Johannes Kepler

Johannes Kepler (December 27 1571 – November 15 1630) was a German Lutheran mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. He is best known for his laws of planetary motion, which provided one of the foundations of Isaac Newton's law of universal gravitation.

Quotes



 * Nature uses as little as possible of anything.
 * As quoted by W. H. Auden, Louis Kronenberger, Viking Book of Aphorisms: A Personal Selection (1920) p. 98; also by Joseph Silk, The Infinite Cosmos: Questions from the Frontiers of Cosmology (2006)


 * We do not ask for what useful purpose the birds do sing, for song is their pleasure since they were created for singing. Similarly, we ought not to ask why the human mind troubles to fathom the secrets of the heavens. The diversity of the phenomena of nature is so great and the treasures hidden in the heavens so rich precisely in order that the human mind shall never be lacking in fresh nourishment.
 * As quoted by Carl Sagan Cosmos (1980)


 * In Terra inest virtus, quae Lunam del.
 * There is a force in the earth which causes the moon to move.
 * Essay dedicated to the Archduke Ferdinand, as quoted by Max Caspar, Kepler (1993) Sect. II, Ch. 9, p. 110.


 * I much prefer the sharpest criticism of a single intelligent man to the thoughtless approval of the masses.
 * As quoted by Gloria Pierre, (K)new Words: Redefine Your Communication (2005) p. 147.


 * I used to measure the heavens, now I measure the shadows of Earth. Although my mind was heaven-bound, the shadow of my body lies here.
 * Epitaph he composed for himself a few months before he died, as quoted by Steven G. Krantz, Brian E. Blank, Calculusː Multivariable (2006) p. 126.


 * Temporis filia veritas; cui me obstetricari non pudet.
 * Truth is the daughter of time, and I feel no shame in being her midwife.
 * As quoted by Shafique N. Virani, The Ismailis in the Middle Ages: A History of Survival, A search for Salvation (2007) p. 28.

Mysterium Cosmographicum (1596)
Full title: Prodromus dissertationum cosmographicarum, continens mysterium cosmographicum, de admirabili proportione orbium coelestium, de que causis coelorum numeri, magnitudinis, motuumque periodicorum genuinis & proprijs, demonstratum, per quinque regularia corpora geometrica Title translations: Forerunner of the Cosmological Essays, Which Contains the Secret of the Universe; on the Marvelous Proportion of the Celestial Spheres, and on the True and Particular Causes of the Number, Magnitude, and Periodic Motions of the Heavens; Established by Means of the Five Regular Geometric Solids] See
 * Unless otherwise noted, translations & quotations by Walter W. Bryant,


 * The earth is the sphere, the measure of all; round it describe a dodecahedron; the sphere including this will be Mars. Round Mars describe a tetrahedron; the sphere including this will be Jupiter. Describe a cube round Jupiter; the sphere including this will be Saturn. Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury.
 * pp. 16–17.


 * Either... the moving intelligences of the planets are weakest in those that are farthest from the sun, or... there is one moving intelligence in the sun, the common center, forcing them all round, but those most violently which are nearest, and that it languishes in some sort and grows weaker at the most distant, because of the remoteness and the attenuation of the virtue.
 * p. 17.


 * Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel.
 * As translated and quoted by Karl Fink in Geschichte der Elementar-Mathematik (1890) A Brief History of Mathematics (1903) Beman, W.W. & Smith, D.E., transl., at an unspecified page.
 * I propose to show that God, in creating the universe and arranging the spheres, had in view the five regular solids of geometry, and fixed by their dimensions the number, proportions and motions of the spheres. Take the sphere of the earth as a unit and circumscribe it with a regular dodecahedron. The sphere that contains this dodecahedron is the sphere of Mars.
 * Translation & quotation by staff writer (April 29, 1911) See p. 278.

De fundamentis astrologiae certioribus (1601)

 * Written 1601, published 1602. Title translations: On the more Certain Fundamentals of Astrology or On Giving Astrology Sounder Foundations.


 * Vim coeli reserate viri: venit agnita ad usus: Ignotae videas commoda nulla rei.
 * Discover the force of the heavens O Men: Once recognised it can be put to use: No use could be seen in unknown things.
 * Unknown translator, from De fundamentis astrologiae certioribus, in Opera Omnia, Vol. 1, Heyder & Zimmer, 1858, p. 417-438.


 * He who will please the crowd and for the sake of the most ephemeral renown will either proclaim those things which nature does not display or even will publish genuine miracles of nature without regard to deeper causes is a spiritually corrupt person… With the best of intentions I publicly speak to the crowd (which is eager for things new) on the subject of what is to come.
 * Unknown translator, from De fundamentis astrologiae certioribus, ibid., Foreword.

Astronomia nova (1609)
Unless otherwise noted, translations and quotations by Walter W. Bryant, Full tile: Astronomia Nova ΑΙΤΙΟΛΟΓΗΤΟΣ seu physica coelestis, tradita commentariis de motibus stellae Martis ex observationibus G.V. Tychonis Brahe. Title translations: New Astronomy, Based upon Causes, or Celestial Physics, Treated by Means of Commentaries on the Motions of the Star Mars, from the Observations of Tycho Brahe, Gent; also known as Commentaries on the Motions of Mars.


 * Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it.
 * p. 35.


 * Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth.
 * p. 35, f.


 * ...wheresoever the earth may be placed, or whithersoever it may be carried by its animal faculty, heavy bodies will always be carried towards it.
 * p. 36.


 * If the earth were not round, heavy bodies would not tend from every side in a straight line towards the center of the earth, but to different points from different sides.
 * p. 36.


 * If two stones were placed... near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other.
 * p. 36.


 * If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density.
 * p. 36.


 * If the earth should cease to attract its waters to itself all the waters of the sea would be raised and would flow to the body of the moon.
 * p. 36.


 * The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward.
 * p. 36.


 * If the attractive virtue of the moon extends as far as the earth, it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther; and, in short, nothing which consists of earthly substance anyhow constituted although thrown up to any height, can ever escape the powerful operation of this attractive virtue.
 * pp. 36-37.


 * Nothing which consists of corporeal matter is absolutely light, but that is comparatively lighter which is rarer, either by its own nature, or by accidental heat. And it is not to be thought that light bodies are escaping to the surface of the universe while they are carried upwards, or that they are not attracted by the earth. They are attracted, but in a less degree, and so are driven outwards by the heavy bodies; which being done, they stop, and are kept by the earth in their own place.
 * p. 37.


 * But although the attractive virtue of the earth extends upwards, as has been said, so very far, yet if any stone should be at a distance great enough to become sensible compared with the earth’s diameter, it is true that on the motion of the earth such a stone would not follow altogether; its own force of resistance would be combined with the attractive force of the earth, and thus it would extricate itself in some degree from the motion of the earth.
 * p. 37.


 * I was almost driven to madness in considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit. Oh, ridiculous me! As the liberation in the diameter could not also be the way to the ellipse. So this notion brought me up short, that the ellipse exists because of the liberation. With reasoning derived from physical principles, agreeing with experience, there is no figure left for the orbit of the planet but a perfect ellipse.
 * Translation & quotation by John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2012), Ch. 58.

Dissertatio cum Nuncio Sidereo (1610)



 * It is not improbable, I must point out, that there are inhabitants not only on the moon but on Jupiter too, or (as was delightfully remarked at a recent gathering of certain philosophers) that those areas are now being unveiled for the first time. But as soon as somebody demonstrates the art of flying, settlers from our species of man will not be lacking. Who would once have thought that the crossing of the wide ocean was calmer and safer than of the narrow Adriatic Sea, Baltic Sea, or English Channel? Given ships or sails adapted to the breezes of heaven, there will be those who will not shrink from even that vast expanse. Therefore, for the sake of those who, as it were, will presently be on hand to attempt this voyage, let us establish the astronomy, Galileo, you of Jupiter, and me of the moon.
 * Translated by Edward Rosen, Kepler's Conversation with Galileo's Sidereal Messenger (1965), p. 39.

Harmonices Mundi (1618)

 * Title translations: The Harmony of the World.




 * No operation of addition or subtraction gives rise to diversity, but all are equally related to their pair of Terms, or Elements.
 * Book I, sect. XX, as translated by Aiton, Duncan and Field, American Philosophical Society (1997), p 25.


 * Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God.
 * Book III, Ch. 1 as quoted by, "Astrology in Kepler's Cosmology" in Astrology, Science, and Society: Historical Essays (1987) ed. P. Curry, p. 154.
 * Geometry, coeternal with God and shining in the divine Mind, gave God the pattern... by which he laid out the world so that it might be best and most beautiful and finally most like the Creator.
 * As quoted by, Kepler's Geometrical Cosmology (1988), p. 123.


 * Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world (for what is there in God which is not God?), and he with his own image reached down to humanity.
 * Book IV, Ch. 1, as quoted in "Kepler's Astrology"in Kepler, Four Hundred Years (1975) ed. Arthur and Peter Beer.




 * Now because 18 months ago the first dawn, 3 months ago broad daylight but a very few days ago the full sun of the most highly remarkable spectacle has risen — nothing holds me back. I can give myself up to the sacred frenzy, I can have the insolence to make a full confession to mortal men that I have stolen the golden vessel of the Egyptians to make from them a tabernacle for my God far from the confines of the land of Egypt. If you forgive me I shall rejoice; if you are angry, I shall bear it; I am indeed casting the die and writing the book, either for my contemporaries or for posterity to read, it matters not which: let the book await its reader for a hundred years; God himself has waited six thousand years for his work to be seen.
 * Book V, Introduction.
 * Variant translation: It may well wait a century for a reader, as God has waited six thousand years for an observer.
 * As quoted by David Brewster, The Martyrs of Science; or, the Lives of Galileo, Tycho Brahe, and Kepler (1841) p. 197. Sometimes misquoted as: "It may be well to wait a century for a reader, as God has waited six thousand years for an observer."
 * Variant translation: I feel carried away and possessed by an unutterable rapture over the divine spectacle of heavenly harmony... I write a book for the present time, or for posterity. It is all the same to me. It may wait a hundred years for its readers, as God has also waited six thousand years for an onlooker.
 * As quoted by Steven G. Krantz, Brian E. Blank, Calculus. Multivariable (2006) p. 126.


 * If you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labour of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that "the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances..."
 * Book V, Ch. 3 dates that his Third Law of Planetary Motion occurred to him, translation by E. J. Aiton, A. M. Duncan, and J. V. Field, The Harmony of the World (1997), Vol. 209, p. 411.
 * Variant translation: A fresh assault overcame the darkness of my reason...
 * As quoted by Steven G. Krantz, Brian E. Blank, Calculus. Multivariable (2006) p. 126.


 * The Earth sings Mi-Fa-Mi, so we can gather ever from this that Misery and Famine reign in our habitat.
 * Book V, Ch. 6, as quoted by Arthur Koestler, The Sleepwalkers (1959)


 * The heavenly bodies are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which... sets landmarks in the immeasurable flow of time. It is therefore, no longer surprising that man, in imitation of his creator, has at last discovered the art of figured song, which was unknown to the ancients. Man wanted to reproduce the continuity of cosmic time... to obtain a sample test of the delight of the Divine Creator in His works, and to partake of his joy by making music in the imitation of God.
 * Book V, Ch. 7, as quoted by Arthur Koestler, The Sleepwalkers (1959)


 * The wisdom of the Lord is infinite as are also His glory and His power. Ye heavens, sing His praises: sun, moon, and planets, glorify Him in your ineffable language! Praise Him, celestial harmonies, and all ye who can comprehend them! And thou, my soul, praise thy Creator! It is by Him and in Him that all exist.
 * As quoted in Forty Thousand Sublime and Beautiful Thoughts (1904) ed. Charles Noel Douglas, p. 845. See also Methodist Review (1873), vol. 55, pp. 187–88.

Joannis Kepleri Astronomi Opera Omnia (1858)

 * Unless otherwise noted, translations and quotations by Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925) citing the Latin source by Christian Frisch, ed., Joannis Kepleri Astronomi Opera Omnia (1858-1871) 8 Volumes. Sorted by Frisch references.


 * Just as the eye was made to see colours, and the ear to hear sounds, so the human mind was made to understand, not whatever you please, but quantity.
 * Frisch, Vol. I, p. 31; Burtt, p. 57.


 * [N]either this nor that supposition is worthy of the name of an astronomical hypothesis, but rather that which is implied in both alike.
 * Frisch, Vol. I, p. 238; Burtt describes this summary statement as describing Kepler's thought: [O]f a number of variant hypotheses about the same facts, that one is true which shows why facts, which in the other hypotheses remain unrelated, are as they are, i.e., which demonstrates their orderly and rational mathematical connexion. p. 54.


 * Indeed I reply in a single word to the sentiments of the saints on these questions about nature; in theology, to be sure, the force of authorities is to be weighed, in philosophy, however, that of causes. Therefore, a saint is Lactantius, who denied the rotundity of the earth; a saint is Augustine, who, admitting the rotundity, yet denied the antipodes; worthy of sainthood is the dutiful performance of moderns who, admitting the meagreness of the earth, yet deny its motion. But truth is more saintly for me, who demonstrate by philosophy, without violating my due respect for the doctors of the church, that the earth is both round and inhabited at the antipodes, and of the most despicable size, and finally is moved among the stars.
 * Frisch, Vol. III. p. 156; Burtt, pp. 55-56.


 * [W]ithout proper experiments I conclude nothing...
 * Frisch, Vol. V. p. 224 Cf. also Vol. I, p. 143; Burtt, p. 50.


 * I certainly know that I owe it [the Copernican theory] this duty, that as I have attested it as true in my deepest soul, and as I contemplate its beauty with incredible and ravishing delight, I should also publicly defend it to my readers with all the force at my command.
 * Frisch, Vol. VI, p. 116 & Cf. also Vol. VIII, p. 266, ff; Burtt, p. 47.


 * Wherever there are qualities there are likewise quantities, but not always vice versa.
 * Frisch, Vol. VIII, p. 147, ff; Burtt, p. 57.


 * There are, in fact, as I began to say above, not a few principles which are the special property of mathematics, such principles as are discovered by the common light of nature, require no demonstration, and which concern quantities primarily; then they are applied to other things, so far as the latter have something in common with quantities. Now there are more of these principles in mathematics than in the other theoretical sciences because of that very characteristic of the human understanding which seems to be such from the law of creation, that nothing can be known completely except quantities or by quantities. And so it happens that the conclusions of mathematics are most certain and indubitable.
 * Frisch, Vol. VIII, p. 148; Burtt, p. 57.


 * [Quantity is the fundamental feature of things,] the 'primarium accidens substantiae,' ...prior to the other categories.
 * Frisch, Vol. VIII, p. 150; Burtt, p. 57.


 * God gives every animal the means of saving its life—why object if he gives astrology to the astronomer?
 * Vol. VIII, p. 705; Burtt, p. 58.

Disputed

 * I was merely thinking God's thoughts after Him. Since we astronomers are priests of the highest God in regard to the book of nature, it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God.
 * Google search of the second sentence, in quotes, yields a trio of 2019 books alone, most (there and in following) attributing it to Kepler—e.g., see Prof Basden's 2019 work, (page 222).
 * While most citations of Kepler have been traced back to a translation of an original work, this quotation appears broadly without any such sourcing (e.g., Basden). Where it is sourced, the sources are either spurious (e.g., to the "New World Encyclopedia", a Paragon House/Unification Church product, wherein it is likewise unsourced), or to such sources as Henry Morris' 1988 creationist work, (page 21f).
 * Until a scholarly source is found that ties these statements to an original text from Kepler, they formally must be considered unattributed to Kepler.


 * The laws of nature are but the mathematical thoughts of God.
 * Attributed to Kepler in some sources (though more recent sources often attribute it to Euclid), such as Mathematically Speaking: A Dictionary of Quotations edited by Carl C. Gaither and Alma E. Cavazos-Gaither (1998), p. 214. The earliest publication located that attributes the quote to Kepler is the piece "The Mathematics of Elementary Chemistry" by Principal J. McIntosh of Fowler Union High School in California, which appeared in School Science and Mathematics, Volume VII (1907), p. 383. Neither this nor any other source located gives a source in Kepler's writings, however, and in an earlier source, the 1888 Notes and Queries, Vol V., it is attributed on p. 165 to Plato. Expressions that relate geometry to the divine "mind of God" include comments in the Harmonices Mundi, e.g., "Geometry is one and eternal shining in the mind of God", and "Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world".

Quotes about Kepler

 * Alphabetized by author

A-B

 * Kepler was the first to discover the art of successfully inquiring [into] her laws of nature, since his predecessors merely constructed explanatory concepts which they endeavoured to apply to the course of nature.
 * E. F. Apelt, Epcchen der Geschichte der Menscheit, Vol. I, p. 243 as quoted by Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * It is not known so generally that Kepler was... a geometrician and algebraist of considerable power, and that he, Desargues, and perhaps Galileo, may be considered as forming a connecting link between the mathematicians of the renaissance and those of modern times. Kepler's work in geometry consists rather in certain general principles enunciated, and illustrated by a few cases, than in any systematic exposition of the subject. In a short chapter on conics inserted in his Paralipomena, published in 1604, he lays down what has been called the principle of continuity, and gives as an example the statement that a parabola is at once the limiting case of an ellipse and of a hyperbola; he illustrates the same doctrine by reference to the foci of conics (the word focus was introduced by him); and he also explains that parallel lines should be regarded as meeting at infinity. He introduced the use of the eccentric angle in discussing properties of the ellipse.
 * W. W. Rouse Ball, A Short Account of the History of Mathematics (1888)


 * Kepler's laws were the climax of thousands of years of an empirical geometry of the heavens. They were discovered as the result of about twenty-two years of incessant calculation, without logarithms, one promising guess after another being ruthlessly discarded as it failed to meet the exacting demands of observational accuracy. Only Kepler's Pythagorean faith in a discoverable mathematical harmony in nature sustained him. The story of his persistence in spite of persecution and domestic tragedies that would have broken an ordinary man is one of the most heroic in science.
 * Eric Temple Bell, The Development of Mathematics (1940)


 * After his own fashion, Desargues discussed... Kepler's principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there...
 * Eric Temple Bell, The Development of Mathematics (1940)


 * In his curious tract on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. Prompted to the task by a dispute with the seller of some casks of wine, he studied the measurement of solids formed by the revolution of a curve round any line whatever. In solving some of the simplest of these problems, he conceived a circle to be formed of an infinite number of triangles having all their vertices in the centre, and their infinitely small bases in the circumference of the circle, and by thus rendering familiar the idea of quantities infinitely great and infinitely small, he gave an impulse to this branch of mathematics. The failure of Kepler, too, in solving some of the more difficult of the problems which he himself proposed roused the attention of geometers, and seems particularly to have attracted the notice of Cavaleri.
 * Sir David Brewster, The Life of Sir Isaac Newton (1832)


 * When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humors, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets.
 * Walter William Bryant, Kepler (1920), p. 35 Note reference: William Gilbert's New Philosophy about our Sublunary World or De Mundo Nostro Sublunari Philosophia Nova (1651)


 * Now, if the Earth move, it is a Planet, and shines to them in the Moone, and to the other Planetary inhabitants, as the Moone and they doe vs upon the Earth: but shine she doth, as Galilie, Kepler, and others prove, and then they per consequens, the rest of the Planets are inhabited, as well as the Moone, which he grants in his dissertation with Galilies Nuncius Siderius, that there be Joiviall and Saturnine Inhabitants, &tc. and that those severall Planets, have their severall Moones about them, as the Earth hath hers, as Galileus hath already evinced by his glasses... yet Kepler, the Emperours Mathematitian, confirms out of his experience, that he saw as much, by the same helpe. Then (I say) the Earth and they be Planets alike, inhabited alike, moved about by the Sunne, the common center of the World alike, and it may be those two greene children... that fell from Heaven, came from thence. We may likewise insert with Campanella and Brunus, that which Melissus, Democritus, Leucipus maintained in their ages, there be infinite Worlds, and infinite Earths, or systemes, because infinite starres and planets, like unto this of ours. Kepler betwixtiest and earnest in his Perspectives, Lunar Geography, dissertat cum nunc:syder seemes in part to agree with this, and partly to contradict; for the Planets he yeelds them to be inhabited, he doubts of the Starres: and so doth Tycho in his Astronomicall Epistles, out of consideration of their variety and greatnesse... that he will never beleeve those great and huge Bodies were made to no other use, then this that we perceave, to illuminate the Earth, a point insensible, in respect of the whole. But who shall dwell in these vast Bodies, Earths, Worlds, if they be inhabited? rational creatures, as Kepler demands? Or have they soules to be saved? Or do they inhabit a better part of the World then we doe? Are we or they Lords of the World? ...this only he proves, that we are in the best place, best World, nearest the Heart of the Sun. Thomas Campanella... subscribes to this of Keplerus, that they are inhabited hee certainly supposeth... and that there are infinite worlds, having made an Apologie for Galileus...
 * Robert Burton, The Anatomy of Melancholy (1628) Part 2, Sect. 2, pp. 239-240.


 * Kepler's achievements in mathematics would alone have been sufficient to win for him enduring fame; he first enunciated clearly the principle of continuity in mathematics, treating the parabola as at once the limiting case of the ellipse and the hyperbola, and showing that parallel lines can be regarded as meeting at infinity; he introduced the word 'focus' into geometry; while in his Stereometria Dolorum, published 1615, he applied the conception to the solution of certain volumes and areas by the use of infinitesimals, thus preparing the way for Desargues, Cavalieri, Barrow, and the developed calculus of Newton and Leibniz.
 * Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * The Neo-Platonic background, which furnished the metaphysical justification for much of this mathematical development (at least as regards its bearing on astronomy) awoke Kepler's full conviction and sympathy. Especially did the aesthetic satisfactions gained by this conception of the universe as a simple, mathematical harmony, appeal vigorously to his artistic nature.
 * Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * Founder of exact modern science though he was, Kepler combined with his exact methods and indeed found his motivation for them in certain long discredited superstitions, including what it is not unfair to describe as sunworship.
 * Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * The sun, according to Kepler, is God the Father, the sphere of the fixed stars is God the Son, the intervening ethereal medium, through which the power of the sun is communicated to impel the planets around their orbits, is the Holy Ghost.
 * Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * Kepler in the first thirty years of the seventeenth century "reduced to order the chaos of data" left by, and added to them just the thing that was needed—mathematical genius. Like Copernicus he created another world-system which, since it did not ultimately prevail, merely remains as a strange monument of colossal intellectual power working on insufficient materials; and even more than Copernicus he was driven by semi-religious fervour—a passion to uncover the magic of mere numbers and to demonstrate the music of the spheres. ...He has to his credit a collection of discoveries and conclusions—some of them more ingenious than useful—from which we today can pick out three that have a permanent importance in the history of astronomy.
 * Herbert Butterfield, The Origins of Modern Science (1949)

C-D
Other points of mathematical interest in Kepler's works are (1) the assertion that the circumference of an ellipse, whose axes are 2a and 2b, is nearly &#x3C0; (a + b); (2) a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear; (3) the assumption of the principle of continuity (which differentiates modern from ancient geometry), when he shows that a has a focus at infinity, that lines radiating from this "cæcus focus" are parallel and have no other point at infinity. The Stereometria led Cavalieri... to the consideration of of infinitely small quantities.
 * Johannes Kepler... imbibed Copernican principles while at the University of Tubingen. His pursuit of science was repeatedly interrupted by war, religious persecution, pecuniary embarrassments, frequent changes of residence, and family troubles. In 1600 he became for one year assistant to... ... His first attempt to explain the solar system was made in 1596, when he thought he had discovered a curious relation between the five regular solids and the number and distance of the planets. The publication of this pseudo-discovery brought him much fame. At one time he tried to represent the orbit of Mars by the oval curve which we now write in polar coördinates, $$\rho = 2r cos^3\theta$$. Maturer reflection and intercourse with Tycho Brahe and Galileo led him to investigations and results worthy of his genius—"Kepler's laws." He enriched pure mathematics as well as astronomy. It is not strange that he was interested in the mathematical science which had done him so much service; for "if the Greeks had not cultivated s, Kepler could not have superseded Ptolemy." The Greeks never dreamed that these curves would ever be of practical use; Aristaeus and Apollonius studied them merely to satisfy their intellectual cravings after the ideal; yet the conic sections assisted Kepler in tracing the march of the planets in their elliptic orbits. Kepler made also extended use of logarithms and decimal fractions, and was enthusiastic in diffusing a knowledge of them. At one time, while purchasing wine, he was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum [Vinariorum] in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the, which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria.
 * Florian Cajori, A History of Mathematics (1893, 1919) 2nd edition, revised and enlarged, citing (Greeks' conic sections quote) William Whewell, History of the Inductive Sciences 3rd Ed., (1858) Vol. I, p 311.


 * A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed.
 * John L. Casti, "Correlations, Causes, and Chance," Searching for Certainty: How Scientists Predict the Future (1990).

Kepler later found a similar law for the satellites of Jupiter. Today we know that such a law holds for any system of bodies that circulates around a central parent body. There are many applications of Kepler's law; for instance, half a century later it gave Isaac Newton the clue to his discovery of the law of universal gravitation.
 * In his 1619 book The Harmony of the World he tells us that he discovered a harmonic law while delivering a lecture on astronomy to his students. Kepler found that for each planet, the cube of the average distance from the sun is proportional to the square of the period of revolution.
 * I. Bernard Cohen, The Triumph of Numbers: How Counting Shaped Modern Life (2005)


 * More than two hundred years before Poncelet, the important concept of a occurred independently to... Johann Kepler... and the French architect Girard Desargues... Kepler (in his Paralipomena in Vitellionem, 1604) declared that a parabola has two foci, one of which is infinitely distant in two opposite directions, and that any point on the curve is joined to this "blind focus" by a line parallel to the axis.
 * , Projective Geometry (1987)


 * The effective inventor of the telescope and compound microscope was Galileo... Galileo's account of the path of the rays through the concave eye-piece and convex objective which he used was not satisfactory and was considerably improved by Kepler, who suggested the use of two convex lenses which became the basis of later instruments. Kepler had already written an important optical treatise in the form of a commentary on Witelo's Perspectiva... His improvements to the telescope may be regarded as what he had learned from the thirteenth-century writer.
 * A.C. Crombie, Robert Grosssteste and the Origins of Experimental Science (1953)


 * With the discovery of the law of inertia and the subsequent downfall of the Aristotelian theory of motion on which Kepler had based his work, his physical theories soon became outmoded and were then rendered obsolete by Newton's work. Yet Kepler's laws of planetary motion remained, so that Edmond Halley could write in his review of Newton's Principia that the first eleven propositions were found to agree with the phenomena of celestial motions, as discovered by the great sagacity and diligence of Kepler.
 * A.M. Duncan, J.V. Field, The Harmony of the World (1997), Preface, Vol.209


 * Although the concept of heavenly harmony was a theme mentioned in the literature of the time... Kepler's world harmony had little influence on his contemporaries. ...With the rise of the experimental science advocated by Francis Bacon and greatly facilitated by the invention and development of scientific instruments, the general trend of the seventeenth century was towards a mechanical natural philosophy in which metaphysical speculation would play little part. Another factor... may possibly be recognized in the nature of developments that had taken place in mathematics during the sixteenth century, for the advances in algebra and the introduction of symbolism favored a nominalist view of mathematics in contrast to the realist Platonic view of geometry that Kepler adopted as a foundation for his theory of a world harmony.
 * A.M. Duncan, J.V. Field, ibid.


 * When he discovered the polyhedral hypothesis soon after being sent to teach mathematics in Graz, he changed his mind [about becoming a Lutheran minister], indicating... that he now saw his work in astronomy as an exercise of a priestly vocation. ...he claimed that, in the Harmonice mundi, he offered to the world nothing less than the plan of creation, which God himself had waited six thousand years for someone to comprehend.
 * A.M. Duncan, J.V. Field, ibid.

E-G

 * Kepler is the first who ventured here [into] an exact mathematical treatment of the problems (of astronomical science), the first to establish natural laws in the specific sense of the new science.
 * R. Eucken, "Kepler als Philosoph" in Philosophische Monatshefte (1878), p. 42ff as quoted by Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science (1925)


 * Dumbleton was one of the first to express functional relationships in graphical form. ...Dumbleton also gave a proof of the Merton mean-speed rule... stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint." He used the method in the Suma [Suma logicæ et philosophiæ naturalis] to study the problem of the variation in the strength of light as a function of the distance from its source. ...He realized that that the decrease in intensity of illumination was not linearly proportional to the distance... But he did not succeed in finding the exact quantitative relationship, which is that the intensity of illumination due to a luminous source is inversely proportional to the square of the distance, a law discovered by Johannes Kepler in 1604.
 * John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2012)


 * I esteem myself happy to have as great an ally as you in my search for truth. I will read your work … all the more willingly because I have for many years been a partisan of the Copernican view because it reveals to me the causes of many natural phenomena that are entirely incomprehensible in the light of the generally accepted hypothesis. To refute the latter I have collected many proofs, but I do not publish them, because I am deterred by the fate of our teacher Copernicus who, although he had won immortal fame with a few, was ridiculed and condemned by countless people (for very great is the number of the stupid).
 * Galileo Galilei, Letter to Kepler (1596), as quoted in The Story of Civilization : The Age of Reason Begins, 1558-1648 (1935) by Will Durant, p. 603

This I shall do the more willingly because many years ago I became a convert to the opinions of Copernicus, and by his theory have succeeded in explaining many phenomena which on the contrary hypothesis are altogether inexplicable. I have arranged many arguments and confutations of the opposite opinions, which, however, I have not yet dared to publish, fearing the fate of our master, Copernicus, who, although he has earned immortal fame among a few, yet by an infinite number (for so only can the number of fools be measured) is hissed and derided. If there were many such as you I would venture to publish my speculations, but since that is not so I shall take time to consider of it.
 * I have as yet read nothing beyond the preface of your book, from which, however, I catch a glimpse of your meaning, and feel great joy on meeting with so powerful an associate in the pursuit of truth, and consequently, such a friend to truth itself; for it is deplorable that there should be so few who care about truth, and who do not persist in their perverse mode of philosophising. But as this is not the fit time for lamenting the melancholy condition of our times, but for congratulating you on your elegant discoveries in confirmation of the truth, I shall only add a promise to peruse your book dispassionately, and with the conviction that I shall find in it much to admire.
 * Galileo Galilei, Letter to Kepler (Aug. 4, 1597) upon receipt of  and as quoted by John Joseph Fahie, Galileo: His Life and Work (1903) pp. 39-40.


 * I thank you because you are the first one, and practically the only one, to have complete faith in my assertions.
 * Galileo Galilei, Letter to Kepler (1610) in gratitude for Kepler's Dissertio cum Nuncio sidereal (Answer to the Sidereal Messenger) after Galileo sent Kepler a copy of the Sidereal Messenger, as quoted in John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2012)


 * To say... that the motion of the Earth meeting with the motion of the Lunar Orb, the concurrence of them occasioneth the Ebbing and Flowing [of the seas], is an absolute vanity, not onely be­cause it is not exprest, nor seen how it should so happen, but the falsity is obvious, for that the Revolution of the Earth is not con­trary to the motion of the Moon, but is towards the same way. So that all that hath been hitherto said, and imagined by others, is, in my judgment, altogether invalid. But amongst all the famous men that have philosophated upon this admirable effect of Nature, I more wonder at Kepler than any of the rest, who being of a free and piercing wit, and having the motion ascri­bed to the Earth, before him, hath for all that given his ear and assent to the Moons predominancy over the Water, and to oc­cult properties, and such like trifles.
 * Galileo Galilei, Diaologo sopra i Due Massi Sistemi del Mondo (1632) as quoted in Salusbury tr. The Systeme of the World: in Four Dialogues (1661)


 * J. Kepler was the first (that I know of) that discover'd the true cause of the Tide, and he explains it largely in his Introduction to the Physics of the Heavens, given in his Commentaries to the Motion of the Planet Mars, where after he has shewn the Gravity or Gravitation of all Bodies towards another, he thus writes: "The Orb of the attracting Power, which is in the Moon is extended as far as the Earth, and draws the Waters under the Torrid Zone, acting upon places where it is vertical, insensibly on included Seas, but sensibly on the Ocean, whose Beds are large, and the Waters have the liberty of reciprocation, that is, of rising and falling"; and in the 70th Page of his Lunar Astronomy,—"But the cause of the Tides of the Sea appear to be the Bodies of the Sun and Moon drawing the Waters of the Sea."
 * David Gregory, The Elements of Astronomy, Physical and Geometrical J. Nicholson (1715) Vol.2, p. 668


 * Afterwards that incomparable Philosopher Sir Isaac Newton, improv'd the hint, and wrote so amply upon this Subject as to make the Theory of the Tides his own, by shewing that the Waters of the Sea rise under the Moon and the Place opposite to it: For Kepler believ'd "that the Impetus occasion'd by the presence of the Moon, by the absence of the Moon, occasions another Impetus; till the Moon returning, stops and moderates the Force of that Impetus, and carries it round with its motion." Therefore this Spheroidical Figure which stands out above the Sphere (like two Mountains, the one under the Moon and the other in the place opposite to it) together with the Moon (which it follows) is carried by the Diurnal Motion, (or rather, according to the truth of the matter, as the Earth turns towards the East it leaves those Eminencies of Water, which being carried by their own motion slowly towards the East, are as it were unmov'd) in its journey makes the Water swell twice and sink twice in the space of 25 Hours, in which time the Moon being gone from the Meridian of any Place, returns to it again.
 * David Gregory, ibid., pp. 668–9


 * Galileo argued that nature, God's second book, is written in mathematical letters... Kepler is even more explicit in his work on world harmony; he says: God created the world in accordance with his ideas of creation. These ideas are the pure archetypal forms which Plato termed Ideas, and they can be understood by man as mathematical constructs. They can be understood by Man, because Man was created in the spiritual image of God. Physics is reflection on the divine Ideas of Creation, therefore physics is divine service.
 * Werner Heisenberg, Tradition in Science (1983) also published as Encounters with Einstein: And Other Essays on People, Places, and Particles

H-K

 * One wonders how many modern scientists faced by a similar situation in their work would fail to be impressed by such remarkable numerical coincidences.
 * Fred Hoyle on Kepler's attention to the apparent harmonics by which he deduced his planetary laws, as quoted in "Kepler's Astrology and Mysticism" by Arthur Beer in Vistas in Astronomy vol. 18 (1975).


 * If Kepler had been a mathematician of the twentieth century, he would have stopped his laborious observational inductions after noting his first law, and deduced the other two analytically.
 * Edward Kasner, "Differential-Geometric Aspects of Dynamics," (1909) in The Princeton Colloquium: Lectures on Mathematics, Delivered September 15 to 17, 1909 (1913) Vol. 3, Part 1.


 * Copernicus, Kepler and Galileo were ‘revisionists’ in rejecting the geocentric system of Ptolemy (which held sway for some 1500 years) and, against an oppressive and repressive mainstream opinion (and officialdom), reinstated—with improvements—the heliocentric system of Aristarchos of Samos (3rd cent BCE).
 * Kazanas, N. (2002). Indigenous Indo-Aryans and the Rigveda: Indo-Aryan migration debate. Journal of Indo-European Studies, 30(3-4), 275-334.


 * Kepler (and Desargues) regarded the two "ends" of the ["straight"] line as meeting at "infinity" so that the line has the structure of a circle. In fact, Kepler actually thought of a line as a circle with its center at infinity.
 * Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)


 * Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance.
 * Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)


 * The Pythagorean dream of musical harmony governing the motion of the stars never lost its mysterious impact, its power to call forth responses from the depth of the unconscious mind. ...But, one might ask, was the "Harmony of the Spheres" a poetic conceit or a scientific concept. A working hypothesis or a dream dreamt through a mystic's ear? ...Even Aristotle laughed "harmony, heavenly harmony" out of the courts of earnest, exact science. Yet... Johannes Kepler became enamoured with the Pythagorean dream, and on this foundation of fantasy, by methods of reasoning equally unsound, built the solid edifice of modern astronomy. It is one of the most astonishing episodes in the history of thought, and an antidote to the pious belief that the Progress of Science is governed by logic.
 * Arthur Koestler, The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959, 1963)


 * The Harmony of the World is the continuation of the Cosmic Mystery, and the climax of his lifelong obsession. What Kepler attempted here is, simply, to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. It was the first attempt of this kind since Plato, and it is the last to our day. After Kepler, fragmentation of experience sets in again, science is divorced from religion, religion from art, substance from form, matter from wind.
 * Arthur Koestler, The Sleepwalkers (1959)

L-M

 * Kepler made free use if indivisibles in both astronomical work and a treatise on measuring volumes of wine casks. He went far beyond the practical needs... and wrote an extensive tract on indivisible methods. Two illustrative examples are his approaches to the areas of a circle and an ellipse.
 * Reinhard Labenbacher, David Pengelley, Mathematical Expeditions: Chronicles by the Explorers (1999)


 * But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles.
 * Colin Maclaurin, An Account of Sir Isaac Newton's Philosophical Discoveries (1750), pp. 54–55


 * He [Kepler] supposes, in that treatise [epitome of astronomy], that the motion of the sun on his axis is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets and carries them along with it in the same direction; as a load-stone turned round in the neighborhood of a magnetic needle makes it turn round at the same time. The planet, according to him, by its inertia endeavors to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like to his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelium to the perihelium, and receding from the sun while it ascends to the aphelium again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part which is attracted is turned towards the sun in the descent, and that the other part is towards the sun in the ascent. By suppositions of this kind he endeavored to account for all the other varieties of the celestial motions.
 * Colin Maclaurin, ibid., p. 55


 * Luckily, Napier came on the scene with his logarithms just when Johannes Kepler, the discoverer of the laws of planetary motion, was deeply immersed in mind-numbing, tedious calculations, filling hundreds of folio pages with lengthy arithmetic operations, in his construction of the orbit of Mars from the observational data of Tycho Brahe. To Kepler, this discovery was a gift from heaven, for logarithms reduced considerably the time he had to spend just doing arithmetic calculations, a task which he detested.
 * Lloyd Motz, Jefferson Hane Weaver, Conquering Mathematics: From Arithmetic to Calculus (1991)

N-R

 * As living bodies have hair, so does the earth have grass and trees, the cicadas being its dandruff; as living creatures secrete urine in a bladder, so do the mountains make springs; sulphur and volcanic products correspond to excrement, metals and rainwater to blood and sweat; the sea water is the earth's nourishment … At the same time the anima terrae [soul of the earth] is also a formative power (facultas formatrix) in the earth's interior and expresses, for example, the five regular bodies in precious stones and fossils ..... It is important that in Kepler's view the anima terrae is responsible for the weather and also for meteoric phenomena. Too much rain, for instance, is an illness of the earth.
 * Wolfgang Pauli in "The Influence of Archetypal Ideas on the Scientific Theories of Kepler" in The Interpretation of Nature and the Psyche (1955) by Carl Jung and Wolfgang Pauli, as translated by Priscilla Silz

intensity of radiation $$=\frac {S}{4\pi}\cdot\frac {1}{R^2}$$ i.e., the intensity is inversely proportional to the square of the distance between the planet P and the Sun. ...Kepler thought carefully about the possibility, but was dubious... to his credit; he mistrusted the idea for a very good reason. ...although during a solar eclipse the Moon blocks the Sun's radiation to part of the Earth, there is no discontinuity in the Earth's motion. If gravitational attraction were radiated as light is radiated, this too would be temporarily blocked by the Moon, so that during the eclipse it would discontinue its eliptical orbit...
 * Kepler also thought of the Inverse Square Law; he thought of it first. ...Kepler regarded gravitational attraction as analogous to propagation of light... Consider now the intensity of light falling on a planet P at a distance R from the Sun. Let S be the total amount if light emitted by the Sun. ...the intensity will be the same at all points distance R from the Sun. But these points constitute a spherical sheet (with center the Sun) whose radius is R and whose surface area, therefore, is 4&#x3C0;R2. Consequently,
 * George Pólya, Mathematical Methods in Science (1977)

S-Z

 * Kepler was a brilliant thinker and a lucid writer, but he was a disaster as a classroom teacher. He mumbled. He digressed. He was at times utterly incomprehensible. He drew only a handful of students his first year at Graz; the next year there were none. He was distracted by an incessant interior clamour of associations and speculations vying for his attention. And one pleasant summer afternoon, deep in the interstices of one of his interminable lectures, he was visited by a revelation that was to alter radically the future of astronomy. Perhaps he stopped in mid-sentence. His inattentive students, longing for the end of the day, took little notice, I suspect, of the historic moment.
 * Carl Sagan in Cosmos (1980)


 * Kepler's project in '' was to give 'true and perfect reasons for the numbers, quantities, and periodic motions of celestial orbits.' The perfect reasons must be based on the simple mathematical principles, which had been discovered by Kepler in the solar system, by using geometric demonstrations. The general scheme of his model was borrowed... from Plato's 'Timaeus', but the mathematical relations for the s (pyramid, cube,, , ) were taken by Kepler from the works of Euclid and Ptolemy. Kepler followed Proclus and believed that 'the main goal of Euclid was to build a geometric theory of the so-called Platonic solids.' Kepler was fascinated by Proclus and often quotes him calling him a 'Pythagorean'.
 * C. Smorinsky, History of Mathematics: A Supplement (2008)


 * Nearly three thousand years ago, the ancient Egyptians knew that a glass lens can make an object look bigger. Nero... is said to have looked through an emerald to watch his gladiators fighting... By the ninth century, people were using 'reading stones' to assist their failing eyesight. These were polished lumps of clear glass, rounded on one side and flat on the other; you sat them on top of the document you were trying to read... The first true spectacles were almost certainly invented in Italy between 1280 and 1300. They acted like a magnifying glass and corrected long-sightedness; it would be another 300 years before lenses able to correct short-sightedness would be developed, in part because these were much harder to make. Johannes Kepler (astronomer, astrologer and mathematician) was the first to explain how convex and concave lenses corrected eyesight. ...lenses were (and still are) made by grinding glass using various types of abrasive material, which in Kepler's time were already being used by jewellers.
 * Ian Stewart, The Mathematics of Life (2011)

A letter of Henry Briggs to Kepler [dated Mar 10, 1625] suggest improvements in the Paralipomena ad Vitellionem. In this letter Briggs... comprehended and accepted Kepler's way of looking at parallels as lines to or from a point at infinity in one direction or its opposite.
 * Kepler states expressly that he gave the name Foci to certain points related to the conic sections which had previously "no name." With their new name he associated his new views about the points themselves, and his doctrines of Continuity (under the name Analogy) and Parallelism, which would soon have become known, and would after a time have been taken up by competent mathematicians....
 * Charles Taylor, "The Geometry of Kepler and Newton," (1899) Transactions of the Cambridge Philosophical Society (1900) pp. 197-219

...The printing ...was finished on time in September 1627 ...but he was not optimistic ...noting, "There will be few purchasers, as is always the case with mathematical works, especially in the present chaos."
 * It was only the third new set of planetary tables in European history. And whereas Copernicus's and Ptolemy's tables were more or less equally accurate, Kepler's were some 50 times more so. Within a few years, it was possible to pinpoint the time of transit of Mercury across the face of the sun so that it was possible to observe it in transit for the first time in human history. Of course, Kepler's theories were more difficult, especially since he had incorporated logarithms, which had only been invented a few years earlier. Much of the book, therefore, was made up of explanatory text that told the reader how to use the tables.
 * James R. Voelkel, Johannes Kepler and the New Astronomy (2001) p. 117