About the Work
Ad Vitellionem paralipomena is a substantial treatise by the German astronomer Johannes Kepler, printed in Frankfurt in the year 1605 by the publishers Claudius Marnius and the heirs of Ioannes Aubrius. The work is dedicated to the Holy Roman Emperor Rudolphus II, who is praised for his support of astronomical science. The book is presented as a supplement or complement to the earlier optical writings of the medieval scholar Vitellio, hence its title, which translates roughly to “Supplements to Vitellio.” The volume includes a title page, a dedication, a table of chapters, dedicatory epigrams, a prooemium (preface), and a series of chapters, beginning with a section on the nature of light. The work is a dense, technical treatise that blends geometry, physics, and astronomy, and it is written in the scholarly Latin of the early seventeenth century. It engages deeply with the observational legacy of Tycho Brahe, whose data Kepler frequently cites and analyzes, and it also enters into dialogue with ancient and medieval authorities such as Aristotle, Ptolemy, Alhazen, and Vitellio, as well as more recent figures like Giambattista della Porta and the astronomer Michael Mästlin.
Context
The book emerges from a specific set of intellectual and observational problems that were pressing in the decades around 1600. Kepler explicitly states that the work originated from questions raised by Tycho Brahe’s astronomical observations. Two optical problems in particular prompted the investigation: the refraction of starlight as it passes through the Earth’s atmosphere, and the puzzling observation that the Moon’s diameter appears diminished during eclipses. Kepler’s task, as he frames it, is to address these anomalies through a rigorous optical theory. The work is thus situated at the intersection of astronomy and physics, a field that Kepler calls the “optical” part of astronomy. He divides astronomy into three parts: the mechanical, which concerns instruments; the historical, which concerns observations; and the optical, which he undertakes to treat in this volume. The book is also a response to a live scientific dispute between Tycho Brahe and Christoph Rothmannus regarding the existence and cause of atmospheric refraction, with Rothmannus denying any difference between air and aether and attributing refraction to vapors, while Tycho argued for two causes: density differences and vapors. Kepler enters this debate, criticizing both positions for lacking a true measure of refraction. The work is further contextualized by Kepler’s engagement with the camera obscura, a technique popularized by Giambattista della Porta, which Kepler uses to analyze the projection of images and to measure solar and lunar diameters. The book also reflects Kepler’s broader cosmological and metaphysical commitments, including his belief in the Sun as the source of light and motion in the universe, a view he supports with references to Moses, Gilbert’s magnetic philosophy, and ancient philosophers.
Structure of the Book
The book is organized into a series of chapters that proceed from general principles of light and vision to specific applications in astronomy. The opening chapter, “De Natura Lucis” (On the Nature of Light), lays out a series of propositions about light: that it emanates from a source, that a point emits infinite lines, that light extends infinitely, that these lines are straight and called rays, that light’s motion is instantaneous, that light attenuates laterally with distance but not longitudinally, that a ray is nothing besides the light itself moving, that the density of light varies inversely with spherical surface area, that light is not impeded by the solidity of bodies, and that reflection occurs at equal angles while refraction occurs toward the perpendicular when light passes into a denser medium. This chapter also includes a critique of Aristotle’s tenets on light and transparency, arguing that light is not merely the “act” or form of the transparent medium but something additional to it. Kepler also introduces his theory of the projection of the Sun’s image through a small hole, crediting Vitellio, Johannes Pisanus, and Hartmannus for prior attempts but noting their failures, and presenting his own experiment with a thread and a multi-angled hole to prove that the projected figure is due to the Sun’s rotundity, not the hole’s shape.
The subsequent chapters address reflection and refraction in detail. Kepler discusses the equality of angles of incidence and reflection, and introduces the concept of the “Plane of Refraction,” which is perpendicular to the refracting surface. He presents propositions on the location of images, arguing that the genuine place of an image is where the visual rays from both eyes, extended through their respective points of reflection or refraction, converge. He concludes that the image is placed on the perpendicular from the object to the surface, and that it is the “latitude of vision” (the distance between the two eyes) that causes this. He then moves to a chapter on the measure of refractions, where he refutes various methods of measuring refraction and presents his own, which uses an analogy with concave mirrors. This leads to a detailed analysis of refraction angles, using Vitellio’s data and Tycho Brahe’s observations to construct refraction tables for water and to find the proportions of media such as air to water. He also derives the altitude of the air, finding it to be about half a German mile at Uraniburg.
The book then applies these optical principles to astronomical phenomena. Kepler discusses astronomical refraction, explaining why the equinox appears to occur twice on the same day, a phenomenon observed by Tycho Brahe and noted by Hipparchus and Ptolemy. He proposes that refraction may explain why Hipparchus found a greater solar eccentricity than Tycho or Albategnius. The work then turns to the nature of the Sun and Moon. Kepler argues that the Sun’s body is made of the densest material in the universe, and that it is like a heart in the world, dispensing life and motion to the planets. He discusses the Moon’s illumination, refuting the idea that we should see the Sun’s image in it as in a convex mirror, and introducing a “fourth species” of light, “communicated light,” distinct from direct reflection and refraction. He calculates the circle of illumination on the Moon and asserts that a perfect full moon can never be seen. The book also addresses the question of whether planets have their own light, with Kepler arguing that Venus and other planets must have proper light, and he discusses the colors of planets, attributing them to the translucency and essential color causes in their matter. He also treats the light of comets, proposing that the Sun’s rays pass through the comet’s dense, impure translucent body, becoming colored and concentrated. The chapter on the Earth’s shadow argues that refraction of sunlight through the air shortens the shadow to about 45 Earth semi-diameters, not 268, and that the Moon’s redness in eclipse comes from refracted solar rays through the air. The book concludes with a chapter on the Moon’s shadow and diurnal darkness, and an appendix on methods for measuring the lunar diameter and observing eclipses, including the use of a camera obscura and various instruments.
Detailed Section-by-Section Summary
Title Page and Dedication to Emperor Rudolphus II
The volume opens with its title page, printed in Frankfurt in 1605 by Claudius Marnius and the heirs of Ioannes Aubrius. Kepler dedicates the work to Emperor Rudolphus II, praising the Emperor’s support of astronomy. He credits Tycho Brahe’s observations as the foundation of the work, noting that Brahe’s astronomy raised two specific optical problems: the refraction of starlight and the diminished diameter of the Moon during eclipses. Kepler outlines his plan to address refraction in Caput IV, lunar and solar diameters in Caput VI, vision in Caput V, the nature of light and colors in Caput I, and parallax in Caput IX, which he identifies as the most difficult part of astronomy. The front matter also includes a table of chapters and dedicatory epigrams.
The Prooemium and the Division of Astronomy
Kepler’s Prooemium divides astronomy into three parts: mechanical (concerning instruments), historical (concerning observations), and optical, which he will treat in this work. He praises solar and lunar eclipses as ancient and marvelous phenomena that have driven the need for precise optical understanding.
Caput I: On the Nature of Light
Kepler begins Caput I, “De Natura Lucis,” with a series of propositions. Light emanates from a source (Prop. I); a point emits infinite lines (II); light extends infinitely (III); these lines are straight and called rays (IV); light’s motion is instantaneous, not temporal (V); light attenuates laterally with distance from the center (VI), but not longitudinally (VII); a ray is nothing besides the light itself moving (VIII); the density of light varies inversely with the spherical surface area (IX); light is not impeded by the solidity of bodies (X); a transparent body is one whose parts are geometrically continuous (XI); light is affected by the surfaces of bodies (XII); dense bodies have dense surfaces (XIII); light passes less easily through dense surfaces (XIV); color is light in potency (XV); light and color do not act on each other (XVI); the opaque is that which is rough with many surfaces (XVII); light is reflected to the opposite region (XVIII); reflection occurs at equal angles (XIX); and light falling obliquely on a denser medium is refracted toward the perpendicular (XX).
Critique of Aristotle’s Tenets on Light and Transparency
Kepler disputes Aristotle’s claim that light is merely the “act” or form of the transparent medium. He argues that light is something additional to the transparent body’s presence, since the term’s origin relates to light’s motion through a body, not the body itself. He rejects the idea that transparency implies color-capacity, noting that a mere negation does not posit a habit. He affirms that colors exist in bodies even in darkness, and that illumination of colors enables the medium to be truly transparent, not vice-versa.
On Vision and Species
Kepler argues against the Aristotelian model where the transparent medium is altered by color to move the eye. He asserts that species or luminous rays flow from the Sun and illuminated colors, are attenuated, and depict their source on the opaque eye. Vision is confused when images of various colors are confounded, and distinct when they are not. He refutes the need for a medium to “inform” the sense, arguing that media exist only to prevent a vacuum. He cites the smell of Alpine valleys and the propagation of sound as analogies.
Refutation of Aristotle’s Arguments
Kepler systematically dismantles six Aristotelian arguments, asserting that the medium does not move the eye, but rather species and rays descend through it. He defends the possibility of species acting without inhering in a subject, comparing light’s incorporeal motion to a mathematical point. He challenges the notion that a medium is required for vision, citing examples of seeing with the eye pressed.
On the Figure of Light
Kepler introduces his theory on the projection of the Sun’s image through a small hole. He credits Vitellio, Johannes Pisanus, and Hartmannus for prior attempts but notes their failures. He recounts his own experiment with a thread and a multi-angled hole, proving that the projected figure is due to the Sun’s rotundity, not the hole’s shape. He notes that Tycho Brahe was forced to conclude the Moon’s diameter is smaller in conjunctions due to this unproven theorem. Kepler then presents definitions, a postulate, and geometric propositions demonstrating that the illuminated figure on a wall is similar to the aperture, and that the image is a composite of the inverted figure of the luminous body and the direct figure of the window. He credits I. Baptista Porta with the camera obscura technique but notes that Porta lacked a demonstration.
The Plane of Reflection and Refraction
The text continues with a discussion of reflection, asserting that the angles of incidence (FAD) and reflection (GAE) must be equal relative to the secondary line DE, not just to the reflecting surface. Therefore, the plane BFGC must be perpendicular to the reflecting surface DCEB. For refraction, the refracted ray BG lies in the same plane as the perpendicular AC and the incident ray. This plane is called the “Plane of Refraction,” which is perpendicular to the refracting surface.
Proposition XVII: The Location of Images
Kepler addresses the location of images, arguing that the genuine place of an image is where the “visual rays” from both eyes, extended through their respective points of reflection or refraction, converge. Since each eye has its own plane of reflection/refraction, and these planes intersect, the image and the visible object lie on their common intersection, which is a straight line perpendicular to the reflecting/refracting surface. Thus, the image is placed on the perpendicular from the object to the surface. He concludes that it is not nature or universal reason, but the “latitude of vision” (the distance between the two eyes) that causes the image to appear on the perpendicular.
Propositions XVIII and XIX: Deviations from the Perpendicular
Proposition XVIII shows that if both eyes lie in the same reflecting/refracting surface (contrary to nature), the image deviates from the perpendicular. For a convex spherical mirror, the image appears closer to the eye than the radiating point. The proof involves a complex geometric demonstration showing that the apex of the visual pyramid cannot lie on the perpendicular, leading to an absurdity. This holds for all inclinations except the perpendicular and horizontal, where the image approaches the points of reflection. Proposition XIX demonstrates the same for refraction: when both eyes view an object very obliquely through a denser medium, the image departs from the perpendicular and approaches the eye. The proof uses a circle and lines to show that unequal refraction at two points causes the image to move closer to the perpendicular from the visible point.
Proposition XX: Against the Common Optical Theorem
Proposition XX concludes that the common optical theorem is not universally true. Kepler contrasts his method with that of Alhazen and Vitellio, who used the perpendicular to the conoid surface. He argues that one must consider the circle of curvature at the point of reflection to find the true image location.
Chapter IV: On the Measure of Refractions
The text moves to Chapter IV, discussing the dispute between Tycho Brahe and Rothmannus. Rothmannus denied any difference between air and aether, attributing refraction to vapors near the horizon. Tycho argued for two causes: the density difference and vapors. Kepler criticizes both for lacking a true measure of refraction. He refutes various methods of measuring refraction, including those based solely on the length of the path through the denser medium, or solely on the angle of incidence. He argues the cause lies in the surface, not the body’s density. He also rejects the idea that the image’s elevation can be used to measure refraction, as it is based on visual deception. Finally, he describes his own method, which uses an analogy with concave mirrors, where the denser medium corresponds to a concave mirror, leading to a new approach to measuring refraction.
The Hyperbolic Construction of Refraction
The analysis continues with hyperboles, where perpendiculars BH, DI, FK are drawn from points B, D, F to line AS, with parallels BC, DE, FG touching the surface. Three refraction angles from Vitellio, with their inclinations, give three separate number classes. If connected so hyperboles submit to their laws, the surface will collect refractions. Using Vitellio’s angles, AH=100000 gives HB=57735 and CH=68806; AI=100000 gives ID=4473 and 46376; AK=100000 gives KF=35412 and KG=30249. The proportion of the figure’s sides and AP distance must be found via hyperbola laws, where HB², ID², KF² relate to rectangles CH·P, EI·P, GK·P, using Apollonius propositions 37, 29, 30.
Propositions on Refraction
Proposition I states that light refracted more obliquely makes a greater angle (Vitellio’s 14th, defectively demonstrated). Kepler’s proof: if the refraction angle did not increase with obliquity, the universal cause would fail. A corollary notes that if only density were considered, refractions would be proportional to incidence angles. Proposition II shows that oblique light encounters a denser medium—using triangle BLM with a right angle at L, LM measures perpendicular width, BM oblique width, so there is more density in BM. Proposition III states that refraction angles grow with greater increments than obliquity, combining proportional parts with secant-based parts. Proposition IV finds no difference between weak and strong light for refraction; Tycho Brahe found the same refractions for the Sun and Moon. Proposition V states that refractions increase most near the horizon. Proposition VI shows that secants of incidence in a denser medium measure refractions. Proposition VII states that in the densest medium all refractions go to perpendiculars. Proposition VIII constructs refraction tables for water, comparing with Vitellio’s. Proposition IX derives refractions from two given ones, using Tycho’s observations. Proposition X finds medium proportions (air to water). Proposition XI finds air’s altitude as half a German mile at Uraniburg.
Refraction and the Apparent Doubling of the Equinox
Kepler opens this discussion by asserting that atmospheric refraction of light from both the Sun and Moon sufficiently explains a puzzling phenomenon observed in Italy. He notes that Ptolemy, in Book 11 of his Magni operis, recorded an equinox observed twice on the same day using Alexandrian armillary spheres, blaming the instrument’s collapse for the discrepancy. However, Kepler points out that Hipparchus, who used the same armillaries closer to their original placement, is drawn into the same blame by Ptolemy, who concluded the two annotations disagreed by five hours. Hipparchus insisted his observation was optimal, defining the true equinoctial hour as the moment when both armillary surfaces are equally illuminated. Kepler observes that while instrument error could explain this, Tycho Brahe frequently noted with corrected instruments that the equinox appeared twice on the same day: in the morning, the rising Sun appears higher due to refraction, seeming already at the equator while still in the south; after noon, free from refraction, it truly completes the semicircle. The opposite applies to the autumnal equinox. Thus, Kepler leaves it uncertain whether instrument defect or refraction caused authors to believe they saw the equinox twice.
Hipparchus’s Eccentricity Explained by Refraction
Kepler then proposes that refraction may explain why Hipparchus found a greater solar eccentricity than Tycho or Albategnius. He notes that Ptolemy accommodated Hipparchus’s observations to his own demonstrations, and that the consensus between them in enumerating the year’s quadrants is greater than their instruments could possibly allow. Kepler suspects that Hipparchus defined the equinox day by the Sun touching opposite horizon points, a method refraction would alter, lengthening the summer semicircle and thereby increasing the apparent eccentricity.
The Sun’s Dense and Vital Matter
Kepler argues that the Sun’s body is made of the densest material in the universe, as required by its intense, far-reaching force. Citing Moses, he posits that all things were made from water and that light was created on the first day. He agrees with an ancient philosopher, mentioned in Diogenes Laertius, that the Sun is a “candentem lapidem” (glowing stone), a view supported by Gilbert’s magnetic philosophy. The Sun’s matter is simple, pure, and pellucid despite its density, being condensed and purified water. Kepler describes the Sun’s office as like a heart in the world, dispensing life and motion to the planets. A vital faculty resides in the Sun’s body, producing light, akin to how flames are generated by animal faculties such as oil or resin. This light is generated in the Sun’s center and spreads, being reflected internally, causing intense central incandescence. The Sun’s surface disperses light to the world; the center is the origin of light, the surface its dispenser.
The Moon’s Illumination and the Fourth Species of Light
Turning to the Moon’s illumination, Kepler recalls that Berosus the Chaldean held the Moon was half-lucid, while Thales of Miletus correctly stated that the Moon is illuminated by the Sun, a view Plutarch attributes to Anaxagoras. Some argue that if the Moon reflected sunlight, we would see the Sun’s image in it as in a convex mirror. Kepler refutes this by introducing a “fourth species” of light, “communicated light,” distinct from direct reflection and refraction. The Moon’s body is dense, colored, and opaque. He discusses Posidonius’s opinion, via Macrobius, that the Moon emits light like a mirror, which Kepler rejects. The circle of illumination on the Moon is smaller than a great circle; Reinholdus calculated its maximum arc as 180° 31′ 20″, which Kepler corrects to 180° 28′ 40″ for a new moon at apogee, and 180° 27′ 16″ for a full moon. The Earth’s illumination circle is 180° 2′ 56″. Kepler asserts that a perfect full moon can never be seen, as the Moon must be within 30′ of opposition to avoid the Earth’s shadow. The line bounding the Moon’s phases is an ellipse, as defined by Apollonius, not a circle.
Venus’s Proper Light and Colored Planets
Kepler notes that Venus’s own light can be seen at the moment of conjunction like a large star, as recorded in Brahe’s observations, and argues that Venus must have its own light. In his 1602 theses on astrology’s foundations, he gave four arguments for planets having proper light: first, Venus would change face like the Moon if only reflecting sunlight; second, many sublunary creatures have innate light; third, a combination of geometrical differences and celestial light’s office shows stars need twofold light, proper and borrowed; and fourth, brilliance and scintillation indicate proper light, while nebulosity indicates borrowed light, citing Cleomedes. He asserts that planetary light is colored, so planets must be translucent with essential color causes in their matter. He compares planet colors to surfaces: black gives Mars, red-gold Jupiter, leaden or white Saturn, blue Venus, and cerulean Mercury. He believes stars are ignited by innate force like the Sun. Kepler describes observing Venus scintillating through a window in December 1602, with undulating rays; later, comparing Venus and a three-day-old Moon, he found Venus’s light ratio to the Moon’s as 4 to 5. He suggests using walls to observe stars’ azimuths and altitudes.
Comets’ Light and the Earth’s Shadow
On comets’ light, Kepler asks what ignites them, noting that their tails always oppose the Sun but curve and decline. He rejects parallax and refraction as causes, proposing instead that the Sun’s rays pass through the comet’s dense, impure translucent body, becoming colored and concentrated. He concludes that comets consist of a humor denser than air, not hard like glass, dissolving gradually; this is confirmed by the nova of 1572, which showed iris colors, indicating humor. He cites Tycho Brahe’s hesitation about Venus illuminating comet tails. On the Earth’s shadow, Kepler argues that refraction of sunlight through the air shortens the shadow to about 45 Earth semi-diameters, not 268. The Moon at perigee passes above this shadow, and the lunar eclipse shadow is the air’s shadow, not the Earth’s. The Moon’s redness in eclipse comes from refracted solar rays through the air, not proper lunar light. He cites Theon, Cleomedes, Reinhold, Plutarch, Frodoardus (eclipse of 926), and Cornelius Gemma (eclipse of 1575) on colors, and describes his own observation at Gratium in 1598 of a lunar eclipse with a red rim. He concludes that the dark part is in Earth’s true shadow, the red part in the air’s shadow transmitting refracted rays, and the bright part in pure sunlight. He discusses the Moon’s pallor, citing Plutarch that stars around the Sun illuminate it, and then solves the problem of refraction in distant regions using eclipse observations, calculating refraction angles and regions.
The Moon’s Shadow and Diurnal Darkness
Kepler ends this chapter and begins the next on the Moon’s shadow and diurnal darkness, noting Tycho Brahe’s claim that the Moon’s diameter appears smaller in solar eclipses due to light, a view corrected by the 1601 solar eclipse.
Clavius’s Lunar Eclipse and the Moon’s Atmosphere
Kepler returns to Clavius’s 1567 observation, arguing that if a luminous limb was seen on the Moon, it must have been a diluted glow without a sharp boundary, caused by refraction in a lunar atmosphere. If Clavius insists the light was sharply defined, Kepler would concede a more geometric cause. Recalling his earlier audacity in attributing Earth-like features to the Moon, he now considers surrounding it with air. As with terrestrial air, rays from the Sun’s extremities could be refracted around the lunar body, making the Sun appear larger rather than the Moon’s diameter. Though a patron suggested the Moon’s edges seemed translucent, Kepler prefers the lunar air explanation, though he admits it is rarely deep enough for this effect.
Corollaries on Eclipses and Occultations
Kepler clarifies that while a solar eclipse darkening the day proves the Moon hid the Sun, it does not predict future darkness. In 1567 in Suevia, a yellow turbid sky caused no complaint, but a more lasting darkness would. The higher air is more deprived of the Sun’s aspect, so night or the Moon’s shadow falls on our heads. For a star to be eclipsed like the Moon, it must lack its own light. Kepler suspects Mars enters Earth’s shadow, and he extends the analogy: Venus’s shadow falls on the Moon, Mercury’s on Venus, Earth’s on Mars, and so on through the planets. The darkness of 47 years might have been a comet’s diluted shadow. He catalogues historical occultations: Jupiter occulted the Southern Ass in 141 BC, Aristotle saw Jupiter cover a star in the Twins’ feet, Mars was hidden by Scorpius in 272 BC, and Venus comprehended a star in 271 BC. Mästlin saw Mars covered by Venus in 1574, and Kepler himself saw this in 1598. Jupiter covering Saturn occurred in 1444, and a comet was thought covered by Jupiter. Mästlin and Kepler saw Jupiter eclipsed by Mars in 1591. Proclus observed Venus under Mars and Mercury under Venus. In 1599, Kepler saw no Mercury near Venus. Venus cannot occult the Sun this century but could have two hundred years ago. Mercury’s nodes are in Taurus and Scorpio beginnings or Aries and Libra ends. In 807, Mercury was seen in the Sun as a small black spot, though the year was likely 808. Kepler lists observers of occultations: Agrippa and Timochares on the Pleiades, Menelaus on the Scorpion’s forehead, Tycho and Copernicus on the Palllicium, Kepler on the Scorpion’s heart, and Mästlin on many lunar occultations. Saturn was covered by the Moon for Waltherus, Jupiter by a monk in 807, Mars by Aristotle and Mästlin, and Venus by Copernicus. Aristotle’s account of Mars being covered by the Moon is dated to 357 BC, the night of April 4.
On Parallaxes and the Measurement of Positions
Astronomy considers the apparent species of the Sun, Moon, and stars, plus Earth’s and the Moon’s shadows. Their motions are sought, and their positions measured by instruments. As points determine a line in geometry, astronomy compares apparent positions to find the form of motion. Position is relative to the world’s diameter or its surface; the latter concerns vision, as we see by the angle of rays to the eye’s center. The immense sphere of fixed stars measures these visual angles from Earth, so stars’ distances cannot be seen directly; we note conjunctions with fixed stars. The Sun and Moon appear equal because they subtend equal angles, though the Sun is farther. Astronomers use instruments like radii and sextants to measure arcs, a mechanical art cultivated by Tycho. Since stars are hidden by day, astronomers use the horizon and its pole, the zenith, shown by the perpendicular line. Navigators take the Sun’s altitude. Kepler defends Tycho against an artisan who disparaged his observations: Tycho does not claim absolute distances but uses visual arcs to find true positions in the world’s diameter.
Parallax and the Two Eyes
Nature gave two eyes to comprehend distance, as the eyes’ axes converge more for nearer objects. With one eye closed, an object appears to jump, which Opticians call “commutatio visus” or parallax. The same happens with stars: the eyes’ diameters are parallel for distant objects, so distance is not discerned. Two observers at different places on Earth, as in a diagram with A in Europe and B at Africa’s extreme promontory, see the Moon C against different fixed stars, O and N. This difference, the arc NO or angle, is the Moon’s parallax. Astronomers compute where the Moon should appear from these observations.
Measuring the Lunar Diameter
Kepler presents methods for measuring the lunar diameter and observing eclipses. He compares lunar and solar diameters, noting Hipparchus’s dioptra, which Ptolemy also used, showed the Moon at apogee equals the solar diameter but exceeds it at perigee. Kepler records his own observations: on 28 March 1585, he saw the Moon near the Pleiades; on 17 July 1587, it stood near them with horns turned. Tycho Brahe’s observers measured the lunar diameter at apogee as 31 or 31.5 minutes using Gemma’s method, though this method tends to overestimate. Other observations from 1587–1591 give values around 30–33 minutes. Kepler concludes the lunar diameter at apogee is 30.5 minutes, but at perigee it is uncertain due to the Moon’s double equation.
Estimating Eclipse Digits
Kepler presents Problem VI: estimating eclipse digits without instruments by dividing the diameter into 12 parts. He cites Gemma Frisius observing 8 digits at Louvain, and Ambrosius Rhodius estimating 10 digits at the 1601 eclipse. For solar eclipses, he describes using a camera obscura with a paper receiving the solar image, dividing its diameter into 12 digits, crediting Reinhold for this method. Problems VII–IX detail correcting the distorted image by accounting for the aperture’s semidiameter. Problems X–XII explain using a transverse rule and index to capture and compute the true eclipse geometry.
Final Problems on Eclipse Observation
Problem XIII urges observers to measure the lunar diameter during the 1605 solar eclipse. Kepler’s own 1601 Prague observation gave the Moon’s diameter as at least 30.5 minutes at apogee, confirming it cannot cover the Sun at perigee. Problems XIV–XXI concern recording eclipse inclinations using various instruments, including a quadrant, and timing phases with clocks, citing Regiomontanus, Walther, and Tycho’s methods.
Main Arguments
The central argument of the work is that a proper understanding of light, vision, and refraction is essential for accurate astronomy. Kepler argues that light emanates from a source in straight lines, that it is instantaneous in its motion, and that it attenuates with distance according to the inverse square law. He refutes the Aristotelian model of vision, in which the transparent medium is altered by color to move the eye, and instead asserts that species or luminous rays flow from the Sun and illuminated colors, are attenuated, and depict their source on the opaque eye. He argues that the medium does not move the eye, but rather species and rays descend through it, and he defends the possibility of species acting without inhering in a subject, comparing light’s incorporeal motion to a mathematical point.
A major argument concerns the projection of images through a small hole. Kepler proves that the projected figure of the Sun through a small aperture is due to the Sun’s rotundity, not the hole’s shape, and that the image is a composite of the inverted figure of the luminous body and the direct figure of the window. This has important consequences for astronomy, as it explains why the Moon’s diameter appears smaller in conjunctions, a conclusion that Tycho Brahe was forced to accept.
Kepler also argues for a new method of measuring refraction. He refutes methods based solely on the length of the path through the denser medium or solely on the angle of incidence, arguing that the cause of refraction lies in the surface, not the body’s density. He presents his own method, which uses an analogy with concave mirrors, and he constructs refraction tables using Vitellio’s angles and Tycho’s observations. He argues that refraction angles grow with greater increments than obliquity, and that refractions increase most near the horizon.
In the astronomical sections, Kepler argues that atmospheric refraction explains a range of phenomena, including the apparent doubling of the equinox and the discrepancy in Hipparchus’s solar eccentricity. He argues that the Sun is the source of light and motion in the universe, and that it is made of a dense, pure, pellucid material, likely condensed water. He introduces the concept of “communicated light” to explain the Moon’s illumination, distinct from reflection and refraction. He argues that planets have their own proper light in addition to borrowed light from the Sun, and that their colors are due to their translucent matter. He argues that comets consist of a humor denser than air, and that their tails are caused by the Sun’s rays passing through this dense, impure body. Finally, he argues that the Earth’s shadow is shortened by refraction to about 45 Earth semi-diameters, and that the Moon’s redness in eclipse comes from refracted solar rays through the air, not from proper lunar light.
Key Concepts
The work introduces and develops several specific named concepts and motifs. One of the most important is the “Plane of Refraction,” which Kepler defines as the plane containing the incident ray, the refracted ray, and the perpendicular to the refracting surface. This plane is perpendicular to the refracting surface, and it is a fundamental tool for analyzing refraction geometrically.
Another key concept is the “latitude of vision,” which refers to the distance between the two eyes. Kepler argues that it is this latitude, not nature or universal reason, that causes an image to appear on the perpendicular from the object to the reflecting or refracting surface. This is a novel explanation for a long-standing problem in optics.
Kepler also introduces a “fourth species” of light, which he calls “communicated light.” This is distinct from direct reflection and refraction, and it explains how the Moon is illuminated by the Sun without acting as a mirror. This concept is crucial for understanding the Moon’s phases and its appearance during eclipses.
The work also develops the concept of the camera obscura, building on the work of Giambattista della Porta. Kepler provides a geometric demonstration of how the image of the Sun is projected through a small hole, showing that the image is a composite of the inverted figure of the luminous body and the direct figure of the aperture. This is used as a tool for measuring solar and lunar diameters and for observing eclipses.
Another key concept is the “commutatio visus,” or parallax, which Kepler explains as the apparent shift in an object’s position when viewed from two different points. He applies this to astronomy, explaining that the Moon’s parallax is the difference in its apparent position against the fixed stars when viewed from two different places on Earth. He also discusses the “dioptra,” an instrument used by Hipparchus and Ptolemy for measuring diameters.
Kepler also develops the idea of the Sun as a “glowing stone” (candentem lapidem), a view he attributes to an ancient philosopher and supports with Gilbert’s magnetic philosophy. He describes the Sun as a heart in the world, dispensing life and motion to the planets, and he locates a vital faculty in the Sun’s body that produces light.
Themes
A recurring theme in the work is the relationship between the senses and the intellect, particularly the way in which visual perception can deceive and must be corrected by reason and geometry. Kepler repeatedly shows how the eye is fooled—by refraction, by the projection of images, by the latitude of vision—and how mathematical analysis is required to arrive at the true positions and sizes of celestial bodies. This theme is grounded in his critique of the Aristotelian model of vision, which he replaces with a mechanistic account of rays and species.
Another theme is the unity of the cosmos and the interconnectedness of its parts. Kepler’s argument that the Sun is a heart dispensing life and motion to the planets, and that the Moon and planets participate in a hierarchy of light, reflects a worldview in which celestial bodies are not inert but are animated by forces and faculties. This is connected to his belief that the Sun is made of the densest material in the universe, and that light is generated in its center and dispersed through its surface.
A related theme is the idea of light as a fundamental principle of nature, with both physical and metaphysical dimensions. Kepler treats light as a physical entity that obeys geometric laws, but he also invests it with vital and generative power. The Sun’s light is not merely a physical phenomenon but a source of life and motion, and the Moon’s “communicated light” is a distinct species that participates in this cosmic economy.
The theme of the fallibility of ancient authorities is also prominent. Kepler repeatedly corrects Aristotle, Ptolemy, Alhazen, and Vitellio, showing where their theories fail and offering his own demonstrations in their place. He is respectful of tradition but insistent on the primacy of observation and geometric proof. This is part of a broader theme of the progress of knowledge, in which Kepler builds on the work of his predecessors—Tycho Brahe, Copernicus, Porta—while also surpassing them.
Finally, the work reflects a theme of the wonder and complexity of the natural world, particularly the phenomena of eclipses, which Kepler calls ancient and marvelous. His detailed analysis of the Moon’s shadow, the Earth’s shadow, and the colors of eclipses is driven not only by a desire for accurate prediction but also by a sense of awe at the intricate workings of the cosmos.
The Author’s Conclusions
Kepler concludes that light is a physical entity that emanates from a source in straight lines, that its motion is instantaneous, and that it attenuates with distance according to the inverse square law. He concludes that vision occurs not through the alteration of a transparent medium, as Aristotle held, but through the reception of species or rays that depict their source on the eye. He concludes that the projection of the Sun’s image through a small hole is due to the Sun’s rotundity, not the hole’s shape, and that this explains why the Moon’s diameter appears smaller in conjunctions.
Kepler concludes that refraction occurs at the surface between two media, not in the body of the medium itself, and that the angle of refraction is not proportional to the angle of incidence but grows with greater increments, especially near the horizon. He provides a method for measuring refraction and constructs tables using the observations of Vitellio and Tycho Brahe. He concludes that atmospheric refraction explains a range of astronomical phenomena, including the apparent doubling of the equinox and the discrepancy in Hipparchus’s solar eccentricity.
Kepler concludes that the Sun is made of the densest material in the universe, that it is the source of light and motion for the planets, and that it acts as a heart in the world. He concludes that the Moon is illuminated by a “communicated light” distinct from reflection and refraction, and that a perfect full moon can never be seen. He concludes that planets have their own proper light in addition to borrowed light, and that their colors are due to their translucent matter. He concludes that comets consist of a humor denser than air, and that their tails are caused by the Sun’s rays passing through this dense body. Finally, he concludes that the Earth’s shadow is shortened by refraction to about 45 Earth semi-diameters, and that the Moon’s redness in eclipse comes from refracted solar rays through the air. The book ends with practical instructions for observing the lunar diameter during the 1605 solar eclipse, and for recording eclipse inclinations and timing phases with clocks, citing the methods of Regiomontanus, Walther, and Tycho.
Significance
The significance of Ad Vitellionem paralipomena lies in its foundational contribution to the science of optics and its application to astronomy. Kepler’s work provides a rigorous geometric treatment of light, vision, and refraction that corrects and supersedes the ancient and medieval traditions of Aristotle, Ptolemy, Alhazen, and Vitellio. His demonstration that the projected image of the Sun through a small hole is due to the Sun’s rotundity is a key insight for the use of the camera obscura in astronomy, and it resolves a puzzle that had troubled Tycho Brahe. His method for measuring refraction and his construction of refraction tables represent a significant advance over previous attempts, and his application of refraction to explain astronomical phenomena such as the doubling of the equinox and the shortening of the Earth’s shadow demonstrates the power of optical theory to solve problems in astronomy.
The work is also significant for its cosmological vision, in which the Sun is conceived as a vital, luminous body that dispenses life and motion to the planets. This view, which Kepler supports with references to ancient philosophy and Gilbert’s magnetic philosophy, anticipates later developments in natural philosophy. The introduction of “communicated light” as a distinct species of light is an important conceptual innovation that allows Kepler to explain the Moon’s illumination without recourse to reflection or refraction.
Finally, the book is significant as a document of the scientific revolution, showing how a practicing astronomer of the early seventeenth century combined careful observation, geometric demonstration, and a willingness to challenge authority in order to advance knowledge. Kepler’s engagement with Tycho Brahe’s data, his critiques of his predecessors, and his practical instructions for observing eclipses all reflect a new kind of scientific practice, one in which theory and observation are tightly intertwined. The work thus stands as a landmark in the history of both optics and astronomy, and it had a lasting influence on the development of these fields.
