1911 Encyclopædia Britannica/Light
LIGHT. Introduction.—§ 1. “Light” may be defined subjectively as the sense-impression formed by the eye. This is the most familiar connotation of the term, and suffices for the discussion of optical subjects which do not require an objective definition, and, in particular, for the treatment of physiological optics and vision. The objective definition, or the “nature of light,” is the ultima Thule of optical research. “Emission theories,” based on the supposition that light was a stream of corpuscles, were at first accepted. These gave place during the opening decades of the 19th century to the “undulatory or wave theory,” which may be regarded as culminating in the “elastic solid theory”—so named from the lines along which the mathematical investigation proceeded—and according to which light is a transverse vibratory motion propagated longitudinally though the aether. The mathematical researches of James Clerk Maxwell have led to the rejection of this theory, and it is now held that light is identical with electromagnetic disturbances, such as are generated by oscillating electric currents or moving magnets. Beyond this point we cannot go at present. To quote Arthur Schuster (Theory of Optics, 1904), “So long as the character of the displacements which constitute the waves remains undefined we cannot pretend to have established a theory of light.” It will thus be seen that optical and electrical phenomena are co-ordinated as a phase of the physics of the “aether,” and that the investigation of these sciences culminates in the derivation of the properties of this conceptual medium, the existence of which was called into being as an instrument of research.^{[1]} The methods of the elastic-solid theory can still be used with advantage in treating many optical phenomena, more especially so long as we remain ignorant of fundamental matters concerning the origin of electric and magnetic strains and stresses; in addition, the treatment is more intelligible, the researches on the electromagnetic theory leading in many cases to the derivation of differential equations which express quantitative relations between diverse phenomena, although no precise meaning can be attached to the symbols employed. The school following Clerk Maxwell and Heinrich Hertz has certainly laid the foundations of a complete theory of light and electricity, but the methods must be adopted with caution, lest one be constrained to say with Ludwig Boltzmann as in the introduction to his Vorlesungen über Maxwell’s Theorie der Elektricität und des Lichtes:—
“So soll ich denn mit saurem Schweiss |
The essential distinctions between optical and electromagnetic phenomena may be traced to differences in the lengths of light-waves and of electromagnetic waves. The aether can probably transmit waves of any wave-length, the velocity of longitudinal propagation being about 3.10^{10} cms. per second. The shortest waves, discovered by Schumann and accurately measured by Lyman, have a wave-length of 0.0001 mm.; the ultra-violet, recognized by their action on the photographic plate or by their promoting fluorescence, have a wave-length of 0.0002 mm.; the eye recognizes vibrations of a wave-length ranging from about 0.0004 mm. (violet) to about 0.0007 (red); the infra-red rays, recognized by their heating power or by their action on phosphorescent bodies, have a wave-length of 0.001 mm.; and the longest waves present in the radiations of a luminous source are the residual rays (“Rest-strahlen”) obtained by repeated reflections from quartz (.0085 mm.), from fluorite (0.056 mm.), and from sylvite (0.06 mm.). The research-field of optics includes the investigation of the rays which we have just enumerated. A delimitation may then be made, inasmuch as luminous sources yield no other radiations, and also since the next series of waves, the electromagnetic waves, have a minimum wave-length of 6 mm.
§ 2. The commonest subjective phenomena of light are colour and visibility, i.e. why are some bodies visible and others not, or, in other words, what is the physical significance of the words “transparency,” “colour” and “visibility.” What is ordinarily understood by a transparent substance is one which transmits all the rays of white light without appreciable absorption—that some absorption does occur is perceived when the substance is viewed through a sufficient thickness. Colour is due to the absorption of certain rays of the spectrum, the unabsorbed rays being transmitted to the eye, where they occasion the sensation of colour (see Colour; Absorption of Light). Transparent bodies are seen partly by reflected and partly by transmitted light, and opaque bodies by absorption. Refraction also influences visibility. Objects immersed in a liquid of the same refractive index and dispersion would be invisible; for example, a glass rod can hardly be seen when immersed in Canada balsam; other instances occur in the petrological examination of rock-sections under the microscope. In a complex rock-section the boldness with which the constituents stand out are measures of the difference between their refractive indices and the refractive index of the mounting medium, and the more nearly the indices coincide the less defined become the boundaries, while the interior of the mineral may be most advantageously explored. Lord Rayleigh has shown that transparent objects can only be seen when non-uniformly illuminated, the differences in the refractive indices of the substance and the surrounding medium becoming inoperative when the illumination is uniform on all sides. R. W. Wood has performed experiments which confirm this view.
The analysis of white light into the spectrum colours, and the re-formation of the original light by transmitting the spectrum through a reversed prism, proved, to the satisfaction of Newton and subsequent physicists until late in the 19th century, that the various coloured rays were present in white light, and that the action of the prism was merely to sort out the rays. This view, which suffices for the explanation of most phenomena, has now been given up, and the modern view is that the prism or grating really does manufacture the colours, as was held previously to Newton. It appears that white light is a sequence of irregular wave trains which are analysed into series of more regular trains by the prism or grating in a manner comparable with the analytical resolution presented by Fourier’s theorem. The modern view points to the mathematical existence of waves of all wave-lengths in white light, the Newtonian view to the physical existence. Strictly, the term “monochromatic” light is only applicable to light of a single wave-length (which can have no actual existence), but it is commonly used to denote light which cannot be analysed by the instruments at our disposal; for example, with low-power instruments the light emitted by sodium vapour would be regarded as homogeneous or monochromatic, but higher power instruments resolve this light into two components of different wave-lengths, each of which is of a higher degree of homogeneity, and it is not impossible that these rays may be capable of further analysis.
§ 3. Divisions of the Subject.—In the early history of the science of light or optics a twofold division was adopted: Catoptrics (from Gr. κάτοπτρον, a mirror), embracing the phenomena of reflection, i.e. the formation of images by mirrors; and Dioptrics (Gr. διά, through), embracing the phenomena of refraction, i.e. the bending of a ray of light when passing obliquely through the surface dividing two media.^{[2]} A third element, Chromatics (Gr. χρῶμα, colour), was subsequently introduced to include phenomena involving colour transformations, such as the iridescence of mother-of-pearl, feathers, soap-bubbles, oil floating on water, &c. This classification has been discarded (although the terms, particularly “dioptric” and “chromatic,” have survived as adjectives) in favour of a twofold division: geometrical optics and physical optics. Geometrical optics is a mathematical development (mainly effected by geometrical methods) of three laws assumed to be rigorously true: (1) the law of rectilinear propagation, viz. that light travels in straight lines or rays in any homogeneous medium; (2) the law of reflection, viz. that the incident and reflected rays at any point of a surface are equally inclined to, and coplanar with, the normal to the surface at the point of incidence; and (3) the law of refraction, viz. that the incident and refracted rays at a surface dividing two media make angles with the normal to the surface at the point of incidence whose sines are in a ratio (termed the “refractive index”) which is constant for every particular pair of media, and that the incident and refracted rays are coplanar with the normal. Physical optics, on the other hand, has for its ultimate object the elucidation of the question: what is light? It investigates the nature of the rays themselves, and, in addition to determining the validity of the axioms of geometrical optics, embraces phenomena for the explanation of which an expansion of these assumptions is necessary.
Of the subordinate phases of the science, “physiological optics” is concerned with the phenomena of vision, with the eye as an optical instrument, with colour-perception, and with such allied subjects as the appearance of the eyes of a cat and the luminosity of the glow-worm and firefly; “meteorological optics” includes phenomena occasioned by the atmosphere, such as the rainbow, halo, corona, mirage, twinkling of stars and colour of the sky, and also the effects of atmospheric dust in promoting such brilliant sunsets as were seen after the eruption of Krakatoa; “magneto-optics” investigates the effects of electricity and magnetism on optical properties; “photo-chemistry,” with its more practical development photography, is concerned with the influence of light in effecting chemical action; and the term “applied optics” may be used to denote, on the one hand, the experimental investigation of material for forming optical systems, e.g. the study of glasses with a view to the formation of a glass of specified optical properties (with which may be included such matters as the transparency of rock-salt for the infra-red and of quartz for the ultra-violet rays), and, on the other hand, the application of geometrical and physical investigations to the construction of optical instruments.
§ 4. Arrangement of the Subject.—The following three divisions of this article deal with: (I.) the history of the science of light; (II.) the nature of light; (III.) the velocity of light; but a summary (which does not aim at scientific precision) may here be given to indicate to the reader the inter-relation of the various optical phenomena, those phenomena which are treated in separate articles being shown in larger type.
The simplest subjective phenomena of light are Colour and intensity, the measurement of the latter being named Photometry. When light falls on a medium, it may be returned by Reflection or it may suffer Absorption; or it may be transmitted and undergo Refraction, and, if the light be composite, Dispersion; or, as in the case of oil films on water, brilliant colours are seen, an effect which is due to Interference. Again, if the rays be transmitted in two directions, as with certain crystals, “double refraction” (see Refraction, Double) takes place, and the emergent rays have undergone Polarization. A Shadow is cast by light falling on an opaque object, the complete theory of which involves the phenomenon of Diffraction. Some substances have the property of transforming luminous radiations, presenting the phenomena of Calorescence, Fluorescence and Phosphorescence. An optical system is composed of any number of Mirrors or Lenses, or of both. If light falling on a system be not brought to a focus, i.e. if all the emergent rays be not concurrent, we are presented with a Caustic and an Aberration. An optical instrument is simply the setting up of an optical system, the Telescope, Microscope, Objective, optical Lantern, Camera Lucida, Camera Obscura and the Kaleidoscope are examples; instruments serviceable for simultaneous vision with both eyes are termed Binocular Instruments; the Stereoscope may be placed in this category; the optical action of the Zoétrope, with its modern development the Cinematograph, depends upon the physiological persistence of Vision. Meteorological optical phenomena comprise the Corona, Halo, Mirage, Rainbow, colour of Sky and Twilight, and also astronomical refraction (see Refraction, Astronomical); the complete theory of the corona involves Diffraction, and atmospheric Dust also plays a part in this group of phenomena.
I. History
§ 1. There is reason to believe that the ancients were more familiar with optics than with any other branch of physics; and this may be due to the fact that for a knowledge of external things man is indebted to the sense of vision in a far greater degree than to other senses. That light travels in straight lines—or, in other words, that an object is seen in the direction in which it really lies—must have been realized in very remote times. The antiquity of mirrors points to some acquaintance with the phenomena of reflection, and Layard’s discovery of a convex lens of rock-crystal among the ruins of the palace of Nimrud implies a knowledge of the burning and magnifying powers of this instrument. The Greeks were acquainted with the fundamental law of reflection, viz. the equality of the angles of incidence and reflection; and it was Hero of Alexandria who proved that the path of the ray is the least possible. The lens, as an instrument for magnifying objects or for concentrating rays to effect combustion, was also known. Aristophanes, in the Clouds (c. 424 B.C.), mentions the use of the burning-glass to destroy the writing on a waxed tablet; much later, Pliny describes such glasses as solid balls of rock-crystal or glass, or hollow glass balls filled with water, and Seneca mentions their use by engravers. A treatise on optics (Κατοπτρικά), assigned to Euclid by Proclus and Marinus, shows that the Greeks were acquainted with the production of images by plane, cylindrical and concave and convex spherical mirrors, but it is doubtful whether Euclid was the author, since neither this work nor the Ὀπτικά, a work treating of vision and also assigned to him by Proclus and Marinus, is mentioned by Pappus, and more particularly since the demonstrations do not exhibit the precision of his other writings.
Reflection, or catoptrics, was the key-note of their explanations of optical phenomena; it is to the reflection of solar rays by the air that Aristotle ascribed twilight, and from his observation of the colours formed by light falling on spray, he attributes the rainbow to reflection from drops of rain. Although certain elementary phenomena of refraction had also been noted—such as the apparent bending of an oar at the point where it met the water, and the apparent elevation of a coin in a basin by filling the basin with water—the quantitative law of refraction was unknown; in fact, it was not formulated until the beginning of the 17th century. The analysis of white light into the continuous spectrum of rainbow colours by transmission through a prism was observed by Seneca, who regarded the colours as fictitious, placing them in the same category as the iridescent appearance of the feathers on a pigeon’s neck.
§ 2. The aversion of the Greek thinkers to detailed experimental inquiry stultified the progress of the science; instead of acquiring facts necessary for formulating scientific laws and correcting hypotheses, the Greeks devoted their intellectual energies to philosophizing on the nature of light itself. In their search for a theory the Greeks were mainly concerned with vision—in other words, they sought to determine how an object was seen, and to what its colour was due. Emission theories, involving the conception that light was a stream of concrete particles, were formulated. The Pythagoreans assumed that vision and colour were caused by the bombardment of the eye by minute particles projected from the surface of the object seen. The Platonists subsequently introduced three elements—a stream of particles emitted by the eye (their “divine fire”), which united with the solar rays, and, after the combination had met a stream from the object, returned to the eye and excited vision.
In some form or other the emission theory—that light was a longitudinal propulsion of material particles—dominated optical thought until the beginning of the 19th century. The authority of the Platonists was strong enough to overcome Aristotle’s theory that light was an activity (ἐνέργεια) of a medium which he termed the pellucid (διαφανές); about two thousand years later Newton’s exposition of his corpuscular theory overcame the undulatory hypotheses of Descartes and Huygens; and it was only after the acquisition of new experimental facts that the labours of Thomas Young and Augustin Fresnel indubitably established the wave-theory.
§ 3. The experimental study of refraction, which had been almost entirely neglected by the early Greeks, received more attention during the opening centuries of the Christian era. Cleomedes, in his Cyclical Theory of Meteors, c. A.D. 50, alludes to the apparent bending of a stick partially immersed in water, and to the rendering visible of coins in basins by filling up with water; and also remarks that the air may refract the sun’s rays so as to render that luminary visible, although actually it may be below the horizon. The most celebrated of the early writers on optics is the Alexandrian Ptolemy (2nd century). His writings on light are believed to be preserved in two imperfect Latin manuscripts, themselves translations from the Arabic. The subjects discussed include the nature of light and colour; the formation of images by various types of mirrors, refractions at the surface of glass and of water, with tables of the angle of refraction corresponding to given angles of incidence for rays passing from air to glass and from air to water; and also astronomical refractions, i.e. the apparent displacement of a heavenly body due to the refraction of light in its passage through the atmosphere. The authenticity of these manuscripts has been contested: the Almagest contains no mention of the Optics, nor is the subject of astronomical refractions noticed, but the strongest objection, according to A. de Morgan, is the fact that their author was a poor geometer.
§ 4. One of the results of the decadence of the Roman empire was the suppression of the academies, and few additions were made to scientific knowledge on European soil until the 13th century. Extinguished in the West, the spirit of research was kindled in the East. The accession of the Arabs to power and territory in the 7th century was followed by the acquisition of the literary stores of Greece, and during the following five centuries the Arabs, both by their preservation of existing works and by their original discoveries (which, however, were but few), took a permanent place in the history of science. Pre-eminent among Arabian scientists is Alhazen, who flourished in the 11th century. Primarily a mathematician and astronomer, he also investigated a wide range of optical phenomena. He examined the anatomy of the eye, and the functions of its several parts in promoting vision; and explained how it is that we see one object with two eyes, and then not by a single ray or beam as had been previously held, but by two cones of rays proceeding from the object, one to each eye. He attributed vision to emanations from the body seen; and on his authority the Platonic theory fell into disrepute. He also discussed the magnifying powers of lenses; and it may be that his writings on this subject inspired the subsequent invention of spectacles. Astronomical observations led to the investigation of refraction by the atmosphere, in particular, astronomical refraction; he explained the phenomenon of twilight, and showed a connexion between its duration and the height of the atmosphere. He also treated optical deceptions, both in direct vision and in vision by reflected and refracted light, including the phenomenon known as the horizontal moon, i.e. the apparent increase in the diameter of the sun or moon when near the horizon. This appearance had been explained by Ptolemy on the supposition that the diameter was actually increased by refraction, and his commentator Theon endeavoured to explain why an object appears larger when viewed under water. But actual experiment showed that the diameter did not increase. Alhazen gave the correct explanation, which, however, Friar Bacon attributes to Ptolemy. We judge of distance by comparing the angle under which an object is seen with its supposed distance, so that if two objects be seen under nearly equal angles and one be supposed to be more distant than the other, then the former will be supposed to be the larger. When near the horizon the sun or moon, conceived as very distant, are intuitively compared with terrestrial objects, and therefore they appear larger than when viewed at elevations.
§ 5. While the Arabs were acting as the custodians of scientific knowledge, the institutions and civilizations of Europe were gradually crystallizing. Attacked by the Mongols and by the Crusaders, the Bagdad caliphate disappeared in the 13th century. At that period the Arabic commentaries, which had already been brought to Europe, were beginning to exert great influence on scientific thought; and it is probable that their rarity and the increasing demand for the originals and translations led to those forgeries which are of frequent occurrence in the literature of the middle ages. The first treatise on optics written in Europe was admitted by its author Vitello or Vitellio, a native of Poland, to be based on the works of Ptolemy and Alhazen. It was written in about 1270, and first published in 1572, with a Latin translation of Alhazen’s treatise, by F. Risner, under the title Thesaurus opticae. Its tables of refraction are more accurate than Ptolemy’s; the author follows Alhazen in his investigation of lenses, but his determinations of the foci and magnifying powers of spheres are inaccurate. He attributed the twinkling of stars to refraction by moving air, and observed that the scintillation was increased by viewing through water in gentle motion; he also recognized that both reflection and refraction were instrumental in producing the rainbow, but he gave no explanation of the colours.
The Perspectiva Communis of John Peckham, archbishop of Canterbury, being no more than a collection of elementary propositions containing nothing new, we have next to consider the voluminous works of Vitellio’s illustrious contemporary, Roger Bacon. His writings on light, Perspectiva and Specula mathematica, are included in his Opus majus. It is conceivable that he was acquainted with the nature of the images formed by light traversing a small orifice—a phenomenon noticed by Aristotle, and applied at a later date to the construction of the camera obscura. The invention of the magic lantern has been ascribed to Bacon, and his statements concerning spectacles, the telescope, and the microscope, if not based on an experimental realization of these instruments, must be regarded as masterly conceptions of the applications of lenses. As to the nature of light, Bacon adhered to the theory that objects are rendered visible by emanations from the eye.
The history of science, and more particularly the history of inventions, constantly confronts us with the problem presented by such writings as Friar Bacon’s. Rarely has it been given to one man to promote an entirely new theory or to devise an original instrument; it is more generally the case that, in the evolution of a single idea, there comes some stage which arrests our attention, and to which we assign the dignity of an “invention.” Furthermore, the obscurity that surrounds the early history of spectacles, the magic lantern, the telescope and the microscope, may find a partial solution in the spirit of the middle ages. The natural philosopher who was bold enough to present to a prince a pair of spectacles or a telescope would be in imminent danger of being regarded in the eyes of the church as a powerful and dangerous magician; and it is conceivable that the maker of such an instrument would jealously guard the secret of its actual construction, however much he might advertise its potentialities.^{[3]}
§ 6. The awakening of Europe, which first manifested itself in Italy, England and France, was followed in the 16th century by a period of increasing intellectual activity. The need for experimental inquiry was realized, and a tendency to dispute the dogmatism of the church and to question the theories of the established schools of philosophy became apparent. In the science of optics, Italy led the van, the foremost pioneers being Franciscus Maurolycus (1494–1575) of Messina, and Giambattista della Porta (1538–1615) of Naples. A treatise by Maurolycus entitled Photismi de Lumine et Umbra prospectivum radiorum incidentium facientes (1575), contains a discussion of the measurement of the intensity of light—an early essay in photometry; the formation of circular patches of light by small holes of any shape, with a correct explanation of the phenomenon; and the optical relations of the parts of the eye, maintaining that the crystalline humour acts as a lens which focuses images on the retina, explaining short- and long-sight (myopia and hyper-metropia), with the suggestion that the former may be corrected by concave, and the latter by convex, lenses. He observed the spherical aberration due to elements beyond the axis of a lens, and also the caustics of refraction (diacaustics) by a sphere (seen as the bright boundaries of the luminous patches formed by receiving the transmitted light on a screen), which he correctly regarded as determined by the intersections of the refracted rays. His researches on refraction were less fruitful; he assumed the angles of incidence and refraction to be in the constant ratio of 8 to 5, and the rainbow, in which he recognized four colours, orange, green, blue and purple, to be formed by rays reflected in the drops along the sides of an octagon. Porta’s fame rests chiefly on his Magia naturalis sive de miraculis rerum naturalium, of which four books were published in 1558, the complete work of twenty books appearing in 1589. It attained great popularity, perhaps by reason of its astonishing medley of subjects—pyrotechnics and perfumery, animal reproduction and hunting, alchemy and optics,—and it was several times reprinted, and translated into English (with the title Natural Magick, 1658), German, French, Spanish, Hebrew and Arabic. The work contains an account of the camera obscura, with the invention of which the author has sometimes been credited; but, whoever the inventor, Porta was undoubtedly responsible for improving and popularizing that instrument, and also the magic lantern. In the same work practical applications of lenses are suggested, combinations comparable with telescopes are vaguely treated and spectacles are discussed. His De Refractione, optices parte (1593) contains an account of binocular vision, in which are found indications of the principle of the stereoscope.
§ 7. The empirical study of lenses led, in the opening decade of the 17th century, to the emergence of the telescope from its former obscurity. The first form, known as the Dutch or Galileo telescope, consisted of a convex and a concave lens, a combination which gave erect images; the later form, now known as the “Keplerian” or “astronomical” telescope (in contrast with the earlier or “terrestrial” telescope) consisted of two convex lenses, which gave inverted images. With the microscope, too, advances were made, and it seems probable that the compound type came into common use about this time. These single instruments were followed by the invention of binoculars, i.e. instruments which permitted simultaneous vision with both eyes. There is little doubt that the experimental realization of the telescope, opening up as it did such immense fields for astronomical research, stimulated the study of lenses and optical systems. The investigations of Maurolycus were insufficient to explain the theory of the telescope, and it was Kepler who first determined the principle of the Galilean telescope in his Dioptrice (1611), which also contains the first description of the astronomical or Keplerian telescope, and the demonstration that rays parallel to the axis of a plano-convex lens come to a focus at a point on the axis distant twice the radius of the curved surface of the lens, and, in the case of an equally convex lens, at an axial point distant only once the radius. He failed, however, to determine accurately the case for unequally convex lenses, a problem which was solved by Bonaventura Cavalieri, a pupil of Galileo.
Early in the 17th century great efforts were made to determine the law of refraction. Kepler, in his Prolegomena ad Vitellionem (1604), assiduously, but unsuccessfully, searched for the law, and can only be credited with twenty-seven empirical rules, really of the nature of approximations, which he employed in his theory of lenses. The true law—that the ratio of the sines of the angles of incidence and refraction is constant—was discovered in 1621 by Willebrord Snell (1591–1626); but was published for the first time after his death, and with no mention of his name, by Descartes. Whereas in Snell’s manuscript the law was stated in the form of the ratio of certain lines, trigonometrically interpretable as a ratio of cosecants, Descartes expressed the law in its modern trigonometrical form, viz. as the ratio of the sines. It may be observed that the modern form was independently obtained by James Gregory and published in his Optica promota (1663). Armed with the law of refraction, Descartes determined the geometrical theory of the primary and secondary rainbows, but did not mention how far he was indebted to the explanation of the primary bow by Antonio de Dominis in 1611; and, similarly, in his additions to the knowledge of the telescope the influence of Galileo is not recorded.
§ 8. In his metaphysical speculations on the system of nature, Descartes formulated a theory of light at variance with the generally accepted emission theory and showing some resemblance to the earlier views of Aristotle, and, in a smaller measure, to the modern undulatory theory. He imagined light to be a pressure transmitted by an infinitely elastic medium which pervades space, and colour to be due to rotatory motions of the particles of this medium. He attempted a mechanical explanation of the law of refraction, and came to the conclusion that light passed more readily through a more highly refractive medium. This view was combated by Pierre de Fermat (1601–1665), who, from the principle known as the “law of least time,” deduced the converse to be the case, i.e. that the velocity varied inversely with the refractive index. In brief, Fermat’s argument was as follows: Since nature performs her operations by the most direct routes or shortest paths, then the path of a ray of light between any two points must be such that the time occupied in the passage is a minimum. The rectilinear propagation and the law of reflection obviously agree with this principle, and it remained to be proved whether the law of refraction tallied.
Although Fermat’s premiss is useless, his inference is invaluable, and the most notable application of it was made in about 1824 by Sir William Rowan Hamilton, who merged it into his conception of the “characteristic function,” by the help of which all optical problems, whether on the corpuscular or on the undulator theory, are solved by one common process. Hamilton was in possession of the germs of this grand theory some years before 1824, but it was first communicated to the Royal Irish Academy in that year, and published in imperfect instalments some years later. The following is his own description of it. It is of interest as exhibiting the origin of Fermat’s deduction, its relation to contemporary and subsequent knowledge, and its connexion with other analytical principles. Moreover, it is important as showing Hamilton’s views on a very singular part of the more modern history of the science to which he contributed so much.
“Those who have meditated on the beauty and utility, in theoretical mechanics, of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced, in the Mécanique analytique . . ., must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics . . ., when it shall possess an appropriate method, and become the unfolding of a central idea. . . . It appears that if a general method in deductive optics can be attained at all, it must flow from some law or principle, itself of the highest generality, and among the highest results of induction. . . . [This] must be the principle, or law, called usually the Law of Least Action; suggested by questionable views, but established on the widest induction, and embracing every known combination of media, and every straight, or bent, or curved line, ordinary or extraordinary, along which light (whatever light may be) extends its influence successively in space and time: namely, that this linear path of light, from one point to another, is always found to be such that, if it be compared with the other infinitely various lines by which in thought and in geometry the same two points might be connected, a certain integral or sum, called often Action, and depending by fixed rules on the length, and shape, and position of the path, and on the media which are traversed by it, is less than all the similar integrals for the other neighbouring lines, or, at least, possesses, with respect to them, a certain stationary property. From this Law, then, which may, perhaps, be named the Law of Stationary Action, it seems that we may most fitly and with best hope set out, in the synthetic or deductive process and in the search of a mathematical method.
“Accordingly, from this known law of least or stationary action I deduced (long since) another connected and coextensive principle, which may be called by analogy the Law of Varying Action, and which seems to offer naturally a method such as we are seeking; the one law being as it were the last step in the ascending scale of induction, respecting linear paths of light, while the other law may usefully be made the first in the descending and deductive way.
“The former of these two laws was discovered in the following manner. The elementary principle of straight rays showed that light, under the most simple and usual circumstances, employs the direct, and therefore the shortest, course to pass from one point to another. Again, it was a very early discovery (attributed by Laplace to Ptolemy), that, in the case of a plane mirror, the bent line formed by the incident and reflected rays is shorter than any other bent line having the same extremities, and having its point of bending on the mirror. These facts were thought by some to be instances and results of the simplicity and economy of nature; and Fermat, whose researches on maxima and minima are claimed by the Continental mathematicians as the germ of the differential calculus, sought anxiously to trace some similar economy in the more complex case of refraction. He believed that by a metaphysical or cosmological necessity, arising from the simplicity of the universe, light always takes the course which it can traverse in the shortest time. To reconcile this metaphysical opinion with the law of refraction, discovered experimentally by Snellius, Fermat was led to suppose that the two lengths, or indices, which Snellius had measured on the incident ray prolonged and on the refracted ray, and had observed to have one common projection on a refracting plane, are inversely proportional to the two successive velocities of the light before and after refraction, and therefore that the velocity of light is diminished on entering those denser media in which it is observed to approach the perpendicular; for Fermat believed that the time of propagation of light along a line bent by refraction was represented by the sum of the two products, of the incident portion multiplied by the index of the first medium and of the refracted portion multiplied by the index of the second medium; because he found, by his mathematical method, that this sum was less, in the case of a plane refractor, than if light went by any other than its actual path from one given point to another, and because he perceived that the supposition of a velocity inversely as the index reconciled his mathematical discovery of the minimum of the foregoing sum with his cosmological principle of least time. Descartes attacked Fermat’s opinions respecting light, but Leibnitz zealously defended them; and Huygens was led, by reasonings of a very different kind, to adopt Fermat’s conclusions of a velocity inversely as the index, and of a minimum time of propagation of light, in passing from one given point to another through an ordinary refracting plane. Newton, however, by his theory of emission and attraction, was led to conclude that the velocity of light was directly, not inversely, as the index, and that it was increased instead of being diminished on entering a denser medium; a result incompatible with the theorem of the shortest time in refraction. This theorem of shortest time was accordingly abandoned by many, and among the rest by Maupertuis, who, however, proposed in its stead, as a new cosmological principle, that celebrated law of least action which has since acquired so high a rank in mathematical physics, by the improvements of Euler and Lagrange.”
§ 9. The second half of the 17th century witnessed developments in the practice and theory of optics which equal in importance the mathematical, chemical and astronomical acquisitions of the period. Original observations were made which led to the discovery, in an embryonic form, of new properties of light, and the development of mathematical analysis facilitated the quantitative and theoretical investigation of these properties. Indeed, mathematical and physical optics may justly be dated from this time. The phenomenon of diffraction, so named by Grimaldi, and by Newton inflection, which may be described briefly as the spreading out, or deviation, from the strictly rectilinear path of light passing through a small aperture or beyond the edge of an opaque object, was discovered by the Italian Jesuit, Francis Maria Grimaldi (1619–1663), and published in his Physico-Mathesis de Lumine (1665); at about the same time Newton made his classical investigation of the spectrum or the band of colours formed when light is transmitted through a prism,^{[4]} and studied interference phenomena in the form of the colours of thin and thick plates, and in the form now termed Newton’s rings; double refraction, in the form of the dual images of a single object formed by a rhomb of Iceland spar, was discovered by Bartholinus in 1670; Huygens’s examination of the transmitted beams led to the discovery of an absence of symmetry now called polarization; and the finite velocity of light was deduced in 1676 by Ole Roemer from the comparison of the observed and computed times of the eclipses of the moons of Jupiter.
These discoveries had a far-reaching influence upon the theoretical views which had been previously held: for instance, Newton’s recombination of the spectrum by means of a second (inverted) prism caused the rejection of the earlier view that the prism actually manufactured the colours, and led to the acceptance of the theory that the colours were physically present in the white light, the function of the prism being merely to separate the physical mixture; and Roemer’s discovery of the finite velocity of light introduced the necessity of considering the momentum of the particles which, on the accepted emission theory, composed the light. Of greater moment was the controversy concerning the emission or corpuscular theory championed by Newton and the undulatory theory presented by Huygens (see section II. of this article). In order to explain the colours of thin plates Newton was forced to abandon some of the original simplicity of his theory; and we may observe that by postulating certain motions for the Newtonian corpuscles all the phenomena of light can be explained, these motions aggregating to a transverse displacement, translated longitudinally, and the corpuscles, at the same time, becoming otiose and being replaced by a medium in which the vibration is transmitted. In this way the Newtonian theory may be merged into the undulatory theory. Newton’s results are collected in his Opticks, the first edition of which appeared in 1704. Huygens published his theory in his Traité de lumière (1690), where he explained reflection, refraction and double refraction, but did not elucidate the formation of shadows (which was readily explicable on the Newtonian hypothesis) or polarization; and it was this inability to explain polarization which led to Newton’s rejection of the wave theory. The authority of Newton and his masterly exposition of the corpuscular theory sustained that theory until the beginning of the 19th century, when it succumbed to the assiduous skill of Young and Fresnel.
§ 10. Simultaneously with this remarkable development of theoretical and experimental optics, notable progress was made in the construction of optical instruments. The increased demand for telescopes, occasioned by the interest in observational astronomy, led to improvements in the grinding of lenses (the primary aim being to obtain forms in which spherical aberration was a minimum), and also to the study of achromatism, the principles of which followed from Newton’s analysis and synthesis of white light. Kepler’s supposition that lenses having the form of surfaces of revolution of the conic sections would bring rays to a focus without spherical aberration was investigated by Descartes, and the success of the latter’s demonstration led to the grinding of ellipsoidal and hyperboloidal lenses, but with disappointing results.^{[5]} The grinding of spherical lenses was greatly improved by Huygens, who also attempted to reduce chromatic aberration in the refracting telescope by introducing a stop (i.e. by restricting the aperture of the rays); to the same experimenter are due compound eye-pieces, the invention of which had been previously suggested by Eustachio Divini. The so-called Huygenian eye-piece is composed of two plano-convex lenses with their plane faces towards the eye; the field-glass has a focal length three times that of the eye-glass, and the distance between them is twice the focal length of the eye-glass. Huygens observed that spherical aberration was diminished by making the deviations of the rays at the two lenses equal, and Ruggiero Giuseppe Boscovich subsequently pointed out that the combination was achromatic. The true development, however, of the achromatic refracting telescope, which followed from the introduction of compound object-glasses giving no dispersion, dates from about the middle of the 18th century. The difficulty of obtaining lens systems in which aberrations were minimized, and the theory of Newton that colour production invariably attended refraction, led to the manufacture of improved specula which permitted the introduction of reflecting telescopes. The idea of this type of instrument had apparently occurred to Marin Mersenne in about 1640, but the first reflector of note was described in 1663 by James Gregory in his Optica promota; a second type was invented by Newton, and a third in 1672 by Cassegrain. Slight improvements were made in the microscope, although the achromatic type did not appear until about 1820, some sixty years after John Dollond had determined the principle of the achromatic telescope (see Aberration, Telescope, Microscope, Binocular Instrument).
§ 11. Passing over the discovery by Ehrenfried Walther Tschirnhausen (1651–1708) of the caustics produced by reflection (“catacaustics”) and his experiments with large reflectors and refractors (for the manufacture of which he established glass-works in Italy); James Bradley’s discovery in 1728 of the “aberration of light,” with the subsequent derivation of the velocity of light, the value agreeing fairly well with Roemer’s estimate; the foundation of scientific photometry by Pierre Bouguer in an essay published in 1729 and expanded in 1760 into his Traité d’optique sur la graduation de la lumière; the publication of John Henry Lambert’s treatise on the same subject, entitled Photometria, sive de Mensura et Gradibus Luminis, Colorum et Umbrae (1760); and the development of the telescope and other optical instruments, we arrive at the closing decades of the 18th century. During the forty years 1780 to 1820 the history of optics is especially marked by the names of Thomas Young and Augustin Fresnel, and in a lesser degree by Arago, Malus, Sir William Herschel, Fraunhofer, Wollaston, Biot and Brewster.
Although the corpuscular theory had been disputed by Benjamin Franklin, Leonhard Euler and others, the authority of Newton retained for it an almost general acceptance until the beginning of the 19th century, when Young and Fresnel instituted their destructive criticism. Basing his views on the earlier undulatory theories and diffraction phenomena of Grimaldi and Hooke, Young accepted the Huygenian theory, assuming, from a false analogy with sound waves, that the wave-disturbance was longitudinal, and ignoring the suggestion made by Hooke in 1672 that the direction of the vibration might be transverse, i.e. at right angles to the direction of the rays. As with Huygens, Young was unable to explain diffraction correctly, or polarization. But the assumption enabled him to establish the principle of interference,^{[6]} one of the most fertile in the science of physical optics. The undulatory theory was also accepted by Fresnel who, perceiving the inadequacy of the researches of Huygens and Young, showed in 1818 by an analysis which, however, is not quite free from objection, that, by assuming that every element of a wave-surface could act as a source of secondary waves or wavelets, the diffraction bands were due to the interference of the secondary waves formed by each element of a primary wave falling upon the edge of an obstacle or aperture. One consequence of Fresnel’s theory was that the bands were independent of the nature of the diffracting edge—a fact confirmed by experiment and therefore invalidating Young’s theory that the bands were produced by the interference between the primary wave and the wave reflected from the edge of the obstacle. Another consequence, which was first mathematically deduced by Poisson and subsequently confirmed by experiment, is the paradoxical phenomenon that a small circular disk illuminated by a point source casts a shadow having a bright centre.
§ 12. The undulatory theory reached its zenith when Fresnel explained the complex phenomena of polarization, by adopting the conception of Hooke that the vibrations were transverse, and not longitudinal.^{[7]} Polarization by double refraction had been investigated by Huygens, and the researches of Wollaston and, more especially, of Young, gave such an impetus to the study that the Institute of France made double refraction the subject of a prize essay in 1812. E. L. Malus (1775–1812) discovered the phenomenon of polarization by reflection about 1808 and investigated metallic reflection; Arago discovered circular polarization in quartz in 1811, and, with Fresnel, made many experimental investigations, which aided the establishment of the Fresnel-Arago laws of the interference of polarized beams; Biot introduced a reflecting polariscope, investigated the colours of crystalline plates and made many careful researches on the rotation of the plane of polarization; Sir David Brewster made investigations over a wide range, and formulated the law connecting the angle of polarization with the refractive index of the reflecting medium. Fresnel’s theory was developed in a strikingly original manner by Sir William Rowan Hamilton, who interpreted from Fresnel’s analytical determination of the geometrical form of the wave-surface in biaxal crystals the existence of two hitherto unrecorded phenomena. At Hamilton’s instigation Humphrey Lloyd undertook the experimental search, and brought to light the phenomena of external and internal conical refraction.
The undulatory vibration postulated by Fresnel having been generally accepted as explaining most optical phenomena, it became necessary to determine the mechanical properties of the aether which transmits this motion. Fresnel, Neumann, Cauchy, MacCullagh, and, especially, Green and Stokes, developed the “elastic-solid theory.” By applying the theory of elasticity they endeavoured to determine the constants of a medium which could transmit waves of the nature of light. Many different allocations were suggested (of which one of the most recent is Lord Kelvin’s “contractile aether,” which, however, was afterwards discarded by its author), and the theory as left by Green and Stokes has merits other than purely historical. At a later date theories involving an action between the aether and material atoms were proposed, the first of any moment being J. Boussinesq’s (1867). C. Christiansen’s investigation of anomalous dispersion in 1870, and the failure of Cauchy’s formula (founded on the elastic-solid theory) to explain this phenomenon, led to the theories of W. Sellmeier (1872), H. von Helmholtz (1875), E. Ketteler (1878), E. Lommel (1878) and W. Voigt (1883). A third class of theory, to which the present-day theory belongs, followed from Clerk Maxwell’s analytical investigations in electromagnetics. Of the greatest exponents of this theory we may mention H. A. Lorentz, P. Drude and J. Larmor, while Lord Rayleigh has, with conspicuous brilliancy, explained several phenomena (e.g. the colour of the sky) on this hypothesis.
For a critical examination of these theories see section II. of this article; reference may also be made to the British Association Reports: “On Physical Optics,” by Humphrey Lloyd (1834), p. 35; “On Double Refraction,” by Sir G. G. Stokes (1862), p. 253; “On Optical Theories,” by R. T. Glazebrook (1885), p. 157.
§ 13. Recent Developments.—The determination of the velocity of light (see section III. of this article) may be regarded as definitely settled, a result contributed to by A. H. L. Fizeau (1849), J. B. L. Foucault (1850, 1862), A. Cornu (1874), A. A. Michelson (1880), James Young and George Forbes (1882), Simon Newcomb (1880–1882) and Cornu (1900). The velocity in moving media was investigated theoretically by Fresnel; and Fizeau (1859), and Michelson and Morley (1886) showed experimentally that the velocity was increased in running water by an amount agreeing with Fresnel’s formula, which was based on the hypothesis of a stationary aether. The optics of moving media have also been investigated by Lord Rayleigh, and more especially by H. A. Lorentz, who also assumed a stationary aether. The relative motion of the earth and the aether has an important connexion with the phenomenon of the aberration of light, and has been treated with masterly skill by Joseph Larmor and others (see Aether). The relation of the earth’s motion to the intensities of terrestrial sources of light was investigated theoretically by Fizeau, but no experimental inquiry was made until 1903, when Nordmeyer obtained negative results, which were confirmed by the theoretical investigations of A. A. Bucherer and H. A. Lorentz.
Experimental photometry has been greatly developed since the pioneer work of Bouguer and Lambert and the subsequent introduction of the photometers of Ritchie, Rumford, Bunsen and Wheatstone, followed by Swan’s in 1859, and O. R. Lummer and E. Brodhun’s instrument (essentially the same as Swan’s) in 1889. This expansion may largely be attributed to the increase in the number of artificial illuminants—especially the many types of filament- and arc-electric lights, and the incandescent gas light. Colour photometry has also been notably developed, especially since the enunciation of the “Purkinje phenomenon” in 1825. Sir William Abney has contributed much to this subject, and A. M. Meyer has designed a photometer in which advantage is taken of the phenomenon of contrast colours. “Flicker photometry” may be dated from O. N. Rood’s investigations in 1893, and the same principle has been applied by Haycraft and Whitman. These questions—colour and flicker photometry—have important affinities to colour perception and the persistence of vision (see Vision). The spectrophotometer, devised by De Witt Bristol Brace in 1899, which permits the comparison of similarly coloured portions of the spectra from two different sources, has done much valuable work in the determination of absorptive powers and extinction coefficients. Much attention has also been given to the preparation of a standard of intensity, and many different sources have been introduced (see Photometry). Stellar photometry, which was first investigated instrumentally with success by Sir John Herschel, was greatly improved by the introduction of Zöllner’s photometer, E. C. Pickering’s meridian photometer and C. Pritchard’s wedge photometer. Other methods of research in this field are by photography—photographic photometry—and radiometric method (see Photometry, Celestial).
The earlier methods for the experimental determination of refractive indices by measuring the deviation through a solid prism of the substance in question or, in the case of liquids, through a hollow prism containing the liquid, have been replaced in most accurate work by other methods. The method of total reflection, due originally to Wollaston, has been put into a very convenient form, applicable to both solids and liquids, in the Pulfrich refractometer (see Refraction). Still more accurate methods, based on interference phenomena, have been devised. Jamin’s interference refractometer is one of the earlier forms of such apparatus; and Michelson’s interferometer is one of the best of later types (see Interference). The variation of refractive index with density has been the subject of much experimental and theoretical inquiry. The empirical rule of Gladstone and Dale was often at variance with experiment, and the mathematical investigations of H. A. Lorentz of Leiden and L. Lorenz of Copenhagen on the electromagnetic theory led to a more consistent formula. The experimental work has been chiefly associated with the names of H. H. Landolt and J. W. Brühl, whose results, in addition to verifying the Lorenz-Lorentz formula, have established that this function of the refractive index and density is a colligative property of the molecule, i.e. it is calculable additively from the values of this function for the component atoms, allowance being made for the mode in which they are mutually combined (see Chemistry, Physical). The preparation of lenses, in which the refractive index decreases with the distance from the axis, by K. F. J. Exner, H. F. L. Matthiessen and Schott, and the curious results of refraction by non-homogeneous media, as realized by R. Wood may be mentioned (see Mirage).
The spectrum of white light produced by prismatic refraction has engaged many investigators. The infra-red or heat waves were discovered by Sir William Herschel, and experiments on the actinic effects of the different parts of the spectrum on silver salts by Scheele, Senebier, Ritter, Seebeck and others, proved the increased activity as one passed from the red to the violet and the ultra-violet. Wollaston also made many investigations in this field, noticing the dark lines—the “Fraunhofer lines”—which cross the solar spectrum, which were further discussed by Brewster and Fraunhofer, who thereby laid the foundations of modern spectroscopy. Mention may also be made of the investigations of Lord Rayleigh and Arthur Schuster on the resolving power of prisms (see Diffraction), and also of the modern view of the function of the prism in analysing white light. The infra-red and ultra-violet rays are of especial interest since, although not affecting vision after the manner of ordinary light, they possess very remarkable properties. Theoretical investigation on the undulatory theory of the law of reflection shows that a surface, too rough to give any trace of regular reflection with ordinary light, may regularly reflect the long waves, a phenomenon experimentally realized by Lord Rayleigh. Long waves—the so-called “residual rays” or “Rest-strahlen”—have also been isolated by repeated reflections from quartz surfaces of the light from zirconia raised to incandescence by the oxyhydrogen flame (E. F. Nichols and H. Rubens); far longer waves were isolated by similar reflections from fluorite (56 µ) and sylvite (61 µ) surfaces in 1899 by Rubens and E. Aschkinass. The short waves—ultra-violet rays—have also been studied, the researches of E. F. Nichols on the transparency of quartz to these rays, which are especially present in the radiations of the mercury arc, having led to the introduction of lamps made of fused quartz, thus permitting the convenient study of these rays, which, it is to be noted, are absorbed by ordinary clear glass. Recent researches at the works of Schott and Genossen, Jena, however, have resulted in the production of a glass transparent to the ultra-violet.
Dispersion, i.e. that property of a substance which consists in having a different refractive index for rays of different wave-lengths, was first studied in the form known as “ordinary dispersion” in which the refrangibility of the ray increased with the wave-length. Cases had been observed by Fox Talbot, Le Roux, and especially by Christiansen (1870) and A. Kundt (1871–1872) where this normal rule did not hold; to such phenomena the name “anomalous dispersion” was given, but really there is nothing anomalous about it at all, ordinary dispersion being merely a particular case of the general phenomenon. The Cauchy formula, which was founded on the elastic-solid theory, did not agree with the experimental facts, and the germs of the modern theory, as was pointed out by Lord Rayleigh in 1900, were embodied in a question proposed by Clerk Maxwell for the Mathematical Tripos examination for 1869. The principle, which occurred simultaneously to W. Sellmeier (who is regarded as the founder of the modern theory) and had been employed about 1850 by Sir G. G. Stokes to explain absorption lines, involves an action between the aether and the molecules of the dispersing substance. The mathematical investigation is associated with the names of Sellmeier, Hermann Helmholtz, Eduard Ketteler, P. Drude, H. A. Lorentz and Lord Rayleigh, and the experimental side with many observers—F. Paschen, Rubens and others; absorbing media have been investigated by A. W. Pflüger, a great many aniline dyes by K. Stöckl, and sodium vapour by R. W. Wood. Mention may also be made of the beautiful experiments of Christiansen (1884) and Lord Rayleigh on the colours transmitted by white powders suspended in liquids of the same refractive index. If, for instance, benzol be gradually added to finely powdered quartz, a succession of beautiful colours—red, yellow, green and finally blue—is transmitted, or, under certain conditions, the colours may appear at once, causing the mixture to flash like a fiery opal. Absorption, too, has received much attention; the theory has been especially elaborated by M. Planck, and the experimental investigation has been prosecuted from the purely physical standpoint, and also from the standpoint of the physical chemist, with a view to correlating absorption with constitution.
Interference phenomena have been assiduously studied. The experiments of Young, Fresnel, Lloyd, Fizeau and Foucault, of Fresnel and Arago on the measurement of refractive indices by the shift of the interference bands, of H. F. Talbot on the “Talbot bands” (which he insufficiently explained on the principle of interference, it being shown by Sir G. B. Airy that diffraction phenomena supervene), of Baden-Powell on the “Powell bands,” of David Brewster on “Brewster’s bands,” have been developed, together with many other phenomena—Newton’s rings, the colours of thin, thick and mixed plates, &c.—in a striking manner, one of the most important results being the construction of interferometers applicable to the determination of refractive indices and wave-lengths, with which the names of Jamin, Michelson, Fabry and Perot, and of Lummer and E. Gehrcke are chiefly associated. The mathematical investigations of Fresnel may be regarded as being completed by the analysis chiefly due to Airy, Stokes and Lord Rayleigh. Mention may be made of Sir G. G. Stokes’ attribution of the colours of iridescent crystals to periodic twinning; this view has been confirmed by Lord Rayleigh (Phil. Mag., 1888) who, from the purity of the reflected light, concluded that the laminae were equidistant by the order of a wave-length. Prior to 1891 only interference between waves proceeding in the same direction had been studied. In that year Otto H. Wiener obtained, on a film 120th of a wave-length in thickness, photographic impressions of the stationary waves formed by the interference of waves proceeding in opposite directions, and in 1892 Drude and Nernst employed a fluorescent film to record the same phenomenon. This principle is applied in the Lippmann colour photography, which was suggested by W. Zenker, realized by Gabriel Lippmann, and further investigated by R. G. Neuhauss, O. H. Wiener, H. Lehmann and others.
Great progress has been made in the study of diffraction, and “this department of optics is precisely the one in which the wave theory has secured its greatest triumphs” (Lord Rayleigh). The mathematical investigations of Fresnel and Poisson were placed on a dynamical basis by Sir G. G. Stokes; and the results gained more ready interpretation by the introduction of “Babinet’s principle” in 1837, and Cornu’s graphic methods in 1874. The theory also gained by the researches of Fraunhofer, Airy, Schwerd, E. Lommel and others. The theory of the concave grating, which resulted from H. A. Rowland’s classical methods of ruling lines of the necessary nature and number on curved surfaces, was worked out by Rowland, E. Mascart, C. Runge and others. The resolving power and the intensity of the spectra have been treated by Lord Rayleigh and Arthur Schuster, and more recently (1905), the distribution of light has been treated by A. B. Porter. The theory of diffraction is of great importance in designing optical instruments, the theory of which has been more especially treated by Ernst Abbe (whose theory of microscopic vision dates from about 1870) by the scientific staff at the Zeiss works, Jena, by Rayleigh and others. The theory of coronae (as diffraction phenomena) was originally due to Young, who, from the principle involved, devised the eriometer for measuring the diameters of very small objects; and Sir G. G. Stokes subsequently explained the appearances presented by minute opaque particles borne on a transparent plate. The polarization of the light diffracted at a slit was noted in 1861 by Fizeau, whose researches were extended in 1892 by H. Du Bois, and, for the case of gratings, by Du Bois and Rubens in 1904. The diffraction of light by small particles was studied in the form of very fine chemical precipitates by John Tyndall, who noticed the polarization of the beautiful cerulean blue which was transmitted. This subject—one form of which is presented in the blue colour of the sky—has been most auspiciously treated by Lord Rayleigh on both the elastic-solid and electromagnetic theories. Mention may be made of R. W. Wood’s experiments on thin metal films which, under certain conditions, originate colour phenomena inexplicable by interference and diffraction. These colours have been assigned to the principle of optical resonance, and have been treated by Kossonogov (Phys. Zeit., 1903). J. C. Maxwell Garnett (Phil. Trans. vol. 203) has shown that the colours of coloured glasses are due to ultra-microscopic particles, which have been directly studied by H. Siedentopf and R. Zsigmondy under limiting oblique illumination.
Polarization phenomena may, with great justification, be regarded as the most engrossing subject of optical research during the 19th century; the assiduity with which it was cultivated in the opening decades of that century received a great stimulus when James Nicol devised in 1828 the famous “Nicol prism,” which greatly facilitated the determination of the plane of vibration of polarized light, and the facts that light is polarized by reflection, repeated refractions, double refraction and by diffraction also contributed to the interest which the subject excited. The rotation of the plane of polarization by quartz was discovered in 1811 by Arago; if white light be used the colours change as the Nicol rotates—a phenomenon termed by Biot “rotatory dispersion.” Fresnel regarded rotatory polarization as compounded from right- and left-handed (dextro- and laevo-) circular polarizations; and Fresnel, Cornu, Dove and Cotton effected their experimental separation. Legrand des Cloizeaux discovered the enormously enhanced rotatory polarization of cinnabar, a property also possessed—but in a lesser degree—by the sulphates of strychnine and ethylene diamine. The rotatory power of certain liquids was discovered by Biot in 1815; and at a later date it was found that many solutions behaved similarly. A. Schuster distinguishes substances with regard to their action on polarized light as follows: substances which act in the isotropic state are termed photogyric; if the rotation be associated with crystal structure, crystallogyric; if the rotation be due to a magnetic field, magnetogyric; for cases not hitherto included the term allogyric is employed, while optically inactive substances are called isogyric. The theory of photogyric and crystallogyric rotation has been worked out on the elastic-solid (MacCullagh and others) and on the electromagnetic hypotheses (P. Drude, Cotton, &c.). Allogyrism is due to a symmetry of the molecule, and is a subject of the greatest importance in modern (and, more especially, organic) chemistry (see Stereoisomerism).
The optical properties of metals have been the subject of much experimental and theoretical inquiry. The explanations of MacCullagh and Cauchy were followed by those of Beer, Eisenlohr, Lundquist, Ketteler and others; the refractive indices were determined both directly (by Kundt) and indirectly by means of Brewster’s law; and the reflecting powers from λ = 251 µµ to λ = 1500 µµ were determined in 1900–1902 by Rubens and Hagen. The correlation of the optical and electrical constants of many metals has been especially studied by P. Drude (1900) and by Rubens and Hagen (1903).
The transformations of luminous radiations have also been studied. John Tyndall discovered calorescence. Fluorescence was treated by John Herschel in 1845, and by David Brewster in 1846, the theory being due to Sir G. G. Stokes (1852). More recent studies have been made by Lommel, E. L. Nichols and Merritt (Phys. Rev., 1904), and by Millikan who discovered polarized fluorescence in 1895. Our knowledge of phosphorescence was greatly improved by Becquerel, and Sir James Dewar obtained interesting results in the course of his low temperature researches (see Liquid Gases). In the theoretical and experimental study of radiation enormous progress has been recorded. The pressure of radiation, the necessity of which was demonstrated by Clerk Maxwell on the electromagnetic theory, and, in a simpler manner, by Joseph Larmor in his article Radiation in these volumes, has been experimentally determined by E. F. Nichols and Hull, and the tangential component by J. H. Poynting. With the theoretical and practical investigation the names of Balfour Stewart, Kirchhoff, Stefan, Bartoli, Boltzmann, W. Wien and Larmor are chiefly associated. Magneto-optics, too, has been greatly developed since Faraday’s discovery of the rotation of the plane of polarization by the magnetic field. The rotation for many substances was measured by Sir William H. Perkin, who attempted a correlation between rotation and composition. Brace effected the analysis of the beam into its two circularly polarized components, and in 1904 Mills measured their velocities. The Kerr effect, discovered in 1877, and the Zeeman effect (1896) widened the field of research, which, from its intimate connexion with the nature of light and electromagnetics, has resulted in discoveries of the greatest importance.
§ 14. Optical Instruments.—Important developments have been made in the construction and applications of optical instruments. To these three factors have contributed. The mathematician has quantitatively analysed the phenomena observed by the physicist, and has inductively shown what results are to be expected from certain optical systems. A consequence of this was the detailed study, and also the preparation, of glasses of diverse properties; to this the chemist largely contributed, and the manufacture of the so-called optical glass (see Glass) is possibly the most scientific department of glass manufacture. The mathematical investigations of lenses owe much to Gauss, Helmholtz and others, but far more to Abbe, who introduced the method of studying the aberrations separately, and applied his results with conspicuous skill to the construction of optical systems. The development of Abbe’s methods constitutes the main subject of research of the present-day optician, and has brought about the production of telescopes, microscopes, photographic lenses and other optical apparatus to an unprecedented pitch of excellence. Great improvements have been effected in the stereoscope. Binocular instruments with enhanced stereoscopic vision, an effect achieved by increasing the distance between the object glasses, have been introduced. In the study of diffraction phenomena, which led to the technical preparation of gratings, the early attempts of Fraunhofer, Nobert and Lewis Morris Rutherfurd, were followed by H. A. Rowland’s ruling of plane and concave gratings which revolutionized spectroscopic research, and, in 1898, by Michelson’s invention of the echelon grating. Of great importance are interferometers, which permit extremely accurate determinations of refractive indices and wave-lengths, and Michelson, from his classical evaluation of the standard metre in terms of the wave-lengths of certain of the cadmium rays, has suggested the adoption of the wave-length of one such ray as a standard with which national standards of length should be compared. Polarization phenomena, and particularly the rotation of the plane of polarization by such substances as sugar solutions, have led to the invention and improvements of polarimeters. The polarized light employed in such instruments is invariably obtained by transmission through a fixed Nicol prism—the polarizer—and the deviation is measured by the rotation of a second Nicol—the analyser. The early forms, which were termed “light and shade” polarimeters, have been generally replaced by “half-shade” instruments. Mention may also be made of the microscopic examination of objects in polarized light, the importance of which as a method of crystallographic and petrological research was suggested by Nicol, developed by Sorby and greatly expanded by Zirkel, Rosenbusch and others.
Bibliography.—There are numerous text-books which give elementary expositions of light and optical phenomena. More advanced works, which deal with the subject experimentally and mathematically, are A. B. Bassett, Treatise on Physical Optics (1892); Thomas Preston, Theory of Light, 2nd ed. by C. F. Joly (1901); R. W. Wood, Physical Optics (1905), which contains expositions on the electromagnetic theory, and treats “dispersion” in great detail. Treatises more particularly theoretical are James Walker, Analytical Theory of Light (1904); A. Schuster, Theory of Optics (1904); P. Drude, Theory of Optics, Eng. trans. by C. R. Mann and R. A. Millikan (1902). General treatises of exceptional merit are A. Winkelmann, Handbuch der Physik, vol. vi. “Optik” (1904); and E. Mascart, Traité d’optique (1889–1893); M. E. Verdet, Leçons d’optique physique (1869, 1872) is also a valuable work. Geometrical optics is treated in R. S. Heath, Geometrical Optics (2nd ed., 1898); H. A. Herman, Treatise on Geometrical Optics (1900). Applied optics, particularly with regard to the theory of optical instruments, is treated in H. D. Taylor, A System of Applied Optics (1906); E. T. Whittaker, The Theory of Optical Instruments (1907); in the publications of the scientific staff of the Zeiss works at Jena: Die Theorie der optischen Instrumente, vol. i. “Die Bilderzeugung in optischen Instrumenten” (1904); in S. Czapski, Theorie der optischen Instrumente, 2nd ed. by O. Eppenstein (1904); and in A. Steinheil and E. Voit, Handbuch der angewandten Optik (1901). The mathematical theory of general optics receives historical and modern treatment in the Encyklopädie der mathematischen Wissenschaften (Leipzig). Meteorological optics is fully treated in J. Pernter, Meteorologische Optik; and physiological optics in H. v Helmholtz, Handbuch der physiologischen Optik (1896) and in A. Koenig, Gesammelte Abhandlungen zur physiologischen Optik (1903).
The history of the subject may be studied in J. C. Poggendorff, Geschichte der Physik (1879); F. Rosenberger, Die Geschichte der Physik (1882–1890); E. Gerland and F. Traumüller, Geschichte der physikalischen Experimentierkunst (1899); reference may also be made to Joseph Priestley, History and Present State of Discoveries relating to Vision, Light and Colours (1772), German translation by G. S. Klügel (Leipzig, 1775). Original memoirs are available in many cases in their author’s “collected works,” e.g. Huygens, Young, Fresnel, Hamilton, Cauchy, Rowland, Clerk Maxwell, Stokes (and also his Burnett Lectures on Light), Kelvin (and also his Baltimore Lectures, 1904) and Lord Rayleigh. Newton’s Opticks forms volumes 96 and 97 of Ostwald’s Klassiker; Huygens’ Über d. Licht (1678), vol. 20, and Kepler’s Dioptrice (1611), vol. 144 of the same series.
Contemporary progress is reported in current scientific journals, e.g. the Transactions and Proceedings of the Royal Society, and of the Physical Society (London), the Philosophical Magazine (London), the Physical Review (New York, 1893 seq.) and in the British Association Reports; in the Annales de chimie et de physique and Journal de physique (Paris); and in the Physikalische Zeitschrift (Leipzig) and the Annalen der Physik und Chemie (since 1900: Annalen der Physik) (Leipzig). (C. E.*)
II. Nature of Light
1. Newton’s Corpuscular Theory.—Until the beginning of the 19th century physicists were divided between two different views concerning the nature of optical phenomena. According to the one, luminous bodies emit extremely small corpuscles which can freely pass through transparent substances and produce the sensation of light by their impact against the retina. This emission or corpuscular theory of light was supported by the authority of Isaac Newton,^{[8]} and, though it has been entirely superseded by its rival, the wave-theory, it remains of considerable historical interest.
2. Explanation of Reflection and Refraction.—Newton supposed the light-corpuscles to be subjected to attractive and repulsive forces exerted at very small distances by the particles of matter. In the interior of a homogeneous body a corpuscle moves in a straight line as it is equally acted on from all sides, but it changes its course at the boundary of two bodies, because, in a thin layer near the surface there is a resultant force in the direction of the normal. In modern language we may say that a corpuscle has at every point a definite potential energy, the value of which is constant throughout the interior of a homogeneous body, and is even equal in all bodies of the same kind, but changes from one substance to another. If, originally, while moving in air, the corpuscles had a definite velocity v_{0}, their velocity v in the interior of any other substance is quite determinate. It is given by the equation 12mv^{2} − 12mv_{0}^{2}＝A, in which m denotes the mass of a corpuscle, and A the excess of its potential energy in air over that in the substance considered.
A ray of light falling on the surface of separation of two bodies is reflected according to the well-known simple law, if the corpuscles are acted on by a sufficiently large force directed towards the first medium. On the contrary, whenever the field of force near the surface is such that the corpuscles can penetrate into the interior of the second body, the ray is refracted. In this case the law of Snellius can be deduced from the consideration that the projection w of the velocity on the surface of separation is not altered, either in direction or in magnitude. This obviously requires that the plane passing through the incident and the refracted rays be normal to the surface, and that, if α_{1} and α_{2} are the angles of incidence and of refraction, v_{1} and v_{2} the velocities of light in the two media,
sin α_{1}/sin α_{2}＝w/v_{1} : w/v_{2}＝v_{2}/v_{1}. | (1) |
The ratio is constant, because, as has already been observed, v_{1} and v_{2} have definite values.
As to the unequal refrangibility of differently coloured light, Newton accounted for it by imagining different kinds of corpuscles. He further carefully examined the phenomenon of total reflection, and described an interesting experiment connected with it. If one of the faces of a glass prism receives on the inside a beam of light of such obliquity that it is totally reflected under ordinary circumstances, a marked change is observed when a second piece of glass is made to approach the reflecting face, so as to be separated from it only by a very thin layer of air. The reflection is then found no longer to be total, part of the light finding its way into the second piece of glass. Newton concluded from this that the corpuscles are attracted by the glass even at a certain small measurable distance.
3. New Hypotheses in the Corpuscular Theory.—The preceding explanation of reflection and refraction is open to a very serious objection. If the particles in a beam of light all moved with the same velocity and were acted on by the same forces, they all ought to follow exactly the same path. In order to understand that part of the incident light is reflected and part of it transmitted, Newton imagined that each corpuscle undergoes certain alternating changes; he assumed that in some of its different “phases” it is more apt to be reflected, and in others more apt to be transmitted. The same idea was applied by him to the phenomena presented by very thin layers. He had observed that a gradual increase of the thickness of a layer produces periodic changes in the intensity of the reflected light, and he very ingeniously explained these by his theory. It is clear that the intensity of the transmitted light will be a minimum if the corpuscles that have traversed the front surface of the layer, having reached that surface while in their phase of easy transmission, have passed to the opposite phase the moment they arrive at the back surface. As to the nature of the alternating phases, Newton (Opticks, 3rd ed., 1721, p. 347) expresses himself as follows:—“Nothing more is requisite for putting the Rays of Light into Fits of easy Reflexion and easy Transmission than that they be small Bodies which by their attractive Powers, or some other Force, stir up Vibrations in what they act upon, which Vibrations being swifter than the Rays, overtake them successively, and agitate them so as by turns to increase and decrease their Velocities, and thereby put them into those Fits.”
4. The Corpuscular Theory and the Wave-Theory compared.—Though Newton introduced the notion of periodic changes, which was to play so prominent a part in the later development of the wave-theory, he rejected this theory in the form in which it had been set forth shortly before by Christiaan Huygens in his Traité de la lumière (1690), his chief objections being: (1) that the rectilinear propagation had not been satisfactorily accounted for; (2) that the motions of heavenly bodies show no sign of a resistance due to a medium filling all space; and (3) that Huygens had not sufficiently explained the peculiar properties of the rays produced by the double refraction in Iceland spar. In Newton’s days these objections were of much weight.
Yet his own theory had many weaknesses. It explained the propagation in straight lines, but it could assign no cause for the equality of the speed of propagation of all rays. It adapted itself to a large variety of phenomena, even to that of double refraction (Newton says [ibid.]:—“. . . the unusual Refraction of Iceland Crystal looks very much as if it were perform’d by some kind of attractive virtue lodged in certain Sides both of the Rays, and of the Particles of the Crystal.”), but it could do so only at the price of losing much of its original simplicity.
In the earlier part of the 19th century, the corpuscular theory broke down under the weight of experimental evidence, and it received the final blow when J. B. L. Foucault proved by direct experiment that the velocity of light in water is not greater than that in air, as it should be according to the formula (1), but less than it, as is required by the wave-theory.
5. General Theorems on Rays of Light.—With the aid of suitable assumptions the Newtonian theory can accurately trace the course of a ray of light in any system of isotropic bodies, whether homogeneous or otherwise; the problem being equivalent to that of determining the motion of a material point in a space in which its potential energy is given as a function of the coordinates. The application of the dynamical principles of “least and of varying action” to this latter problem leads to the following important theorems which William Rowan Hamilton made the basis of his exhaustive treatment of systems of rays.^{[9]} The total energy of a corpuscle is supposed to have a given value, so that, since the potential energy is considered as known at every point, the velocity v is so likewise.
(a) The path along which light travels from a point A to a point B is determined by the condition that for this line the integral ∫v ds, in which ds is an element of the line, be a minimum (provided A and B be not too near each other). Therefore, since v = μv_{0}, if v_{0} is the velocity of light in vacuo and μ the index of refraction, we have for every variation of the path the points A and B remaining fixed,
δ∫μ ds = 0. | (2) |
(b) Let the point A be kept fixed, but let B undergo an infinitely small displacement BB′ (= q) in a direction making an angle θ with the last element of the ray AB. Then, comparing the new ray AB′ with the original one, it follows that
δ∫μ ds = μ_{Β}q cos θ, | (3) |
where μ_{Β} is the value of μ at the point B.
6. General Considerations on the Propagation of Waves.—“Waves,” i.e. local disturbances of equilibrium travelling onward with a certain speed, can exist in a large variety of systems. In a theory of these phenomena, the state of things at a definite point may in general be defined by a certain directed or vector quantity P,^{[10]} which is zero in the state of equilibrium, and may be called the disturbance (for example, the velocity of the air in the case of sound vibrations, or the displacement of the particles of an elastic body from their positions of equilibrium). The components P_{x}, P_{y}, P_{z} of the disturbance in the directions of the axes of coordinates are to be considered as functions of the coordinates x, y, z and the time t, determined by a set of partial differential equations, whose form depends on the nature of the problem considered. If the equations are homogeneous and linear, as they always are for sufficiently small disturbances, the following theorems hold.
(a) Values of P_{x}, P_{y}, P_{z} (expressed in terms of x, y, z, t) which satisfy the equations will do so still after multiplication by a common arbitrary constant.
(b) Two or more solutions of the equations may be combined into a new solution by addition of the values of P_{x}, those of P_{y}, &c., i.e. by compounding the vectors P, such as they are in each of the particular solutions.
In the application to light, the first proposition means that the phenomena of propagation, reflection, refraction, &c., can be produced in the same way with strong as with weak light. The second proposition contains the principle of the “superposition” of different states, on which the explanation of all phenomena of interference is made to depend.
In the simplest cases (monochromatic or homogeneous light) the disturbance is a simple harmonic function of the time (“simple harmonic vibrations”), so that its components can be represented by
The “phases” of these vibrations are determined by the angles nt + ƒ_{1}, &c., or by the times t + ƒ_{1}/n, &c. The “frequency” n is constant throughout the system, while the quantities ƒ_{1}, ƒ_{2}, ƒ_{3}, and perhaps the “amplitudes” a_{1}, a_{2}, a_{3} change from point to point. It may be shown that the end of a straight line representing the vector P, and drawn from the point considered, in general describes a certain ellipse, which becomes a straight line, if ƒ_{1} = ƒ_{2} = ƒ_{3}. In this latter case, to which the larger part of this article will be confined, we can write in vector notation
P = A cos (nt + ƒ), | (4) |
where A itself is to be regarded as a vector.
We have next to consider the way in which the disturbance changes from point to point. The most important case is that of plane waves with constant amplitude A. Here ƒ is the same at all points of a plane (“wave-front”) of a definite direction, but changes as a linear function as we pass from one such wave-front to the next. The axis of x being drawn at right angles to the wave-fronts, we may write ƒ = ƒ_{0} − kx, where ƒ_{0} and k are constants, so that (4) becomes
P = A cos (nt − kx + ƒ_{0}). | (5) |
This expression has the period 2π/n with respect to the time and the perion 2π/k with respect to x, so that the “time of vibration” and the “wave-length” are given by T = 2π/n, λ = 2π/k. Further, it is easily seen that the phase belonging to certain values of x and t is equal to that which corresponds to x + Δx and t + Δt provided Δx = (n/k) Δt. Therefore the phase, or the disturbance itself, may be said to be propagated in the direction normal to the wave-fronts with a velocity (velocity of the waves) v = n/k, which is connected with the time of vibration and the wave-length by the relation
λ = vT. | (6) |
the same velocity. In anisotropic bodies (crystals), with which the theory of light is largely concerned, the problem is more complicated. As a general rule we can say that, for a given direction of the wave-fronts, the vibrations must have a determinate direction, if the propagation is to take place according to the simple formula given above. It is to be understood that for a given direction of the waves there may be two or even more directions of vibration of the kind, and that in such a case there are as many different velocities, each belonging to one particular direction of vibration.
7. Wave-surface.—After having found the values of v for a particular frequency and different directions of the wave-normal, a very instructive graphical representation can be employed.
Let ON be a line in any direction, drawn from a fixed point O, OA a length along this line equal to the velocity v of waves having ON for their normal, or, more generally, OA, OA′, &c., lengths equal to the velocities v, v ′, &c., which such waves have according to their direction of vibration, Q, Q′, &c., planes perpendicular to ON through A, A^{1}, &c. Let this construction be repeated for all directions of ON, and let W be the surface that is touched by all the planes Q, Q′, &c. It is clear that if this surface, which is called the “wave-surface,” is known, the velocity of propagation of plane waves of any chosen direction is given by the length of the perpendicular from the centre O on a tangent plane in the given direction. It must be kept in mind that, in general, each tangent plane corresponds to one definite direction of vibration. If this direction is assigned in each point of the wave-surface, the diagram contains all the information which we can desire concerning the propagation of plane waves of the frequency that has been chosen.
The plane Q employed in the above construction is the position after unit of time of a wave-front perpendicular to ON and originally passing through the point O. The surface W itself is often considered as the locus of all points that are reached in unit of time by a disturbance starting from O and spreading towards all sides. Admitting the validity of this view, we can determine in a similar way the locus of the points reached in some infinitely short time dt, the wave-surface, as we may say, or the “elementary wave,” corresponding to this time. It is similar to W, all dimensions of the latter surface being multiplied by dt. It may be noticed that in a heterogeneous medium a wave of this kind has the same form as if the properties of matter existing at its centre extended over a finite space.
8. Theory of Huygens.—Huygens was the first to show that the explanation of optical phenomena may be made to depend on the wave-surface, not only in isotropic bodies, in which it has a spherical form, but also in crystals, for one of which (Iceland spar) he deduced the form of the surface from the observed double refraction. In his argument Huygens availed himself of the following principle that is justly named after him: Any point that is reached by a wave of light becomes a new centre of radiation from which the disturbance is propagated towards all sides. On this basis he determined the progress of light-waves by a construction which, under a restriction to be mentioned in § 13, applied to waves of any form and to all kinds of transparent media. Let σ be the surface (wave-front) to which a definite phase of vibration has advanced at a certain time t, dt an infinitely small increment of time, and let an elementary wave corresponding to this interval be described around each point P of σ. Then the envelope σ′ of all these elementary waves is the surface reached by the phase in question at the time t + dt, and by repeating the construction all successive positions of the wave-front can be found.
Huygens also considered the propagation of waves that are laterally limited, by having passed, for example, through an opening in an opaque screen. If, in the first wave-front σ, the disturbance exists only in a certain part bounded by the contour s, we can confine ourselves to the elementary waves around the points of that part, and to a portion of the new wave-front σ′ whose boundary passes through the points where σ′ touches the elementary waves having their centres on s. Taking for granted Huygens’s assumption that a sensible disturbance is only found in those places where the elementary waves are touched by the new wave-front, it may be inferred that the lateral limits of the beam of light are determined by lines, each element of which joins the centre P of an elementary wave with its point of contact P′ with the next wave-front. To lines of this kind, whose course can be made visible by using narrow pencils of light, the name of “rays” is to be given in the wave-theory. The disturbance may be conceived to travel along them with a velocity u = PP′/dt, which is therefore called the “ray-velocity.”
The construction shows that, corresponding to each direction of the wave-front (with a determinate direction of vibration), there is a definite direction and a definite velocity of the ray. Both are given by a line drawn from the centre of the wave-surface to its point of contact with a tangent plane of the given direction. It will be convenient to say that this line and the plane are conjugate with each other. The rays of light, curved in non-homogeneous bodies, are always straight lines in homogeneous substances. In an isotropic medium, whether homogeneous or otherwise, they are normal to the wave-fronts, and their velocity is equal to that of the waves.
By applying his construction to the reflection and refraction of light, Huygens accounted for these phenomena in isotropic bodies as well as in Iceland spar. It was afterwards shown by Augustin Fresnel that the double refraction in biaxal crystals can be explained in the same way, provided the proper form be assigned to the wave-surface.
In any point of a bounding surface the normals to the reflected and refracted waves, whatever be their number, always lie in the plane passing through the normal to the incident waves and that to the surface itself. Moreover, if α_{1} is the angle between these two latter normals, and α_{2} the angle between the normal to the boundary and that to any one of the reflected and refracted waves, and v_{1}, v_{2} the corresponding wave-velocities, the relation
sin α_{1}/sin α_{2} = v_{1}/v_{2} | (7) |
is found to hold in all cases. These important theorems may be proved independently of Huygens’s construction by simply observing that, at each point of the surface of separation, there must be a certain connexion between the disturbances existing in the incident, the reflected, and the refracted waves, and that, therefore, the lines of intersection of the surface with the positions of an incident wave-front, succeeding each other at equal intervals of time dt, must coincide with the lines in which the surface is intersected by a similar series of reflected or refracted wave-fronts.
In the case of isotropic media, the ratio (7) is constant, so that we are led to the law of Snellius, the index of refraction being given by
μ = v_{1}/v_{2} | (8) |
(cf. equation 1).
9. General Theorems on Rays, deduced from Huygens’s Construction.—(a) Let A and B be two points arbitrarily chosen in a system of transparent bodies, ds an element of a line drawn from A to B, u the velocity of a ray of light coinciding with ds. Then the integral ∫u^{−1} ds, which represents the time required for a motion along the line with the velocity u, is a minimum for the course actually taken by a ray of light (unless A and B be too far apart). This is the “principle of least time” first formulated by Pierre de Fermat for the case of two isotropic substances. It shows that the course of a ray of light can always be inverted.
(b) Rays of light starting in all directions from a point A and travelling onward for a definite length of time, reach a surface σ, whose tangent plane at a point B is conjugate, in the medium surrounding B, with the last element of the ray AB.
(c) If all rays issuing from A are concentrated at a point B, the integral ∫u^{−1}ds has the same value for each of them.
(d) In case (b) the variation of the integral caused by an infinitely small displacement q of B, the point A remaining fixed, is given by δ∫u^{−1}ds = q cos θ/v_{B}. Here θ is the angle between the displacement q and the normal to the surface σ, in the direction of propagation, v_{B} the velocity of a plane wave tangent to this surface.
In the case of isotropic bodies, for which the relation (8) holds, we recover the theorems concerning the integral ∫μds which we have deduced from the emission theory (§ 5).
10. Further General Theorems.—(a) Let V_{1} and V_{2} be two planes in a system of isotropic bodies, let rectangular axes of coordinates be chosen in each of these planes, and let x_{1}, y_{1} be the coordinates of a point A in V_{1}, and x_{2}, y_{2} those of a point B in V_{2}. The integral ∫μds, taken for the ray between A and B, is a function of x_{1}, y_{1}, x_{2}, y_{2} and, if ξ_{1} denotes either x_{1} or y_{1}, and ξ_{2} either x_{2} or y_{2}, we shall have
On both sides of this equation the first differentiation may be performed by means of the formula (3). The second differentiation admits of a geometrical interpretation, and the formula may finally be employed for proving the following theorem:
Let ω_{1} be the solid angle of an infinitely thin pencil of rays issuing from A and intersecting the plane V_{2} in an element σ_{2} at the point B. Similarly, let ω_{2} be the solid angle of a pencil starting from B and falling on the element σ_{1} of the plane V_{1} at the point A. Then, denoting by μ_{1} and μ_{2} the indices of refraction of the matter at the points A and B, by θ_{1} and θ_{2} the sharp angles which the ray AB at its extremities makes with the normals to V_{1} and V_{2}, we have
(b) There is a second theorem that is expressed by exactly the same formula, if we understand by σ_{1} and σ_{2} elements of surface that are related to each other as an object and its optical image—by ω_{1}, ω_{2} the infinitely small openings, at the beginning and the end of its course, of a pencil of rays issuing from a point A of σ_{1} and coming together at the corresponding point B of σ_{2}, and by θ_{1}, θ_{2} the sharp angles which one of the rays makes with the normals to σ_{1} and σ_{2}. The proof may be based upon the first theorem. It suffices to consider the section σ of the pencil by some intermediate plane, and a bundle of rays starting from the points of σ_{1} and reaching those of σ_{2} after having all passed through a point of that section σ.
(c) If in the last theorem the system of bodies is symmetrical around the straight line AB, we can take for σ_{1} and σ_{2} circular planes having AB as axis. Let h_{1} and h_{2} be the radii of these circles, i.e. the linear dimensions of an object and its image, ε_{1} and ε_{2} the infinitely small angles which a ray R going from A to B makes with the axis at these points. Then the above formula gives μ_{1}h_{1}ε_{1} = μ_{2}h_{2}ε_{2}, a relation that was proved, for the particular case μ_{1} = μ_{2} by Huygens and Lagrange. It is still more valuable if one distinguishes by the algebraic sign of h_{2} whether the image is direct or inverted, and by that of ε_{2} whether the ray R on leaving A and on reaching B lies on opposite sides of the axis or on the same side.
The above theorems are of much service in the theory of optical instruments and in the general theory of radiation.
11. Phenomena of Interference and Diffraction.—The impulses or motions which a luminous body sends forth through the universal medium or aether, were considered by Huygens as being without any regular succession; he neither speaks of vibrations, nor of the physical cause of the colours. The idea that monochromatic light consists of a succession of simple harmonic vibrations like those represented by the equation (5), and that the sensation of colour depends on the frequency, is due to Thomas Young^{[11]} and Fresnel,^{[12]} who explained the phenomena of interference on this assumption combined with the principle of super-position. In doing so they were also enabled to determine the wave-length, ranging from 0.000076 cm. at the red end of the spectrum to 0.000039 cm. for the extreme violet and, by means of the formula (6), the number of vibrations per second. Later investigations have shown that the infra-red rays as well as the ultra-violet ones are of the same physical nature as the luminous rays, differing from these only by the greater or smaller length of their waves. The wave-length amounts to 0.006 cm. for the least refrangible infra-red, and is as small as 0.00001 cm. for the extreme ultra-violet.
Another important part of Fresnel’s work is his treatment of diffraction on the basis of Huygens’s principle. If, for example, light falls on a screen with a narrow slit, each point of the slit is regarded as a new centre of vibration, and the intensity at any point behind the screen is found by compounding with each other the disturbances coming from all these points, due account being taken of the phases with which they come together (see Diffraction; Interference).
12. Results of Later Mathematical Theory.—Though the theory of diffraction developed by Fresnel, and by other physicists who worked on the same lines, shows a most beautiful agreement with observed facts, yet its foundation, Huygens’s principle, cannot, in its original elementary form, be deemed quite satisfactory. The general validity of the results has, however, been confirmed by the researches of those mathematicians (Siméon Denis Poisson, Augustin Louis Cauchy, Sir G. G. Stokes, Gustav Robert Kirchhoff) who investigated the propagation of vibrations in a more rigorous manner. Kirchhoff^{[13]} showed that the disturbance at any point of the aether inside a closed surface which contains no ponderable matter can be represented as made up of a large number of parts, each of which depends upon the state of things at one point of the surface. This result, the modern form of Huygens’s principle, can be extended to a system of bodies of any kind, the only restriction being that the source of light be not surrounded by the surface. Certain causes capable of producing vibrations can be imagined to be distributed all over this latter, in such a way that the disturbances to which they give rise in the enclosed space are exactly those which are brought about by the real source of light.^{[14]} Another interesting result that has been verified by experiment is that, whenever rays of light pass through a focus, the phase undergoes a change of half a period. It must be added that the results alluded to in the above, though generally presented in the terms of some particular form of the wave theory, often apply to other forms as well.
13. Rays of Light.—In working out the theory of diffraction it is possible to state exactly in what sense light may be said to travel in straight lines. Behind an opening whose width is very large in comparison with the wave-length the limits between the illuminated and the dark parts of space are approximately determined by rays passing along the borders.
This conclusion can also be arrived at by a mode of reasoning that is independent of the theory of diffraction.^{[15]} If linear differential equations admit a solution of the form (5) with A constant, they can also be satisfied by making A a function of the coordinates, such that, in a wave-front, it changes very little over a distance equal to the wave-length λ, and that it is constant along each line conjugate with the wave-fronts. In cases of this kind the disturbance may truly be said to travel along lines of the said direction, and an observer who is unable to discern lengths of the order of λ, and who uses an opening of much larger dimensions, may very well have the impression of a cylindrical beam with a sharp boundary.
A similar result is found for curved waves. If the additional restriction is made that their radii of curvature be very much larger than the wave-length, Huygens’s construction may confidently be employed. The amplitudes all along a ray are determined by, and proportional to, the amplitude at one of its points.
14. Polarized Light.—As the theorems used in the explanation of interference and diffraction are true for all kinds of vibratory motions, these phenomena can give us no clue to the special kind of vibrations in light-waves. Further information, however, may be drawn from experiments on plane polarized light. The properties of a beam of this kind are completely known when the position of a certain plane passing through the direction of the rays, and in which the beam is said to be polarized, is given. “This plane of polarization,” as it is called, coincides with the plane of incidence in those cases where the light has been polarized by reflection on a glass surface under an angle of incidence whose tangent is equal to the index of refraction (Brewster’s law).
The researches of Fresnel and Arago left no doubt as to the direction of the vibrations in polarized light with respect to that of the rays themselves. In isotropic bodies at least, the vibrations are exactly transverse, i.e. perpendicular to the rays, either in the plane of polarization or at right angles to it. The first part of this statement also applies to unpolarized light, as this can always be dissolved into polarized components.
Much experimental work has been done on the production of polarized rays by double refraction and on the reflection of polarized light, either by isotropic or by anisotropic transparent bodies, the object of inquiry being in the latter case to determine the position of the plane of polarization of the reflected rays and their intensity.
In this way a large amount of evidence has been gathered by which it has been possible to test different theories concerning the nature of light and that of the medium through which it is propagated. A common feature of nearly all these theories is that the aether is supposed to exist not only in spaces void of matter, but also in the interior of ponderable bodies.
15. Fresnel’s Theory.—Fresnel and his immediate successors assimilated the aether to an elastic solid, so that the velocity of propagation of transverse vibrations could be determined by the formula v = √(K/ρ), where K denotes the modulus of rigidity and ρ the density. According to this equation the different properties of various isotropic transparent bodies may arise from different values of K, of ρ, or of both. It has, however, been found that if both K and ρ are supposed to change from one substance to another, it is impossible to obtain the right reflection formulae. Assuming the constancy of K Fresnel was led to equations which agreed with the observed properties of the reflected light, if he made the further assumption (to be mentioned in what follows as “Fresnel’s assumption”) that the vibrations of plane polarized light are perpendicular to the plane of polarization.
Let the indices p and n relate to the two principal cases in which the incident (and, consequently, the reflected) light is polarized in the plane of incidence, or normally to it, and let positive directions h and h′ be chosen for the disturbance (at the surface itself) in the incident and for that in the reflected beam, in such a manner that, by a common rotation, h and the incident ray prolonged may be made to coincide with h′ and the reflected ray. Then, if α_{1} and α_{2} are the angles of incidence and refraction, Fresnel shows that, in order to get the reflected disturbance, the incident one must be multiplied by
α_{p} = −sin (α_{1} − α_{2}) / sin (α_{1} + α_{2}) | (9) |
in the first, and by
α_{n} = tan (α_{1} − α_{2}) / tan (α_{1} + α_{2}) | (10) |
in the second principal case.
As to double refraction, Fresnel made it depend on the unequal elasticity of the aether in different directions. He came to the conclusion that, for a given direction of the waves, there are two possible directions of vibration (§ 6), lying in the wave-front, at right angles to each other, and he determined the form of the wave-surface, both in uniaxal and in biaxal crystals.
Though objections may be urged against the dynamic part of Fresnel’s theory, he admirably succeeded in adapting it to the facts.
16. Electromagnetic Theory.—We here leave the historical order and pass on to Maxwell’s theory of light.
James Clerk Maxwell, who had set himself the task of mathematically working out Michael Faraday’s views, and who, both by doing so and by introducing many new ideas of his own, became the founder of the modern science of electricity,^{[16]} recognized that, at every point of an electromagnetic field, the state of things can be defined by two vector quantities, the “electric force” E and the “magnetic force” H, the former of which is the force acting on unit of electricity and the latter that which acts on a magnetic pole of unit strength. In a non-conductor (dielectric) the force E produces a state that may be described as a displacement of electricity from its position of equilibrium. This state is represented by a vector D (“dielectric displacement”) whose magnitude is measured by the quantity of electricity reckoned per unit area which has traversed an element of surface perpendicular to D itself. Similarly, there is a vector quantity B (the “magnetic induction”) intimately connected with the magnetic force H. Changes of the dielectric displacement constitute an electric current measured by the rate of change of D, and represented in vector notation by
C = Ḋ | (11) |
Periodic changes of D and B may be called “electric” and “magnetic vibrations.” Properly choosing the units, the axes of coordinates (in the first proposition also the positive direction of s and n), and denoting components of vectors by suitable indices, we can express in the following way the fundamental propositions of the theory.
(a) Let s be a closed line, σ a surface bounded by it, n the normal to σ. Then, for all bodies,
H_{s}ds = | 1 | C_{n} d_{σ}, E_{s} ds = − | 1 | d | B_{n} d_{σ}, | |
c | c | dt |
where the constant c means the ratio between the electro-magnet and the electrostatic unit of electricity.
From these equations we can deduce:
(α) For the interior of a body, the equations
∂H_{z} | − | ∂H_{y} | = | 1 | C_{x}, | ∂H_{x} | − | ∂H_{z} | = | 1 | C_{y}, | ∂H_{y} | − | ∂H_{x} | = | 1 | C_{z} |
∂y | ∂z | c | ∂z | ∂x | c | ∂x | ∂y | c |
∂E_{z} | − | ∂E_{y} | = − | 1 | ∂B_{z} | , | ∂E_{x} | − | ∂E_{z} | = − | 1 | ∂B_{y} | , | ∂E_{y} | − | ∂E_{x} | = − | 1 | ∂B_{z} | ; | |||
∂y | ∂z | c | ∂t | ∂z | ∂x | c | ∂t | ∂x | ∂y | c | ∂t |
(ß) For a surface of separation, the continuity of the tangential components of E and H;
(γ) The solenoidal distribution of C and B, and in a dielectric that of D. A solenoidal distribution of a vector is one corresponding to that of the velocity in an incompressible fluid. It involves the continuity, at a surface, of the normal component of the vector.
(b) The relation between the electric force and the dielectric displacement is expressed by
the constants ε_{1}, ε_{2}, ε_{3} (dielectric constants) depending on the properties of the body considered. In an isotropic medium they have a common value ε, which is equal to unity for the free aether, so that for this medium D = E.
(c) There is a relation similar to (14) between the magnetic force and the magnetic induction. For the aether, however, and for all ponderable bodies with which this article is concerned, we may write B = H. It follows from these principles that, in an isotropic dielectric, transverse electric vibrations can be propagated with a velocity
v = c / √ε. | (15) |
Indeed, all conditions are satisfied if we put
D_{x} = 0, D_{y} = a cos n (t − xv^{−1} + l), | D_{z} = 0, | |
H_{x} = 0, H_{y} = 0, | H_{z} = avc^{−1} cos n (t − xv^{−1} + l) |
For the free aether the velocity has the value c. Now it had been
found that the ratio c between the two units of electricity agrees
within the limits of experimental errors with the numerical value of
the velocity of light in aether. (The mean result of the most exact
determinations^{[17]} of c is 3,001·10^{10} cm./sec., the largest deviations
being about 0,008·10^{10}; and Cornu^{[18]} gives 3,001·10^{10} ± 0,003·10^{10}
as the most probable value of the velocity of light.) By this Maxwell
was led to suppose that light consists of transverse electromagnetic
disturbances. On this assumption, the equations (16) represent a
beam of plane polarized light. They show that, in such a beam,
there are at the same time electric and magnetic vibrations, both
transverse, and at right angles to each other.
It must be added that the electromagnetic field is the seat of two kinds of energy distinguished by the names of electric and magnetic energy, and that, according to a beautiful theorem due to J. H. Poynting,^{[19]} the energy may be conceived to flow in a direction perpendicular both to the electric and to the magnetic force. The amounts per unit of volume of the electric and the magnetic energy are given by the expressions
and
whose mean values for a full period are equal in every beam of light.
The formula (15) shows that the index of refraction of a body is given by √ε, a result that has been verified by Ludwig Boltzmann’s measurements^{[20]} of the dielectric constants of gases. Thus Maxwell’s theory can assign the true cause of the different optical properties of various transparent bodies. It also leads to the reflection formulae (9) and (10), provided the electric vibrations of polarized light be supposed to be perpendicular to the plane of polarization, which implies that the magnetic vibrations are parallel to that plane.
Following the same assumption Maxwell deduced the laws of double refraction, which he ascribes to the unequality of ε_{1}, ε_{2}, ε_{3}. His results agree with those of Fresnel and the theory has been confirmed by Boltzmann,^{[21]} who measured the three coefficients in the case of crystallized sulphur, and compared them with the principal indices of refraction. Subsequently the problem of crystalline reflection has been completely solved and it has been shown that, in a crystal, Poynting’s flow of energy has the direction of the rays as determined by Huygens’s construction.
Two further verifications must here be mentioned. In the first place, though we shall speak almost exclusively of the propagation of light in transparent dielectrics, a few words may be said about the optical properties of conductors. The simplest assumption concerning the electric current C in a metallic body is expressed by the equation C = σE, where σ is the coefficient of conductivity. Combining this with his other formulae (we may say with (12) and (13)), Maxwell found that there must be an absorption of light, a result that can be readily understood since the motion of electricity in a conductor gives rise to a development of heat. But, though Maxwell accounted in this way for the fundamental fact that metals are opaque bodies, there remained a wide divergence between the values of the coefficient of absorption as directly measured and as calculated from the electrical conductivity; but in 1903 it was shown by E. Hagen and H. Rubens^{[22]} that the agreement is very satisfactory in the case of the extreme infra-red rays.
In the second place, the electromagnetic theory requires that a surface struck by a beam of light shall experience a certain pressure. If the beam falls normally on a plane disk, the pressure is normal too; its total amount is given by c^{−1}(i_{1} + i_{2} − i_{3}), if i_{1}, i_{2} and i_{3} are the quantities of energy that are carried forward per unit of time by the incident, the reflected, and the transmitted light. This result has been quantitatively verified by E. F. Nicholls and G. F. Hull.^{[23]}
Maxwell’s predictions have been splendidly confirmed by the experiments of Heinrich Hertz^{[24]} and others on electromagnetic waves; by diminishing the length of these to the utmost, some physicists have been able to reproduce with them all phenomena of reflection, refraction (single and double), interference, and polarization.^{[25]} A table of the wave-lengths observed in the aether now has to contain, besides the numbers given in § 11, the lengths of the waves produced by electromagnetic apparatus and extending from the long waves used in wireless telegraphy down to about 0.6 cm.
17. Mechanical Models of the Electromagnetic Medium.—From the results already enumerated, a clear idea can be formed of the difficulties which were encountered in the older form of the wave-theory. Whereas, in Maxwell’s theory, longitudinal vibrations are excluded ab initio by the solenoidal distribution of the electric current, the elastic-solid theory had to take them into account, unless, as was often done, one made them disappear by supposing them to have a very great velocity of propagation, so that the aether was considered to be practically incompressible. Even on this assumption, however, much in Fresnel’s theory remained questionable. Thus George Green,^{[26]} who was the first to apply the theory of elasticity in an unobjectionable manner, arrived on Fresnel’s assumption at a formula for the reflection coefficient A_{n} sensibly differing from (10).
In the theory of double refraction the difficulties are no less serious. As a general rule there are in an anisotropic elastic solid three possible directions of vibration (§ 6), at right angles to each other, for a given direction of the waves, but none of these lies in the wave-front. In order to make two of them do so and to find Fresnel’s form for the wave-surface, new hypotheses are required. On Fresnel’s assumption it is even necessary, as was observed by Green, to suppose that in the absence of all vibrations there is already a certain state of pressure in the medium.
If we adhere to Fresnel’s assumption, it is indeed scarcely possible to construct an elastic model of the electromagnetic medium. It may be done, however, if the velocities of the particles in the model are taken to represent the magnetic force H, which, of course, implies that the vibrations of the particles are parallel to the plane of polarization, and that the magnetic energy is represented by the kinetic energy in the model. Considering further that, in the case of two bodies connected with each other, there is continuity of H in the electromagnetic system, and continuity of the velocity of the particles in the model, it becomes clear that the representation of H by that velocity must be on the same scale in all substances, so that, if ξ, η, ζ are the displacements of a particle and g a universal constant, we may write
H_{x} = g | ∂ξ | , H_{y} = g | ∂η | , H_{z} = g | ∂ζ | . |
∂t | ∂t | ∂t |
By this the magnetic energy per unit of volume becomes
12 g^{2} | ∂ξ | ^{2} | + | ∂η | ^{2} | + | ∂ζ | ^{2} | , | |||
∂t | ∂t | ∂t |
and since this must be the kinetic energy of the elastic medium, the density of the latter must be taken equal to g^{2}, so that it must be the same in all substances.
It may further be asked what value we have to assign to the potential energy in the model, which must correspond to the electric energy in the electromagnetic field. Now, on account of (11) and (19), we can satisfy the equations (12) by putting D_{x} = gc (∂ζ/∂y − ∂η/∂z), &c., so that the electric energy (17) per unit of volume becomes
12 g^{2}c^{2} | 1 | ∂ζ | − | ∂η | 2 + | 1 | ∂ξ | − | ∂ζ | 2 + | 1 | ∂η | − | ∂ξ | 2 . | |||
ε_{1} | ∂y | ∂z | ε_{2} | ∂z | ∂x | ε_{3} | ∂x | ∂y |
This, therefore, must be the potential energy in the model.
It may be shown, indeed, that, if the aether has a uniform constant density, and is so constituted that in any system, whether homogeneous or not, its potential energy per unit of volume can be represented by an expression of the form
12 L | ∂ζ | − | ∂η | 2 + M | ∂ξ | − | ∂ζ | 2 + N | ∂η | − | ∂ξ | 2 , |
∂y | ∂z | ∂z | ∂x | ∂x | ∂y |
where L, M, N are coefficients depending on the physical properties
of the substance considered, the equations of motion will exactly
correspond to the equations of the electromagnetic field.
18. Theories of Neumann, Green, and MacCullagh.—A theory of light in which the elastic aether has a uniform density, and in which the vibrations are supposed to be parallel to the plane of polarization, was developed by Franz Ernst Neumann,^{[27]} who gave the first deduction of the formulas for crystalline reflection. Like Fresnel, he was, however, obliged to introduce some illegitimate assumptions and simplifications. Here again Green indicated a more rigorous treatment.
By specializing the formula for the potential energy of an anisotropic body he arrives at an expression which, if some of his coefficients are made to vanish and if the medium is supposed to be incompressible, differs from (20) only by the additional terms
2 L | ∂ζ | ∂η | − | ∂η | ∂ζ | + M | ∂ξ | ∂ζ | − | ∂ζ | ∂ξ | + N | ∂η | ∂ξ | − | ∂ξ | ∂η | . | ||||||
∂y | ∂z | ∂y | ∂z | ∂z | ∂x | ∂z | ∂x | ∂x | ∂y | ∂x | ∂y |
If ξ, η, ζ vanish at infinite distance the integral of this expression
over all space is zero, when L, M, N are constants, and the same
will be true when these coefficients change from point to point,
provided we add to (21) certain terms containing the differential
coefficients of L, M, N, the physical meaning of these terms being
that, besides the ordinary elastic forces, there is some extraneous
force (called into play by the displacement) acting on all those
elements of volume where L, M, N are not constant. We may
conclude from this that all phenomena can be explained if we admit
the existence of this latter force, which, in the case of two contingent
bodies, reduces to a surface-action on their common boundary.
James MacCullagh^{[28]} avoided this complication by simply assuming an expression of the form (20) for the potential energy. He thus established a theory that is perfectly consistent in itself, and may be said to have foreshadowed the electromagnetic theory as regards the form of the equations for transparent bodies. Lord Kelvin afterwards interpreted MacCullagh’s assumption by supposing the only action which is called forth by a displacement to consist in certain couples acting on the elements of volume and proportional to the components 12 {(∂ζ/∂y) − (∂η/∂z)}, &c., of their rotation from the natural position. He also showed^{[29]} that this “rotational elasticity” can be produced by certain hidden rotations going on in the medium.
We cannot dwell here upon other models that have been proposed, and most of which are of rather limited applicability. A mechanism of a more general kind ought, of course, to be adapted to what is known of the molecular constitution of bodies, and to the highly probable assumption of the perfect permeability for the aether of all ponderable matter, an assumption by which it has been possible to escape from one of the objections raised by Newton (§ 4) (see Aether).
The possibility of a truly satisfactory model certainly cannot be denied. But it would, in all probability, be extremely complicated. For this reason many physicists rest content, as regards the free aether, with some such general form of the electromagnetic theory as has been sketched in § 16.
19. Optical Properties of Ponderable Bodies. Theory of Electrons.—If we want to form an adequate representation of optical phenomena in ponderable bodies, the conceptions of the molecular and atomistic theories naturally suggest themselves. Already, in the elastic theory, it had been imagined that certain material particles are set vibrating by incident waves of light. These particles had been supposed to be acted on by an elastic force by which they are drawn back towards their positions of equilibrium, so that they can perform free vibrations of their own, and by a resistance that can be represented by terms proportional to the velocity in the equations of motion, and may be physically understood if the vibrations are supposed to be converted in one way or another into a disorderly heat-motion. In this way it had been found possible to explain the phenomena of dispersion and (selective) absorption, and the connexion between them (anomalous dispersion).^{[30]} These ideas have been also embodied into the electromagnetic theory. In its more recent development the extremely small, electrically charged particles, to which the name of “electrons” has been given, and which are supposed to exist in the interior of all bodies, are considered as forming the connecting links between aether and matter, and as determining by their arrangement and their motion all optical phenomena that are not confined to the free aether.^{[31]}
It has thus become clear why the relations that had been established between optical and electrical properties have been found to hold only in some simple cases (§ 16). In fact it cannot be doubted that, for rapidly alternating electric fields, the formulae expressing the connexion between the motion of electricity and the electric force take a form that is less simple than the one previously admitted, and is to be determined in each case by elaborate investigation. However, the general boundary conditions given in § 16 seem to require no alteration. For this reason it has been possible, for example, to establish a satisfactory theory of metallic reflection, though the propagation of light in the interior of a metal is only imperfectly understood.
One of the fundamental propositions of the theory of electrons is that an electron becomes a centre of radiation whenever its velocity changes either in direction or in magnitude. Thus the production of Röntgen rays, regarded as consisting of very short and irregular electromagnetic impulses, is traced to the impacts of the electrons of the cathode-rays against the anti-cathode, and the lines of an emission spectrum indicate the existence in the radiating body of as many kinds of regular vibrations, the knowledge of which is the ultimate object of our investigations about the structure of the spectra. The shifting of the lines caused, according to Doppler’s law, by a motion of the source of light, may easily be accounted for, as only general principles are involved in the explanation. To a certain extent we can also elucidate the changes in the emission that are observed when the radiating source is exposed to external magnetic forces (“Zeeman-effect”; see Magneto-Optics).
20. Various Kinds of Light-motion.—(a) If the disturbance is represented by
so that the end of the vector P describes an ellipse in a plane perpendicular to the direction of propagation, the light is said to be elliptically, or in special cases circularly, polarized. Light of this kind can be dissolved in many different ways into plane polarized components.
There are cases in which plane waves must be elliptically or circularly polarized in order to show the simple propagation of phase that is expressed by formulae like (5). Instances of this kind occur in bodies having the property of rotating the plane of polarization, either on account of their constitution, or under the influence of a magnetic field. For a given direction of the wave-front there are in general two kinds of elliptic vibrations, each having a definite form, orientation, and direction of motion, and a determinate velocity of propagation. All that has been said about Huygens’s construction applies to these cases.
(b) In a perfect spectroscope a sharp line would only be observed if an endless regular succession of simple harmonic vibrations were admitted into the instrument. In any other case the light will occupy a certain extent in the spectrum, and in order to determine its distribution we have to decompose into simple harmonic functions of the time the components of the disturbance, at a point of the slit for instance. This may be done by means of Fourier’s theorem.
An extreme case is that of the unpolarized light emitted by incandescent solid bodies, consisting of disturbances whose variations are highly irregular, and giving a continuous spectrum. But even with what is commonly called homogeneous light, no perfectly sharp line will be seen. There is no source of light in which the vibrations of the particles remain for ever undisturbed, and a particle will never emit an endless succession of uninterrupted vibrations, but at best a series of vibrations whose form, phase and intensity are changed at irregular intervals. The result must be a broadening of the spectral line.
In cases of this kind one must distinguish between the velocity of propagation of the phase of regular vibrations and the velocity with which the said changes travel onward (see below, iii. Velocity of Light).
(c) In a train of plane waves of definite frequency the disturbance is represented by means of goniometric functions of the time and the coordinates. Since the fundamental equations are linear, there are also solutions in which one or more of the coordinates occur in an exponential function. These solutions are of interest because the motions corresponding to them are widely different from those of which we have thus far spoken. If, for example, the formulae contain the factor
with the positive constant r, the disturbance is no longer periodic with respect to x, but steadily diminishes as x increases. A state of things of this kind, in which the vibrations rapidly die away as we leave the surface, exists in the air adjacent to the face of a glass prism by which a beam of light is totally reflected. It furnishes us an explanation of Newton’s experiment mentioned in § 2. (H. A. L.)
III. Velocity of Light
The fact that light is propagated with a definite speed was first brought out by Ole Roemer at Paris, in 1676, through observations of the eclipses of Jupiter’s satellites, made in different relative positions of the Earth and Jupiter in their respective orbits. It is possible in this way to determine the time required for light to pass across the orbit of the earth. The dimensions of this orbit, or the distance of the sun, being taken as known, the actual speed of light could be computed. Since this computation requires a knowledge of the sun’s distance, which has not yet been acquired with certainty, the actual speed is now determined by experiments made on the earth’s surface. Were it possible by any system of signals to compare with absolute precision the times at two different stations, the speed could be determined by finding how long was required for light to pass from one station to another at the greatest visible distance. But this is impracticable, because no natural agent is under our control by which a signal could be communicated with a greater velocity than that of light. It is therefore necessary to reflect a ray back to the point of observation and to determine the time which the light requires to go and come. Two systems have been devised for this purpose. One is that of Fizeau, in which the vital appliance is a rapidly revolving toothed wheel; the other is that of Foucault, in which the corresponding appliance is a mirror revolving on an axis in, or parallel to, its own plane.
Fig. 1. |
The principle underlying Fizeau’s method is shown in the accompanying figs. 1 and 2. Fig. 1 shows the course of a ray of light which, emanating from a luminous point L, strikes the plane surface of a plate of glass M at an angle of about 45°. A fraction of the light is reflected from the two surfaces of Fizeau. the glass to a distant reflector R, the plane of which is at right angles to the course of the ray. The latter is thus reflected back on its own course and, passing through the glass M on its return, reaches a point E behind the glass. An observer with his eye at E looking through the glass sees the return ray as a distant luminous point in the reflector R, after the light has passed over the course in both directions.
In actual practice it is necessary to interpose the object glass of a telescope at a point O, at a distance from M nearly equal to its focal length. The function of this appliance is to render the diverging rays, shown by the dotted lines, nearly parallel, in order that more light may reach R and be thrown back again. But the principle may be conceived without respect to the telescope, all the rays being ignored except the central one, which passes over the course we have described.
Fig. 2. |
Conceiving the apparatus arranged in such a way that the observer sees the light reflected from the distant mirror R, a fine toothed wheel WX is placed immediately in front of the glass M, with its plane perpendicular to the course of the ray, in such a way that the ray goes out and returns through an opening between two adjacent teeth. This wheel is represented in section by WX in fig. 1, and a part of its circumference, with the teeth as viewed by the observer, is shown in fig. 2. We conceive that the latter sees the luminous point between two of the teeth at K. Now, conceive that the wheel is set in revolution. The ray is then interrupted as every tooth passes, so that what is sent out is a succession of flashes. Conceive that the speed of the mirror is such that while the flash is going to the distant mirror and returning again, each tooth of the wheel takes the place of an opening between the teeth. Then each flash sent out will, on its return, be intercepted by the adjacent tooth, and will therefore become invisible. If the speed be now doubled, so that the teeth pass at intervals equal to the time required for the light to go and come, each flash sent through an opening will return through the adjacent opening, and will therefore be seen with full brightness. If the speed be continuously increased the result will be successive disappearances and reappearances of the light, according as a tooth is or is not interposed when the ray reaches the apparatus on its return. The computation of the time of passage and return is then very simple. The speed of the wheel being known, the number of teeth passing in one second can be computed. The order of the disappearance, or the number of teeth which have passed while the light is going and coming, being also determined in each case, the interval of time is computed by a simple formula.
The most elaborate determination yet made by Fizeau’s method was that of Cornu. The station of observation was at the Paris Observatory. The distant reflector, a telescope with a reflector at its focus, was at Montlhéry, distant 22,910 metres from the toothed wheel. Of the wheels most used one had Cornu. 150 teeth, and was 35 millimetres in diameter; the other had 200 teeth, with a diameter of 45 mm. The highest speed attained was about 900 revolutions per second. At this speed, 135,000 (or 180,000) teeth would pass per second, and about 20 (or 28) would pass while the light was going and coming. But the actual speed attained was generally less than this. The definitive result derived by Cornu from the entire series of experiments was 300,400 kilometres per second. Further details of this work need not be set forth because the method is in several ways deficient in precision. The eclipses and subsequent reappearances of the light taking place gradually, it is impossible to fix with entire precision upon the moment of complete eclipse. The speed of the wheel is continually varying, and it is impossible to determine with precision what it was at the instant of an eclipse.
The defect would be lessened were the speed of the toothed wheel placed under control of the observer who, by action in one direction or the other, could continually check or accelerate it, so as to keep the return point of light at the required phase of brightness. If the phase of complete extinction is chosen for this purpose a definite result cannot be reached; but by choosing the moment when the light is of a certain definite brightness, before or after an eclipse, the observer will know at each instant whether the speed should be accelerated or retarded, and can act accordingly. The nearly constant speed through as long a period as is deemed necessary would then be found by dividing the entire number of revolutions of the wheel by the time through which the light was kept constant. But even with these improvements, which were not actually tried by Cornu, the estimate of the brightness on which the whole result depends would necessarily be uncertain. The outcome is that, although Cornu’s discussion of his experiments is a model in the care taken to determine so far as practicable every source of error, his definitive result is shown by other determinations to have been too great by about 11000 part of its whole amount.
An important improvement on the Fizeau method was made in 1880 by James Young and George Forbes at Glasgow. This consisted in using two distant reflectors which were placed nearly in the same straight line, and at unequal distances. The ratio of the distances was nearly 12 : 13. The phase Young and Forbes. observed was not that of complete extinction of either light, but that when the two lights appeared equal in intensity. But it does not appear that the very necessary device of placing the speed of the toothed wheel under control of the observer was adopted. The accordance between the different measures was far from satisfactory, and it will suffice to mention the result which was
These experimenters also found a difference of 2% between the speed of red and blue light, a result which can only be attributed to some unexplained source of error.
The Foucault system is much more precise, because it rests upon the measurement of an angle, which can be made with great precision.
Fig. 3. |
The vital appliance is a rapidly revolving mirror. Let AB (fig. 3) be a section of this mirror, which we shall first suppose at rest. A ray of light LM emanating from a source at L, is reflected in the direction MQR to a distant mirror R, from which it is perpendicularly reflected back upon its original course. Foucault. This mirror R should be slightly concave, with the centre of curvature near M, so that the ray shall always be reflected back to M on whatever point of R it may fall. Conceiving the revolving mirror M as at rest, the return ray will after three reflections, at M, R and M again, be returned along its original course to the point L from which it emanated. An important point is that the return ray will always follow the fixed line ML no matter what the position of the movable mirror M, provided there is a distant reflector to send the ray back. Now, suppose that, while the ray is going and coming, the mirror M, being set in revolution, has turned from the position in which the ray was reflected to that shown by the dotted line. If α be the angle through which the surface has turned, the course of the return ray, after reflection, will then deviate from ML by the angle 2α, and so be thrown to a point E, such that the angle LME = 2α. If the mirror is in rapid rotation the ray reflected from it will strike the distant mirror as a series of flashes, each formed by the light reflected when the mirror was in the position AB. If the speed of rotation is uniform, the reflected rays from the successive flashes while the mirror is in the dotted position will thus all follow the same direction ME after their second reflection from the mirror. If the motion is sufficiently rapid an eye observing the reflected ray will see the flashes as an invariable point of light so long as the speed of revolution remains constant. The time required for the light to go and come is then equal to that required by the mirror to turn through half the angle LME, which is therefore to be measured. In practice it is necessary on this system, as well as on that of Fizeau, to condense the light by means of a lens, Q, so placed that L and R shall be at conjugate foci. The position of the lens may be either between the luminous point L and the mirror M, or between M and R, the latter being the only one shown in the figure. This position has the advantage that more light can be concentrated, but it has the disadvantage that, with a given magnifying power, the effect of atmospheric undulation, when the concave reflector is situated at a great distance, is increased in the ratio of the focal length of the lens to the distance LM from the light to the mirror. To state the fact in another form, the amplitude of the disturbances produced by the air in linear measure are proportional to the focal distance of the lens, while the magnification required increases in the inverse ratio of the distance LM. Another difficulty associated with the Foucault system in the form in which its originator used it is that if the axis of the mirror is at right angles to the course of the ray, the light from the source L will be flashed directly into the eye of the observer, on every passage of the revolving mirror through the position in which its normal bisects the two courses of the ray. This may be avoided by inclining the axis of the mirror.
In Foucault’s determination the measures were not made upon a luminous point, but upon a reticule, the image of which could not be seen unless the reflector was quite near the revolving mirror. Indeed the whole apparatus was contained in his laboratory. The effective distance was increased by using several reflectors; but the entire course of the ray measured only 20 metres. The result reached by Foucault for the velocity of light was 298,000 kilometres per second.
The first marked advance on Foucault’s determination was made by Albert A. Michelson, then a young officer on duty at the U.S. Naval Academy, Annapolis. The improvement consisted in using the image of a slit through which the rays of the sun passed after reflection from a heliostat. In this way Michelson. it was found possible to see the image of the slit reflected from the distant mirror when the latter was nearly 600 metres from the station of observation. The essentials of the arrangement are those we have used in fig. 3, L being the slit. It will be seen that the revolving mirror is here interposed between the lens and its focus. It was driven by an air turbine, the blast of which was under the control of the observer, so that it could be kept at any required speed. The speed was determined by the vibrations of two tuning forks. One of these was an electric fork, making about 120 vibrations per second, with which the mirror was kept in unison by a system of rays reflected from it and the fork. The speed of this fork was determined by comparison with a freely vibrating fork from time to time. The speed of the revolving mirror was generally about 275 turns per second, and the deflection of the image of the slit about 112.5 mm. The mean result of nearly 100 fairly accordant determinations was:—
Velocity of light in air | 299,828 km. per sec. |
Reduction to a vacuum | +82 |
Velocity of light in a vacuum | 299,910 ± 50 |
While this work was in progress Simon Newcomb obtained the official support necessary to make a determination on a yet larger scale. The most important modifications made in the Foucault-Michelson system were the following:—Newcomb.
1. Placing the reflector at the much greater distance of several kilometres.
2. In order that the disturbances of the return image due to the passage of the ray through more than 7 km. of air might be reduced to a minimum, an ordinary telescope of the “broken back” form was used to send the ray to the revolving mirror.
3. The speed of the mirror was, as in Michelson’s experiments, completely under control of the observer, so that by drawing one or the other of two cords held in the hand the return image could be kept in any required position. In making each measure the receiving telescope hereafter described was placed in a fixed position and during the “run” the image was kept as nearly as practicable upon a vertical thread passing through its focus. A “run” generally lasted about two minutes, during which time the mirror commonly made between 25,000 and 30,000 revolutions. The speed per second was found by dividing the entire number of revolutions by the number of seconds in the “run.” The extreme deviations between the times of transmission of the light, as derived from any two runs, never approached to the thousandth part of its entire amount. The average deviation from the mean was indeed less than 15000 part of the whole.
Fig. 4. |
To avoid the injurious effect of the directly reflected flash, as well as to render unnecessary a comparison between the directions of the outgoing and the return ray, a second telescope, turning horizontally on an axis coincident with that of the revolving mirror, was used to receive the return ray after reflection. This required the use of an elongated mirror of which the upper half of the surface reflected the outgoing ray, and the lower other half received and reflected the ray on its return. On this system it was not necessary to incline the mirror in order to avoid the direct reflection of the return ray. The greatest advantage of this system was that the revolving mirror could be turned in either direction without break of continuity, so that the angular measures were made between the directions of the return ray after reflection when the mirror moved in opposite directions. In this way the speed of the mirror was as good as doubled, and the possible constant errors inherent in the reference to a fixed direction for the sending telescope were eliminated. The essentials of the apparatus are shown in fig. 4. The revolving mirror was a rectangular prism M of steel, 3 in. high and 112 in. on a side in cross section, which was driven by a blast of air acting on two fan-wheels, not shown in the fig., one at the top, the other at the bottom of the mirror. NPO is the object-end of the fixed sending telescope the rays passing through it being reflected to the mirror by a prism P. The receiving telescope ABO is straight, and has its objective under O. It was attached to a frame which could turn around the same axis as the mirror. The angle through which it moved was measured by a divided arc immediately below its eye-piece, which is not shown in the figure. The position AB is that for receiving the ray during a rotation of the mirror in the anti-clockwise direction; the position A′B′ that for a clockwise rotation.
In these measures the observing station was at Fort Myer, on a hill above the west bank of the Potomac river. The distant reflector was first placed in the grounds of the Naval Observatory, at a distance of 2551 metres. But the definitive measures were made with the reflector at the base of the Washington monument, 3721 metres distant. The revolving mirror was of nickel-plated steel, polished on all four vertical sides. Thus four reflections of the ray were received during each turn of the mirror, which would be coincident were the form of the mirror invariable. During the preliminary series of measures it was found that two images of the return ray were sometimes formed, which would result in two different conclusions as to the velocity of light, according as one or the other was observed. The only explanation of this defect which presented itself was a tortional vibration of the revolving mirror, coinciding in period with that of revolution, but it was first thought that the effect was only occasional.
In the summer of 1881 the distant reflector was removed from the Observatory to the Monument station. Six measures made in August and September showed a systematic deviation of +67 km. per second from the result of the Observatory series. This difference led to measures for eliminating the defect from which it was supposed to arise. The pivots of the mirror were reground, and a change made in the arrangement, which would permit of the effect of the vibration being determined and eliminated. This consisted in making the relative position of the sending and receiving telescopes interchangeable. In this way, if the measured deflection was too great in one position of the telescopes, it would be too small by an equal amount in the reverse position. As a matter of fact, when the definitive measures were made, it was found that with the improved pivots the mean result was the same in the two positions. But the new result differed systematically from both the former ones. Thirteen measures were made from the Monument in the summer of 1882, the results of which will first be stated in the form of the time required by the ray to go and come. Expressed in millionths of a second this was:—
Least result of the 13 measures | 24.819 |
Greatest result | 24.831 |
Double distance between mirrors | 7.44242 km. |
Applying a correction of +12 km. for a slight convexity in the face of the revolving mirror, this gives as the mean result for the speed of light in air, 299,778 km. per second. The mean results for the three series were:—
Observatory, 1880–1881 | V in air = 299,627 |
Monument, 1881 | V in air = 299,694 |
Monument, 1882 | V in air = 299,778 |
The last result being the only one from which the effect of distortion was completely eliminated, has been adopted as definitive. For reduction to a vacuum it requires a correction of +82 km. Thus the final result was concluded to be
This result being less by 50 km. than that of Michelson, the latter made another determination with improved apparatus and arrangements at the Case School of Applied Science in Cleveland. The result was
So far as could be determined from the discordance of the separate measures, the mean error of Newcomb’s result would be less than ±10 km. But making allowance for the various sources of systematic error the actual probable error was estimated at ±30 km.
It seems remarkable that since these determinations were made, a period during which great improvements have become possible in every part of the apparatus, no complete redetermination of this fundamental physical constant has been carried out.
The experimental measures thus far cited have been primarily those of the velocity of light in air, the reduction to a vacuum being derived from theory alone. The fundamental constant at the basis of the whole theory is the speed of light in a vacuum, such as the celestial spaces. The question of the relation between the velocity in vacuo, and in a transparent medium of any sort, belongs to the domain of physical optics. Referring to the preceding section for the principles at play we shall in the present part of the article confine ourselves to the experimental results. With the theory of the effect of a transparent medium is associated that of the possible differences in the speed of light of different colours.
The question whether the speed of light in vacuo varies with its wave-length seems to be settled with entire certainty by observations of variable stars. These are situated at different distances, some being so far that light must be several centuries in reaching us from them. Were Velocity and wave-length. there any difference in the speed of light of various colours it would be shown by a change in the colour of the star as its light waxed and waned. The light of greatest speed preceding that of lesser speed would, when emanated during the rising phase, impress its own colour on that which it overtook. The slower light would predominate during the falling phase. If there were a difference of 10 minutes in the time at which light from the two ends of the visible spectrum arrived, it would be shown by this test. As not the slightest effect of the kind has ever been seen, it seems certain that the difference, if any, cannot approximate to 11⋅000⋅000 part of the entire speed. The case is different when light passes through a refracting medium. It is a theoretical result of the undulatory theory of light that its velocity in such a medium is inversely proportional to the refractive index of the medium. This being different for different colours, we must expect a corresponding difference in the velocity.
Foucault and Michelson have tested these results of the undulatory theory by comparing the time required for a ray of light to pass through a tube filled with a refracting medium, and through air. Foucault thus found, in a general way, that there actually was a retardation; but his observations took account only of the mean retardation of light of all the wave-lengths, which he found to correspond with the undulatory theory. Michelson went further by determining the retardation of light of various wave-lengths in carbon bisulphide. He made two series of experiments, one with light near the brightest part of the spectrum; the other with red and blue light. Putting V for the speed in a vacuum and V_{1} for that in the medium, his result was
Yellow light | V : V_{1} = 1.758 |
Refractive index for yellow | 1.64 |
Difference from theory | +0.12 |
The estimated uncertainty was only 0.02, or 16 of the difference between observation and theory.
The comparison of red and blue light was made differentially. The colours selected were of wave-length about 0.62 for red and 0.49 for blue. Putting V_{r} and V_{b} for the speeds of red and blue light respectively in bisulphide of carbon, the mean result compares with theory as follows:—
Observed value of the ratio V_{r}, V_{b} | 1.0245 |
Theoretical value (Verdet) | 1.025 |
This agreement may be regarded as perfect. It shows that the divergence of the speed of yellow light in the medium from theory, as found above, holds through the entire spectrum.
The excess of the retardation above that resulting from theory is probably due to a difference between “wave-speed” and “group-speed” pointed out by Rayleigh. Let fig. 5 represent a short series of progressive undulations of constant period and wave-length. The wave-speed is that required to carry a wave crest A to the position of the crest B in the wave time. But when a flash of light like that measured passes through a refracting medium, the front waves of the flash are continually dying away, as shown at the end of the figure, and the place of each is taken by the wave following. A familiar case of this sort is seen when a stone is thrown into a pond. The front waves die out one at a time, to be followed by others, each of which goes further than its predecessor, while new waves are formed in the rear. Hence the group, as represented in the figure by the larger waves in the middle, moves as a whole more slowly than do the individual waves. When the speed of light is measured the result is not the wave-speed as above defined, but something less, because the result depends on the time of the group passing through the medium. This lower speed is called the group-velocity of light. In a vacuum there is no dying out of the waves, so that the group-speed and the wave-speed are identical. From Michelson’s experiments it would follow that the retardation was about 114 of the whole speed. This would indicate that in carbon bisulphide each individual light wave forming the front of a moving ray dies out in a space of about 15 wave-lengths.
Authorities.—For Foucault’s descriptions of his experiments see Comptes Rendus (September 22 and November 24, 1862), and Recueil de Travaux Scientifiques de Léon Foucault (2 vols., 4to, Paris, 1878). Cornu’s determination is found in Annales de l’Observatoire de Paris, Mémoires, vol. xiii. The works of Michelson and Newcomb are published in extenso in the Astronomical Papers of the American Ephemeris, vols. i. and ii. (S. N.)
- ↑ The invention of “aethers” is to be carried back, at least, to the Greek philosophers, and with the growth of knowledge they were empirically postulated to explain many diverse phenomena. Only one “aether” has survived in modern science—that associated with light and electricity, and of which Lord Salisbury, in his presidential address to the British Association in 1894, said, “For more than two generations the main, if not the only, function of the word ‘aether’ has been to furnish a nominative case to the verb ‘to undulate.’ ” (See Aether.)
- ↑ With the Greeks the word “Optics” or Ὀπτικά (from ὄπτομαι, the obsolete present of ὁρῶ, I see) was restricted to questions concerning vision, &c., and the nature of light.
- ↑ It seems probable that spectacles were in use towards the end of the 13th century. The Italian dictionary of the Accademici della Crusca (1612) mentions a sermon of Jordan de Rivalto, published in 1305, which refers to the invention as “not twenty years since”; and Muschenbroek states that the tomb of Salvinus Armatus, a Florentine nobleman who died in 1317, bears an inscription assigning the invention to him. (See the articles Telescope and Camera Obscura for the history of these instruments.)
- ↑ Newton’s observation that a second refraction did not change the colours had been anticipated in 1648 by Marci de Kronland (1595–1667), professor of medicine at the university of Prague, in his Thaumantias, who studied the spectrum under the name of Iris trigonia. There is no evidence that Newton knew of this, although he mentions de Dominic’s experiment with the glass globe containing water.
- ↑ The geometrical determination of the form of the surface which will reflect, or of the surface dividing two media which will refract, rays from one point to another, is very easily effected by using the “characteristic function” of Hamilton, which for the problems under consideration may be stated in the form that “the optical paths of all rays must be the same.” In the case of reflection, if A and B be the diverging and converging points, and P a point on the reflecting surface, then the locus of P is such that AP + PB is constant. Therefore the surface is an ellipsoid of revolution having A and B as foci. If the rays be parallel, i.e. if A be at infinity, the surface is a paraboloid of revolution having B as focus and the axis parallel to the direction of the rays. In refraction if A be in the medium of index µ, and B in the medium of index µ′, the characteristic function shows that µAP + µ′PB, where P is a point on the surface, must be constant. Plane sections through A and B of such surfaces were originally investigated by Descartes, and are named Cartesian ovals. If the rays be parallel, i.e. A be at infinity, the surface becomes an ellipsoid of revolution having B for one focus, µ′/µ for eccentricity, and the axis parallel to the direction of the rays.
- ↑ Young’s views of the nature of light, which he formulated as Propositions and Hypotheses, are given in extenso in the article Interference. See also his article “Chromatics” in the supplementary volumes to the 3rd edition of the Encyclopaedia Britannica.
- ↑ A crucial test of the emission and undulatory theories, which was realized by Descartes, Newton, Fermat and others, consisted in determining the velocity of light in two differently refracting media. This experiment was conducted in 1850 by Foucault, who showed that the velocity was less in water than in air, thereby confirming the undulatory and invalidating the emission theory.
- ↑ Newton, Opticks (London, 1704).
- ↑ Trans. Irish Acad. 15, p. 69 (1824); 16, part i. “Science,” p. 4 (1830), part ii., ibid. p. 93 (1830); 17, part i., p. 1 (1832).
- ↑ This kind of type will always be used in this article to denote vectors.
- ↑ Phil. Trans. (1802), part i. p. 12.
- ↑ Œuvres complètes de Fresnel (Paris, 1866). (The researches were published between 1815 and 1827.)
- ↑ Ann. Phys. Chem. (1883), 18, p. 663.
- ↑ H. A. Lorentz, Zittingsversl. Akad. v. Wet. Amsterdam, 4 (1896), p. 176.
- ↑ H. A. Lorentz, Abhandlungen über theoretische Physik, 1 (1907), p. 415.
- ↑ Clerk Maxwell, A Treatise on Electricity and Magnetism (Oxford, 1st ed., 1873).
- ↑ H. Abraham, Rapports présentés au congrès de physique de 1900 (Paris), 2, p. 247.
- ↑ Ibid., p. 225.
- ↑ Phil. Trans., 175 (1884), p. 343.
- ↑ Ann. d. Phys. u. Chem. 155 (1875), p. 403.
- ↑ Ibid. 153 (1874), p. 525.
- ↑ Ann. d. Phys. 11 (1903), p. 873.
- ↑ Phys. Review, 13 (1901), p. 293.
- ↑ Hertz, Untersuchungen über die Ausbreitung der elektrischen Kraft (Leipzig, 1892).
- ↑ A. Righi, L’Ottica delle oscillazioni elettriche (Bologna, 1897); P. Lebedew, Ann. d. Phys. u. Chem., 56 (1895), p. 1.
- ↑ “Reflection and Refraction,” Trans. Cambr. Phil. Soc. 7, p. 1 (1837); “Double Refraction,” ibid. p. 121 (1839).
- ↑ “Double Refraction,” Ann. d. Phys. u. Chem. 25 (1832), p. 418; “Crystalline Reflection,” Abhandl. Akad. Berlin (1835), p. 1.
- ↑ Trans. Irish Acad. 21, “Science,” p. 17 (1839).
- ↑ Math. and Phys. Papers (London, 1890), 3, p. 466.
- ↑ Helmholtz, Ann. d. Phys. u. Chem., 154 (1875), p. 582.
- ↑ H. A. Lorentz, Versuch einer Theorie der elektrischen u. optischen Erscheinungen in bewegten Körpern (1895) (Leipzig, 1906); J. Larmor, Aether and Matter (Cambridge, 1900).