upper Austria, the house of the Emperor, and the ready and unexampled bounty of his nobles to my petition. Here is that corner, not the western one of the horoscope, but on the earth whither, by permission of my Imperial master, I have betaken myself from a too uneasy Court; and whence, during these years of my life, which now tends towards its setting, emanate these Harmonics and the other matters on which I am engaged.”
The fifth book contains a great deal of nonsense about the harmony of the spheres; the notes contributed by the several planets are gravely set down, that of Mercury having the greatest resemblance to a melody, though perhaps more reminiscent of a bugle-call. Yet the book is not all worthless for it includes Kepler’s Third Law, which he had diligently sought for years. In his own words, “The proportion existing between the periodic times of any two planets is exactly the sesquiplicate proportion of the mean distances of the orbits,” or as generally given, “the squares of the periodic times are proportional to the cubes of the mean distances.” Kepler was evidently transported with delight and wrote, “What I prophesied two and twenty years ago, as soon as I discovered the five solids among the heavenly orbits,—what I firmly believed long before I had seen Ptolemy’s ‘Harmonics’—what I had promised my friends in the title of this book, which I named before I was sure of my discovery,—what sixteen years ago I urged as a thing to be sought,—that for which I joined Tycho Brahe, for which I settled in Prague, for which I have devoted the best part of my life to astronomical computations, at length I have brought to light, and have recognised its truth beyond my most sanguine expectations. Great as is the absolute nature of Harmonics, with all its details as set forth in my third book, it is all found among the celestial motions, not indeed in the manner which I imagined (that is not the least part of my delight), but in another very different, and yet most perfect and excellent. It is now eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the unveiled sun, most admirable to gaze on, burst out upon me. Nothing holds me; I will indulge in my sacred fury; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my God far away from the confines of Egypt. If you forgive me, I rejoice, if you are angry, I can bear it; the die is cast, the book is written; to be read either now or by posterity, I care not which; it may well wait a century for a reader, as God has waited six thousand years for an observer.” He gives the date 15th May, 1618, for the completion of his discovery. In his “Epitome of the Copernican Astronomy,” he gives his own idea as to the reason for this Third Law. “Four causes concur for lengthening the periodic time. First, the length of the path; secondly, the weight or quantity of matter to be carried; thirdly, the degree of strength of the moving virtue; fourthly, the bulk or space into which is spread out the matter to be moved. The orbital paths of the planets are in the simple ratio of the distances; the weights or quantities of matter in different planets are in the subduplicate ratio of the same distances, as has been already proved; so that with every increase of distance a planet has more matter and therefore is moved more slowly, and accumulates more time in its revolution, requiring already, as it did, more time by reason of the length of the way. The third and fourth causes compensate each other in a comparison of different planets; the simple and subduplicate proportion compound the sesquiplicate proportion, which therefore is the ratio of the periodic times.” The only part of this “explanation” that is true is that the paths are in the simple ratio of the distances, the “proof” so confidently claimed being of the circular kind commonly known as “begging the question”. It was reserved for Newton to establish the Laws of Motion, to find the law of force that would constrain a planet to obey Kepler’s first and second Laws, and to prove that it must therefore also obey the third.