Kepler, Johann

Early years

Johannes Kepler was born in the imperial city of Weil der Stadt (30 kilometers from Stuttgart , now the federal state of Baden-Württemberg ). His father, Heinrich Kepler, served as a mercenary in the Spanish Netherlands . When the young man was 18 years old, his father went on a regular campaign and disappeared forever. Kepler’s mother, Katarina Kepler , kept a tavern , moonlighted as fortune-telling and herbal medicine ^[4] .

Kepler became interested in astronomy as a child, when his mother showed the impressionable boy a bright comet ( 1577 ), and later a lunar eclipse (1580). After smallpox suffered in childhood, Kepler received a lifelong visual impairment, which prevented him from conducting astronomical observations, but he retained his enthusiastic love for astronomy forever ^[5] .

In 1589, Kepler graduated from high school at the Maulbronn Monastery , showing outstanding abilities ^[6] . City officials awarded him a scholarship to help him further his studies. In 1591 he entered the university in Tübingen - at first to the faculty of arts , which then included mathematics and astronomy, then transferred to the theological faculty. Here he first heard (from Michael Mestlin ) about the heliocentric system of the world developed by Nikolai Copernicus and immediately became its staunch supporter ^[7] . Kepler's university friend was Christoph Bezold , future lawyer.

Kepler originally planned to become a Protestant priest , but thanks to his extraordinary mathematical abilities, he was invited in 1594 to give lectures on mathematics at the University of Graz (now in Austria ).

Kepler spent 6 years in Graz. Here his first book The Secret of the Universe ( Mysterium Cosmographicum ) was published (1596). In it, Kepler tried to find the secret harmony of the Universe , for which he compared various “Platonic solids” ( regular polyhedra ) to the orbits of five then known planets (the Earth sphere he singled out especially). He presented the orbit of Saturn as a circle (not yet an ellipse ) on the surface of a ball circumscribed around a cube. In turn, a ball was inserted into the cube, which was supposed to represent the orbit of Jupiter . A tetrahedron was inscribed in this ball, described around a ball representing the orbit of Mars , etc. This work, after Kepler’s further discoveries, lost its original meaning (if only because the orbits of the planets were not circular); Nevertheless, Kepler believed in the existence of hidden mathematical harmony of the Universe until the end of his life, and in 1621 he reprinted The Secret of the World, making numerous changes and additions to it ^[8] .

Kepler sent the book “The Secret of the Universe” to Galileo and Tycho Brahe . Galileo approved Kepler's heliocentric approach, although he did not support mystical numerology . Subsequently, they conducted lively correspondence, and this circumstance (communication with the "heretic" - the Protestant ) at the trial of Galileo was particularly emphasized as aggravating Galileo’s guilt ^[9] .

Tycho Brahe, like Galileo, rejected Kepler’s far-fetched constructions, but praised his knowledge, originality of thought and invited Kepler to his place.

In 1597, Kepler married the widow of Barbara Mueller von Mulek. Their first two children died in infancy, and their wife fell ill with epilepsy . To top off adversity, persecution of Protestants began in Catholic Graz ^[10] . Kepler, included in the list of exiled "heretics", was forced to leave the city and accept the invitation of Tycho Brahe. By this time, Brahe himself was evicted from his observatory and moved to Prague , where he served with Emperor Rudolph II as a court astronomer and astrologer.

Prague

In 1600, both exiles - Kepler and Brahe - met in Prague. Spent 10 years here - the most fruitful period of Kepler’s life.

It soon became clear that the views of Copernicus and Kepler on astronomy Tycho Brahe shared only partially. To preserve geocentrism , Brahe proposed a compromise model: all planets except the Earth revolve around the Sun, and the Sun revolves around a motionless Earth ( geo-heliocentric system of the world ). This theory gained great fame and for several decades was the main competitor of the Copernican world system ^[11] .

After the death of Brahe in 1601, Kepler became his successor in office. Because of the endless wars, the emperor’s treasury was constantly empty, and Kepler’s salary was rarely and poorly paid. He was forced to moonlight by compiling horoscopes . Kepler also had to wage a long-standing lawsuit with the heirs of Tycho Brahe, who tried to take away from him, among other things the property of the deceased, also the results of astronomical observations. In the end, they managed to pay off ^[12] .

In 1604, Kepler published his observations of a supernova , now called by his name .

Being an excellent observer, Tycho Brahe for many years has been a voluminous work on the observation of planets and hundreds of stars , and the accuracy of its measurements was significantly higher than that of all its predecessors. To increase accuracy, Brahe used both technical improvements and a special technique to neutralize observation errors. The systematic nature of the measurements was especially valuable ^[13] .

Over the course of several years, Kepler carefully studied the data of Brahe and, as a result of careful analysis, came to the conclusion that the trajectory of the motion of Mars is not a circle, but an ellipse , in one of whose foci the Sun is located - the position known today as the first Kepler's law . The analysis led to the second law (in fact, the second law was discovered even earlier than the first): the radius vector connecting the planet and the Sun describes equal areas at equal times. This meant that the farther the planet is from the Sun, the slower it moves.

Kepler's laws were formulated by Kepler in 1609 in the book " New Astronomy ", and, for the sake of caution, he attributed them only to Mars ^[14] .

The new movement model aroused great interest among Copernican scholars, although not all of them accepted it. Galileo Keplerian ellipses strongly rejected ^[15] . After Kepler’s death, Galileo noted in a letter: “I always appreciated Kepler’s mind - sharp and free, perhaps even too free, but our ways of thinking are completely different” ^[16] .

In 1610, Galileo informed Kepler of the discovery of the moons of Jupiter . Kepler met this message incredulously and in the polemic “Conversation with the Star Herald” brought a somewhat humorous objection: “it is unclear what [satellites] would be if there is no one on this planet who could admire this sight” ^[17] . But later, having received his copy of the telescope, Kepler changed his mind, confirmed the observation of satellites, and he himself took up the theory of lenses. The result was an advanced telescope and fundamental Diopter work.

In Prague, Kepler had two sons and a daughter. In 1611, the eldest son Friedrich died of smallpox . At the same time, the insane emperor Rudolph II , having lost the war with his own brother Matthew , denied the Czech crown in his favor and soon died ^[18] . Kepler began training to move to Linz , but then, after a long illness, his wife Barbara died.

Recent years

In 1612, collecting scarce funds, Kepler moved to Linz , where he lived for 14 years. He retained the position of court mathematician and astronomer, but in the matter of payment the new emperor was no better than the old. Some income came from teaching and horoscopes.

In 1613, Kepler married the 24-year-old daughter of the joiner Susanna. They had seven children, four survived ^[19] .

In 1615, Kepler received the news that his mother was accused of witchcraft . The accusation is serious: last winter in Leonberg , where Katarina lived, 6 women were burned under the same article. The accusation contained 49 points: connection with the devil , blasphemy , corruption , necromancy , etc. Kepler writes to the city authorities; the mother is initially released, but then arrested again. The investigation lasted 5 years. Finally, in 1620, the trial began. Kepler himself acted as a defender, and a year later the exhausted woman was finally released. The following year, she passed away ^[20] .

Meanwhile, Kepler continued astronomical research and in 1618 discovered the third law : the ratio of the cube of the average distance of the planet from the Sun to the square of the period of revolution of the planet around the Sun is a constant for all planets:

a ³ / T ² = const.

Kepler publishes this result in the final book, “ Harmony of the World, ” and applies it not only to Mars, but to all other planets (including, of course, the Earth), as well as to Galilean satellites ^[21] .

Note that in the book, along with the most valuable scientific discoveries, philosophical discussions about the “ music of the spheres ” and Platonic solids are also presented, which, according to the scientist, constitute the aesthetic essence of the highest project of the universe.

In 1626, during the Thirty Years War, Linz was besieged and soon captured. Looting and fires began; among others, the printing house burned down. Kepler moved to Ulm and in 1628 went to serve with Wallenstein .

In 1630, Kepler went to the emperor in Regensburg to receive at least part of his salary. On the way he caught a cold and soon died.

After Kepler’s death, the heirs inherited: worn clothes, 22 florins in cash, 29,000 florins of unpaid salaries, 27 published manuscripts, and many unpublished ones; they were later published in a 22-volume collection ^[4] .

With the death of Kepler, his misfortunes did not end. At the end of the Thirty Years War , the cemetery where he was buried was completely destroyed, and nothing was left of his grave. Part of Kepler’s archive has disappeared. In 1774, the majority of the archive (18 volumes of 22), on the recommendation of Leonard Euler, was acquired by the St. Petersburg Academy of Sciences ^[22] , which is now stored in the St. Petersburg branch of the RAS archive ^[23] .

Albert Einstein called Kepler “an incomparable man” and wrote about his fate ^[24] :

He lived in an era when there was still no confidence in the existence of a certain general regularity for all natural phenomena. How deep was his faith in such a pattern, if, working alone, not supported by anyone and not understood, for many decades he drew strength in it for a difficult and painstaking empirical study of planetary motion and the mathematical laws of this movement!

Now that these laws are already established, it is hard to imagine how much ingenuity, imagination and tireless, hard work it took to establish these laws and to express them with such great accuracy.

Astronomy

At the end of the 16th century, in astronomy, there was still a struggle between the geocentric system of Ptolemy and the heliocentric system of Copernicus . Opponents of the Copernican system cited the fact that in relation to the calculation error, it is no better than the Ptolemaic one. Recall that in the Copernican model, planets uniformly move in circular orbits: in order to reconcile this assumption with the apparent unevenness of planetary motion, Copernicus had to introduce additional motions along epicycles . Although Copernicus had fewer epicycles than Ptolemy, his astronomical tables, initially more accurate than the Ptolemaic ones, soon diverged significantly from observations, which puzzled and cooled the enthusiastic Copernicans.

The three laws of planetary motion discovered by Kepler fully and with excellent accuracy explained the apparent unevenness of these movements. Instead of numerous contrived epicycles, the Kepler model includes only one curve - an ellipse . The second law established how the speed of the planet changes when moving away or approaching the Sun, and the third allows us to calculate this speed and the period of revolution around the Sun.

Although historically the Kepler system of the world is based on the Copernican model, in fact they have very little in common (only the daily rotation of the Earth). The circular motions of the spheres bearing the planets disappeared, the concept of a planetary orbit appeared. In the Copernican system, the Earth still occupied a somewhat special position, since Copernicus declared the center of the Earth's orbit as the center of the world. Kepler’s Earth is an ordinary planet, the movement of which is subject to the general three laws. All orbits of celestial bodies are ellipses ( Newton later discovered the motion along a hyperbolic trajectory), the common focus of the orbits is the Sun.

Kepler also derived the " Kepler equation " used in astronomy to determine the position of celestial bodies.

The laws of planetary kinematics , discovered by Kepler, later served Newton as the basis for creating the theory of gravity . Newton mathematically proved that all Kepler’s laws are direct consequences of the law of gravity.

Kepler’s views on the structure of the universe outside the solar system stemmed from his mystical philosophy. He considered the sun motionless, and considered the sphere of stars the boundary of the world. Kepler did not believe in the infinity of the Universe, and proposed ( 1610 ) what was later called the photometric paradox : if the number of stars is infinite, then in any direction the gaze would come across a star and there would be no dark areas in the sky ^{[25] [ 26]} .

Strictly speaking, Kepler’s world system claimed not only to reveal the laws of planetary motion, but also much more. Like the Pythagoreans , Kepler considered the world to be a realization of some numerical harmony, both geometric and musical; revealing the structure of this harmony would give answers to the most profound questions ^{[27] [28]} :

I found out that all celestial movements, both in their whole and in all individual cases, are imbued with a general harmony, though not the one that I assumed, but even more perfect.

Original text (lat.)

Totam Harmonices naturam, quantaquanta est, cum omnibus suis partibus, libro III. explicatis, inter Motus cœlestes reperiri; non eo quidem modo, quem ego conceperam animo; pars hæc est non postrema mei gaudij; sed diversissimo alio, simulque & præstantissimo & perfectissimo.

For example, Kepler explains why there are exactly six planets (by that time only six planets of the Solar system were known) and they are placed in space in this way and not otherwise: it turns out that the orbits of the planets are inscribed in regular polyhedra . Interestingly, based on these unscientific considerations, Kepler predicted the existence of two satellites of Mars and an intermediate planet between Mars and Jupiter.

Kepler’s laws combined clarity, simplicity and computational power, but the mystical form of his world system thoroughly clogged the real essence of Kepler’s great discoveries. Nevertheless, Kepler’s contemporaries were already convinced of the accuracy of the new laws, although their deepest meaning until Newton remained incomprehensible. No attempts were made to reanimate the Ptolemy model or suggest a system of motion other than the heliocentric one that was no longer made.

Kepler did a lot for the adoption of the Gregorian calendar by the Protestants (at the Sejm in Regensburg , 1613 , and in Aachen , 1615 ).

Kepler became the author of the first extensive (in three volumes) exposition of Copernican astronomy ( Epitome Astronomiae Copernicanae , 1617 - 1622 ), which immediately was honored to be included in the Index of Prohibited Books . In this book, his main work, Kepler included a description of all his discoveries in astronomy.

In the summer of 1627, Kepler, after 22 years of work, published (at his own expense ^[29] ) astronomical tables, which in honor of the emperor he called the " Rudolphs ". Demand for them was huge, since all the previous tables had long diverged from observations. It is also important that for the first time labor included logarithm tables convenient for calculations. Keplerian tables served astronomers and sailors until the beginning of the 19th century ^[4] .

A year after Kepler’s death, Gassendi observed the passage of Mercury predicted by him on the solar disk ^[30] . In 1665, the Italian physicist and astronomer Giovanni Alfonso Borelli published a book in which Kepler's laws are confirmed for the satellites of Jupiter discovered by Galileo .

Math

Kepler found a method for determining the volumes of various bodies of revolution , which he described in the book “New Stereometry of Wine Barrels” ( 1615 ). The method he proposed contained the first elements of integral calculus ^[31] . Cavalieri later used the same approach to develop an exceptionally fruitful “indivisible method” . The completion of this process was the discovery of mathematical analysis .

In addition, Kepler analyzed the symmetry of snowflakes in great detail. Studies on symmetry led him to the assumption of tight packing of balls, according to which the highest packing density is achieved by pyramidal ordering of balls on top of each other ^[32] . It was not possible to mathematically prove this fact for 400 years - the first report on the proof of the Kepler hypothesis appeared only in 1998 in the work of mathematician . Kepler's pioneering work in the field of symmetry later found application in crystallography and coding theory.

In astronomical research, Kepler contributed to the theory of conic sections . He compiled one of the first tables of logarithms ^[33] .

Kepler first encountered the term " arithmetic mean " ^[34] .

Kepler entered the history of projective geometry : he first introduced the most important concept of an infinitely distant point ^[35] . He introduced the concept of the focus of a conic section and considered projective transformations of conical sections, including changing their type - for example, translating an ellipse into a hyperbola .

Mechanics and Physics

It was Kepler who introduced the term inertia into physics as the inherent property of bodies to resist applied external forces. At the same time, like Galileo, he formulates in a clear form the first law of mechanics: every body that is not affected by other bodies is at rest or makes a uniform rectilinear motion ^[36] .

Kepler came close to the discovery of the law of gravity, although he did not try to express it mathematically. He wrote in the book "New Astronomy" that in nature there is "a mutual bodily desire of similar (related) bodies to unity or union." The source of this force, in his opinion, is magnetism in combination with the rotation of the Sun and planets around its axis ^[37] .

In another book, Kepler specified ^{[37] [38]} :

I define gravity as a force similar to magnetism - mutual attraction. The force of gravity is greater, the two bodies are closer to one another.

Original text (lat.)

Gravitatem ego definio virtute, magneticæ simili, attractionis mutuæ. Hujus verò attractionis major vis est in corporibus inter se vicinis, quàm in remotis.

True, Kepler mistakenly believed that this force spreads only in the plane of the ecliptic . Apparently, he believed that the force of attraction is inversely proportional to distance (and not the square of the distance); however, its wording is not clear enough.

Kepler the first, almost a hundred years earlier than Newton , put forward the hypothesis that the cause of the tides is the effect of the moon on the upper layers of the oceans ^[39] .

Optics

In 1604, Kepler published a substantial optics treatise, “Supplements to Vitellius,” and in 1611, another book, Dioptrica. The history of optics as a science begins with these works ^[40] . In these works, Kepler sets out in detail both geometric and physiological optics. He describes the refraction of light, refraction and the concept of an optical image, the general theory of lenses and their systems. Introduces the terms " optical axis " and " meniscus ", for the first time formulates the law of incidence of illumination inversely proportional to the square of the distance to the light source. For the first time describes the phenomenon of total internal reflection of light upon transition to a less dense medium.

Описанный им физиологический механизм зрения, с современных позиций, принципиально верен. Кеплер выяснил роль хрусталика , верно описал причины близорукости и дальнозоркости .

Глубокое проникновение в законы оптики привело Кеплера к схеме телескопической подзорной трубы ( телескоп Кеплера ), изготовленной в 1613 году Кристофом Шайнером. К 1640-м годам такие трубы вытеснили в астрономии менее совершенный телескоп Галилея.

Отношение Кеплера к астрологии было двойственным. С одной стороны, он допускал, что земное и небесное находятся в некоем гармоничном единстве и взаимосвязи. С другой — скептически оценивал возможность использовать эту гармонию для предсказания конкретных событий.

Кеплер говорил: «Люди ошибаются, думая, что от небесных светил зависят земные дела» ^[41] . Широко известно также другое его откровенное высказывание ^{[42] [43]} :

Конечно, эта астрология глупая дочка; но, Боже мой, куда бы делась её мать, высокомудрая астрономия, если бы у неё не было глупенькой дочки! Свет ведь ещё гораздо глупее и так глуп, что для пользы этой старой разумной матери глупая дочь должна болтать и лгать. И жалованье математиков ( Mathematicorum salaria ) так ничтожно, что мать наверное бы голодала, если бы дочь ничего не зарабатывала.

Оригинальный текст (нем.)

Es ist wol diese Astrologia ein närrisches Töchterlin … aber lieber Gott/wo wolt jhr Mutter die hochvernünfftige Astronomia bleiben/wann sie diese Jhre närrische Tochter nit hette/ist doch die Welt noch viel närrischer/und so närrisch/daß deroselben zu jhren selbst frommen diese alte verständige Mutter die Astronomia durch der Tochter Narrentaydung/weil sie zumal auch einen Spiegel hat/nur eyngeschwärzt und eyngelogen werden muß. Und seynd sonsten die Mathematicorum salaria so seltzam und so gering/daß die Mutter gewißlich Hunger leyden müste/wann die Tochter nichts erwürbe.

Тем не менее, Кеплер не порывал с астрологией никогда. Более того, он имел свой собственный взгляд на природу астрологии, чем выделялся среди астрологов-современников. В труде «Гармония мира» он утверждает, что «в небесах нет светил, приносящих несчастья», но человеческая душа способна «резонировать» с лучами света, исходящими от небесных тел, она запечатлевает в памяти конфигурацию этих лучей в момент своего рождения. Сами же планеты, в представлении Кеплера, были живыми существами, наделёнными индивидуальной душой ^[44] .

Благодаря некоторым удачным предсказаниям Кеплер заработал репутацию искусного астролога. В Праге одной из его обязанностей было составление гороскопов для императора. Следует заметить, вместе с тем, что Кеплер при этом не занимался астрологией исключительно ради заработка и составлял гороскопы для себя и своих близких. Так в своей работе «О себе» он приводит описание собственного гороскопа, а когда в январе 1598 года у него родился сын, Генрих, Кеплер составил гороскоп и для него. По его мнению, ближайшим годом, когда жизни его сына угрожала опасность, был 1601 год, но сын умер уже в апреле 1598 года .

Попытки Кеплера составить гороскоп для полководца Валленштейна также далеко разошлись с реальностью. В 1608 году Кеплер составил гороскоп полководцу, в котором предрекал женитьбу на 33-м году жизни, называл опасными для жизни годы 1613, 1625 и 70-й год жизни Валленштейна, а также описал ряд других событий. Когда стало очевидно, что предсказания потерпели неудачу. Валленштейн вернул гороскоп Кеплеру, который, исправив в нём время рождения на полчаса, получил точное соответствие между предсказанием и течением жизни. Однако и этот вариант содержал промахи. Так, Кеплер полагал, что период с 1632 по 1634 год будет благополучным для полководца, и не сулит опасности. Но в феврале 1634 года Валленштейн был убит.

В честь И. Кеплера были названы:

• Кеплер — ударный кратер на Луне
• Кеплер — ударный кратер на Марсе
• (1134) Кеплер — астероид
• Сверхновая Кеплера — сверхновая звезда № 1604 , описанная им.
• Кеплер (телескоп) — орбитальная обсерватория НАСА , выведена на орбиту в 2009 году для поиска и исследования планет за пределами Солнечной системы.
• Университет в Линце .
• Keplerplatz — площадь и станция метро в городе Вена ^[45] .
• Иоганн Кеплер (ATV) — Европейский грузовой космический корабль, 2011 года.

В Вайль-дер-Штадте ^[46] , Праге (см. ), Граце ^[47] и Регенсбурге функционируют музеи Кеплера.

Другие мероприятия в память о Кеплере:

• В 1971 году к 400-летию со дня рождения Кеплера в ГДР была выпущена памятная монета достоинством 5 марок .
• В 2009 году к 400-летию открытия Кеплеровских законов в Германии выпущена памятная серебряная монета достоинством 10 евро .

Жизни учёного посвящены художественные произведения:

• Опера и симфония «Гармония мира» композитора Пауля Хиндемита (1956 год).
• Историческая повесть Юрия Медведева «Капитан звёздного океана (Кеплер)», Молодая гвардия, 1972.
• Художественный фильм «Иоганнес Кеплер» режиссёра Франка Фогеля (ГДР, 1974 год).
• Роман Джона Бэнвилла Кеплер , переведённый на русский язык в 2008 году.
• Опера «Кеплер» композитора Филипа Гласса (2009 год).
• Художественный фильм «Глаз астронома» режиссёра Стэна Ньюманна (Франция, 2012 год).
• Опера « Суд Кеплера» композитора Тима Уаттса (2016 год).

• Марки в честь 400-летнего юбилея Кеплера (1971)
•  

  1971, ГДР
•  

  1971, Румыния
•  

  1971, ОАЭ
•  

  1971, ФРГ

• Гипотеза Кеплера
• Законы Кеплера
• История астрономии
• История физики
• Кеплеровы элементы орбиты
• Метод неделимых
• Тихо Браге
• Уравнение Кеплера

• Mysterium cosmographicum (Тайна мироздания), 1596
• Astronomiae Pars Optica (Оптика в астрономии), 1604
• Ad Vitellionem paralipomena (Дополнения к Вителлию), физиологическая оптика, 1604
• De Stella nova in pede Serpentarii (О новой звезде в созвездии Змееносца), 1604
• Astronomia nova (Новая астрономия), 1609
• Tertius Interveniens , 1610
• Dissertatio cum Nuncio Sidereo (Разговор со Звёздным вестником), полемика со «Звёздным вестником» Галилея , 1610
• Dioptrice (Диоптрика), 1611
• De nive sexangula (О шестиугольных снежинках), 1611
• De vero Anno, quo aeternus Dei Filius humanam naturam in Utero benedictae Virginis Mariae assumpsit ), 1613
• Eclogae Chronicae ( 1615 )
• Nova stereometria doliorum vinariorum (Новая стереометрия винных бочек), 1615
• Epitome Astronomiae Copernicanae (Коперниканская астрономия, в трёх томах, выходивших в 1618—1621)
• Harmonices Mundi (Гармония мира), 1619
• Mysterium cosmographicum (Тайна мира, 2-е изд.), 1621
• Tabulae Rudolphinae ( Рудольфовы таблицы ), 1627
• Somnium («Сон, или Посмертное сочинение о лунной астрономии», фантастический рассказ о полёте на Луну), 1634
• Библиография научных работ Кеплера с ссылками на оригиналы

Translations into Russian

• Кеплер, Иоганн. Новая стереометрия винных бочек . — М.—Л.: ГТТИ, 1935. — 360 с.
• Кеплер, Иоганн. О шестиугольных снежинках. Сон. Разговор с Звёздным вестником. . — М.: Наука, 1982.
• Разговор с звёздным вестником по изд.: И. Кеплер, О шестиугольных снежинках, М., Наука, 1982.
• Сон, или Посмертное сочинение о лунной астрономии по изд.: И. Кеплер, О шестиугольных снежинках, М., Наука, 1982.

• Белонучкин В. Е. Кеплер, Ньютон и всё-всё-всё. М.: Наука, Серия Библиотечка «Квант», выпуск 78, 1990. 128 с.
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• Kepler's discovery (анимации на тему «Новой астрономии» Кеплера)
• Daniel A. Di Liscia, «Johannes Kepler» (Stanford Encyclopedia of Philosophy) (англ.)
• Johannes Kepler's New Astronomy (англ.)
• John J. O'Connor and Edmund F. Robertson . Johannes Kepler (англ.) — биография в архиве MacTutor .
• Davis AEL , «Kepler's Planetary Laws»