The Principles of Physical Optics is a comprehensive historical and scientific treatise by Ernst Mach, the Austrian physicist and philosopher of science. The work represents the culmination of Mach’s decades-long investigation into the nature of light, continuing the methodological approach he had previously applied to mechanics and heat. The book traces the development of optical knowledge from ancient Greek speculation through the nineteenth-century discoveries that transformed the field, while simultaneously presenting Mach’s own philosophical perspective on how scientific concepts originate and evolve.
The volume emerged from lectures Mach delivered beginning during his tenure as a Privat Dozent at Prague, where he first developed his distinctive approach to geometrical optics. The text bears the marks of its long gestation, combining careful historical scholarship with original experimental observations and theoretical analysis. Mach explicitly states that the work continues the task begun in his earlier volumes on mechanics and heat, and that its method most closely resembles the latter work. He expresses the hope that he has revealed the origins of general optical concepts and the historical threads of their development, extricated from what he terms “metaphysical ballast.”
The book’s translation into English was undertaken by John S. Anderson and Ames V. Young, who preserved the plan of the German edition, including its frequent italics and original illustration lettering. The translators corrected several errors, principally algebraic in nature, and added one figure for clarity. Their translation, dated November 1925, was intended to serve both those interested in Mach’s philosophical aspects and physics students seeking a historically grounded introduction to optics.
Mach’s dedication addresses Dr. Paul Carus and Messrs. Hegeler, recalling a correspondence that began when early issues of the journal Open Court failed to reach him. He credits Carus’s firm with educational work and masterly translations that enabled him to reach a wide audience. The dedication is dated Munich, July 1913, and reflects the personal and intellectual debts Mach accumulated over his long career.
The circumstances of the book’s composition are poignantly documented in its preface. Mach acknowledges that due to age and illness, he yielded to publisher pressure to print this portion of his projected work, with radiation, the decline of emission theory, Maxwell’s theory, and relativity deferred to a subsequent part that never appeared. His son, a long-time colleague, collaborated on the researches underlying these chapters. Mach also takes the opportunity to disclaim being a forerunner of relativity theory, despite publications suggesting this, calling himself “merely an unprejudiced rambler” who withholds from atomistic belief as from relativism. He questions whether relativity will prove more than “a transitory inspiration,” a remark that reveals both his characteristic skepticism toward grand theoretical constructions and his commitment to empirical, physiological, and psychological approaches to scientific questions.
Context
Mach’s Principles of Physical Optics occupies a distinctive position in the history of science, written at a moment when classical optics was being transformed by developments that would eventually lead to relativity and quantum theory. The book represents the culmination of a nineteenth-century tradition of historical and philosophical reflection on physics, a tradition in which Mach was a central figure alongside figures like Pierre Duhem and William Whewell.
The intellectual context of the work is shaped by several converging currents. First, there was the long-standing debate between emission and wave theories of light, which had been effectively settled in favor of the wave theory by Fresnel’s work in the early nineteenth century but which Mach treats with characteristic historical detachment, showing how both approaches contributed to the development of optical knowledge. Second, there was the emergence of physiological optics as a distinct discipline, concerned with the dependence of visual phenomena on the conditions of the individual observer—a field to which Mach himself made significant contributions. Third, there was the ongoing philosophical debate about the nature of scientific theories and their relationship to experience, a debate in which Mach’s own empiricist and anti-metaphysical position was highly influential.
Mach’s approach to optics was shaped by his broader philosophical commitments, particularly his rejection of absolute space and time, his skepticism toward atomism, and his insistence that scientific concepts must be understood in terms of their origins in experience. These commitments had already been expressed in his earlier works on mechanics and heat, and they find their optical application in the present volume. The book is thus not merely a history of optics but an extended argument about how scientific knowledge develops and what it can legitimately claim to know.
The period during which Mach wrote was one of profound transition in physics. The Michelson-Morley experiment had failed to detect the luminiferous ether, Lorentz had developed his electron theory, and Einstein had published his special theory of relativity in 1905. Mach’s attitude toward these developments was characteristically cautious and critical. He explicitly disclaimed being a forerunner of relativity, despite the efforts of some philosophers and physicists to claim him as such. His objections, he states, rest on the physiology of the senses, theoretical ideas, and experimental conceptions—a triple foundation that reflects his conviction that scientific theories must be grounded in the full range of human experience.
The book also reflects the institutional and personal networks of European science in the late nineteenth and early twentieth centuries. Mach acknowledges the assistance of numerous colleagues and collaborators: the late Professor E. Reusch of Tübingen, Karl Fritsch-Prokesch (a Viennese optician), and Dr. Rudolf Steinheil of Munich. He thanks Professor J. Plassmann of Münster and J. G. Hagen of the Specola Vaticana for discovering F. Grimaldi’s picture in the possession of Professor M. Rajna at Bologna University Observatory, and he acknowledges the efforts of those who photographed it and constructed a block of Grimaldi’s signature. These acknowledgments reveal the international and collaborative character of optical research during this period, as well as Mach’s own position within these networks.
Structure of the Book
The book is organized as a series of chapters that trace the historical and conceptual development of optics from antiquity to the nineteenth century. The structure reflects Mach’s stated intention to prioritize salient facts over strict chronological sequence, allowing thematic and conceptual threads to guide the presentation rather than a simple linear narrative.
The opening chapter serves as an introduction to the fundamental phenomena of optics and to Mach’s methodological approach. Mach begins with the basic facts of sensation—brightness, darkness, light, and color cannot be described but only named—and proceeds to distinguish physical optical data, which are observable by all people with normal sight, from physiological optical data, which concern dependence on individual organ conditions. He then summarizes the six fundamental properties of light that the book will examine: rectilinear propagation with definite velocity; reflection and refraction at boundaries; refractive index differing per color; diffraction at edges and slits; periodicity in space and time; and periodic properties representable only by distances perpendicular to the ray. This introductory chapter also credits the major figures in the history of optics—Euclid, Hero, Descartes, Grimaldi, Newton, Young, Fresnel, Huygens, and Malus—with their respective discoveries, providing a map of the territory the book will explore.
The second chapter examines the rectilinear propagation of light, beginning with ancient theories of visual rays. Mach discusses Euclid’s assumption that rays emanate from the eye forming a cone, an approach he notes resembles Faraday’s lines of force in its geometrical character. He examines the optical hypotheses of Damianus, son of Heliodor of Larissa, whose fourteen propositions systematized the emanation theory of vision. The chapter then traces the evidence for rectilinear propagation through the work of Maurolycus, Kepler, and others, before turning to the determination of light’s velocity. Galileo’s proposal for measuring the speed of light using screened lanterns is described, followed by the astronomical methods of Descartes, Huygens, Rømer, and Bradley. The chapter concludes with the establishment of photometry by Bouguer and Lambert, who developed quantitative methods for measuring light intensity.
The third chapter treats reflection and refraction, beginning with Hero’s principle that light reflecting at a plane mirror takes the shortest path between its initial and final points. Fermat’s extension of this principle to refraction—light chooses the path involving the shortest time—is presented as a landmark in the mathematical treatment of optics. The chapter compares the formulations of Snell and Descartes, showing their equivalence while noting the greater convenience of Snell’s construction for geometrical purposes. The historical development from Euclid’s qualitative observations through Descartes’ quantitative formulation to the theoretical interpretations of Fermat and Huygens is traced, culminating in Foucault’s experiments.
The fifth chapter develops the science of dioptrics, beginning with the historical preconditions for the discovery of the telescope. Mach examines the existence of lenses in classical times, the development of spectacles in the thirteenth century, and the gradual accumulation of optical knowledge that made the telescope possible. The chapter then presents the mathematical theory of lenses, including the construction methods of R. Smith, Kepler’s approach to refraction, and the projective relations discovered by Möbius. The treatment extends to systems of lenses, conjugate elements, and principal points, drawing on the work of Lagrange, Gauss, and others.
The sixth chapter concerns the composition of light, presenting Newton’s experiments on reflection, refraction, and spectral separation. Mach describes Newton’s demonstrations that colors pre-exist in white light, his method for producing pure spectra, and his investigations of homogeneous light. Newton’s color-mixing rule, based on the center of gravity of spectral colors arranged around a circle, is presented in detail. The chapter also examines the physiological theories of color vision developed by Hering and others, as well as the objections raised by Goethe and Schopenhauer to Newton’s theory.
The seventh chapter continues the development of the theory of color and dispersion, examining the discovery of fixed spectral lines by Wollaston and Fraunhofer, the development of achromatic combinations, and the phenomenon of anomalous dispersion discovered by Christiansen and Kundt. Mach’s own work with G. v. Osnobischin on total reflection methods is also presented.
The eighth chapter introduces the concept of periodicity, drawing on Grimaldi’s diffraction experiments and his prescient observation that light seems to spread in waves. Hooke’s observations of colors in thin plates and Newton’s experiments on the same phenomena are examined in detail. Newton’s theory of periodic “fits” of reflection and transmission is presented, along with his measurements of the rings produced by pressing lenses together.
The ninth chapter traces the further development of the theory of interference, focusing on Young’s contributions and his demonstration that light propagates more slowly in media where it is refracted toward the normal. The chapter also discusses Ritter and Wollaston’s discovery of non-luminous chemically active rays and the emergence of Fresnel, who independently reached views similar to Young’s while working in isolation as a young road-construction engineer.
The tenth and subsequent chapters treat polarization, beginning with Huygens’ observations of calcite and continuing through Malus’s discoveries. The mathematical representation of the properties of light is developed, and the further explanation of light’s behavior through wave theory is explored. The book concludes with discussions of rectilinear rays, reflection, and the further explanation of optical phenomena.
Main Arguments
The central argument of Mach’s Principles of Physical Optics concerns the nature and development of scientific theories. Throughout the book, Mach presents optics as a discipline in which theoretical concepts emerge from and are continually modified by empirical experience. He argues that optical theories are not mere summaries of observations but comprehensive descriptions that enable prediction: if a phenomenon responds to a certain reaction, it will also respond to another. Progress in science consists of modifying and restricting propositions as new experiences limit earlier generalizations.
Mach’s treatment of the history of optics is shaped by his conviction that scientific ideas are not discovered fully formed but develop through a complex process of accumulation, correction, and conceptual transformation. He shows how the rectilinear propagation of light, first known to Euclid around 300 B.C., was gradually incorporated into more comprehensive theoretical frameworks. Huygens demonstrated that rectilinear propagation, reflection, and refraction could all be derived from the assumption of finite propagation velocity; Fresnel completed this project by incorporating periodicity and polarization, showing Huygens’ laws to be special cases of diffraction. This narrative exemplifies Mach’s view that scientific progress consists not in the replacement of one theory by another but in the progressive modification and restriction of propositions as new experiences are accumulated.
A second major argument concerns the distinction between physical and physiological optics. Mach insists that confusing these two domains led Goethe and Schopenhauer into one-sided ideas. Physical optical data are occurrences observable by all people with normal sight, while physiological optical data concern dependence on individual organ conditions. This distinction is not merely taxonomic but has profound implications for understanding the nature of scientific knowledge. Mach shows how ancient theories of vision, such as Epicurus’s belief that sight comes from emanations from the eye, arose from overestimating physiological factors and drawing false analogies with the sense of touch. The proper separation of physical and physiological considerations is presented as a precondition for genuine scientific progress.
A third argument concerns the role of individual workers in the development of optical knowledge. Mach emphasizes that ideas about light’s nature were shaped by prominent individual scientists, transformed by new facts and associated views. His approach is explicitly critical and psychological, seeking to understand how particular thinkers arrived at their conclusions and why those conclusions took the forms they did. This is evident in his treatment of figures like Damianus, whose strange views are excused by reference to the examples of seeing in darkness and luminous animal eyes, and Newton, whose failure to notice the fixed spectral lines is explained by the fact that he and his followers did not simultaneously fulfill the requisite experimental conditions.
A fourth argument concerns the relationship between theory and experiment in optics. Mach repeatedly shows how theoretical preconceptions can blind scientists to empirical facts. Newton’s assumption that refraction and dispersion ran parallel made achromatic lenses seem impossible, and this prejudiced notion persisted until Dollond and Klingenstjerna demonstrated otherwise. Leroux’s observation of greater deviation in red than violet with an iodine-filled prism was dismissed as illusion and forgotten because it contradicted the expected color order. Christiansen and Kundt’s observations of anomalous dispersion fundamentally reformed dispersion theory precisely because they could not be accommodated within existing frameworks. These episodes support Mach’s broader argument that scientific theories must remain open to revision in the face of new experience.
A fifth argument concerns the continuity of scientific development. Mach emphasizes that discoveries are prepared by long chains of prior work and that the attribution of discoveries to individual genius often obscures the collective nature of scientific progress. The telescope, he argues, could not have been invented without the prior existence of suitable objects such as lenses, which had existed since classical times. Huygens held that only superhuman genius could have invented the telescope theoretically, but three hundred years of spectacle use contributed to its chance invention. Mach suggests that similar observations were probably made previously but kept secret, noting the strong tendency among early investigators to conceal their results, especially when publication entailed danger.
Key Concepts
The book develops several specific concepts that structure Mach’s treatment of optics. The first is the distinction between physical and physiological optical data, which Mach establishes in his introduction and maintains throughout. Physical data are those observable by all people with normal sight, while physiological data depend on individual organ conditions. This distinction is illustrated through examples: chalk appears white near a bright flame but yellow near a sodium flame; after taking santonin, both appear yellow. The same elements are sometimes physical, sometimes physiological features, and confusing these domains leads to one-sided theories.
A second key concept is the notion of the six fundamental properties of light. Mach enumerates these as: rectilinear propagation with definite velocity in homogeneous space; reflection and refraction at boundaries; refractive index differing per color; diffraction at edges and slits; periodicity in space and time subject to disturbances; and periodic properties representable only by distances perpendicular to the ray. These six properties provide the organizational framework for the book’s treatment of optical phenomena.
A third concept is the principle of shortest path or least time, developed by Hero for reflection and extended by Fermat to refraction. Hero proposed that light reflecting at a plane mirror travels between its initial and final points via a point such that the path is shorter than any other path through any other point of the plane. Fermat discovered that for refraction, light chooses the path involving the shortest time, leading to the condition that the sine of the angle of incidence divided by the velocity in the first medium equals the sine of the angle of refraction divided by the velocity in the second medium. This principle, Mach notes, stimulated J. Bernoulli in founding the calculus of variations.
A fourth concept is the distinction between different formulations of the refraction law. Snell finds the ratio of cosecants to be constant, while Descartes expresses the same relationship in terms of sines. Mach notes that Snell’s conception is more convenient for construction, while Descartes’ formula is more convenient for exact calculation. Reflection may be considered a special case of refraction where the refractive index equals negative one. Snell’s construction is used by Gauss in dioptrics research.
A fifth concept is the optical path, defined as the product of refractive index and geometrical path length. The principle that the optical path is a minimum in refraction provides a mechanical analogy, illustrated by a pulley apparatus, that makes Fermat’s principle intuitive.
A sixth concept is the projective relation between object space and image space in optical systems. Mach presents Möbius’s discovery that object space relates projectively to image space, lying in perspective with respect to the center of curvature. The center of curvature corresponds to itself as a double point, and each surface point corresponds to itself. This projective treatment provides the clearest presentation of the relationships between conjugate elements in optical systems.
A seventh concept is the distinction between normal and anomalous dispersion. Normal dispersion, in which refractive index increases with decreasing wavelength, was long thought to be the only possible arrangement. Cauchy developed a mathematical theory showing that refractive index increases as a function of decreasing wavelengths, agreeing well with observation. Anomalous dispersion, discovered by Christiansen and Kundt, occurs in substances with high absorption and marked surface colors. Kundt recognized a connection between absorption and anomalous dispersion: approaching an absorption region from the red end, refractive indices increase above normal values; approaching from the violet end, they decrease below normal. Normal dispersion is a special case where absorption lies beyond the violet end.
An eighth concept is the periodic nature of light, established through the phenomena of interference and diffraction. Grimaldi’s observation that light propagates not only directly, refracted, and reflected, but also by a certain fourth mode—diffracted—led him to compare light to a fluid spreading in waves. Newton’s theory of periodic “fits” of reflection and transmission explained the rings produced by pressing lenses together, with reflection occurring at certain thicknesses and transmission at others. Young demonstrated that homogeneous light possesses opposite qualities at equal distances in its direction of motion, capable of neutralizing each other and extinguishing light where united.
A ninth concept is the distinction between different types of color blindness. Dalton first investigated the phenomenon scientifically, and three types are distinguished: red-green blindness, in which only yellow and blue are seen in the spectrum; yellow-blue blindness, in which red is seen at both extremities and green in the center; and extremely rare total color blindness, in which the spectrum appears entirely white.
A tenth concept is the method of total reflection for measuring refractive indices, developed by Wollaston and applied by Christiansen to discover anomalous dispersion. The sine of the limiting angle equals the reciprocal of the refractive index, providing a convenient experimental method. Mach and Osnobischin improved this method using multiple reflecting prisms with cemented cells and an optical arrangement ensuring that only parallel rays fell on reflecting surfaces.
Themes
Several deeper themes run through Mach’s treatment of optics, giving the book its distinctive character as both a scientific treatise and a philosophical meditation on the nature of knowledge.
The most pervasive theme is the relationship between sensation and reality. Mach opens with the observation that brightness, darkness, light, and color cannot be described but only named. A copper piece visible in sunlight or near a lamp appears red, but the same red can arise from galvanic current through the eye, pressure on the eye, or hallucination. Red is “the terminal link of a chain of interrelated circumstances.” This theme recurs throughout the book, most notably in the treatment of physiological optics and the distinction between physical and physiological data. Mach’s insistence on the primacy of sensation as the foundation of all knowledge reflects his broader empiricist philosophy, but it also serves a specifically optical purpose: understanding the nature of light requires understanding the nature of sight.
A second theme is the historical contingency of scientific discovery. Mach repeatedly shows that discoveries depend on a complex interplay of conceptual preparation, experimental technique, and individual insight. The telescope could not have been invented without the prior existence of lenses; Newton failed to notice the fixed spectral lines because he did not simultaneously fulfill the requisite experimental conditions; Leroux’s observation of anomalous dispersion was dismissed because it contradicted theoretical expectations. These episodes suggest that scientific progress is not inevitable but depends on the fortunate conjunction of many factors.
A third theme is the danger of theoretical preconceptions. Mach shows how Newton’s assumption that refraction and dispersion ran parallel made achromatic lenses seem impossible, and how this prejudiced notion persisted until it was overturned by experiment. The supposed necessity of a particular color order in prismatic deviation was another preconception that mathematical theories reinforced until anomalous dispersion was discovered. Mach’s treatment of these episodes reflects his broader conviction that scientific theories must remain open to revision and that the history of science is largely the history of the correction of errors.
A fourth theme is the unity of optical phenomena. Mach emphasizes that the six fundamental properties of light are not independent but interconnected, and that the history of optics is the history of discovering these interconnections. Huygens showed that rectilinear propagation, reflection, and refraction could all be derived from finite propagation velocity; Fresnel completed this project by incorporating periodicity and polarization. The similarities between color distribution in different phenomena support the assumption of a common origin for light. This theme reflects Mach’s conviction that science seeks comprehensive descriptions that unify diverse phenomena.
A fifth theme is the relationship between theory and practice in science. Mach shows how practical concerns—the improvement of telescopes, the correction of lenses, the measurement of light intensity—drove theoretical developments. Fraunhofer’s interest in improving achromatic telescopes led him to investigate refraction and dispersion in different glasses and ultimately to discover the fixed spectral lines. The development of photometry by Bouguer and Lambert was motivated by practical needs for measuring light. This theme challenges the view of science as purely theoretical inquiry, emphasizing instead the mutual reinforcement of theoretical and practical concerns.
A sixth theme is the role of secrecy and openness in scientific communication. Mach notes that each science has a period of secrecy and magical character, and that knowledge of many optical discoveries was likely kept secret. The tendency to keep results secret was strong, especially if publication entailed danger, as Roger Bacon’s fate proves. The first burning for witchcraft took place at Toulouse in 1275, and a witchcraft trial occurred as late as 1782. This theme reflects Mach’s interest in the social conditions of scientific knowledge and his conviction that the progress of science depends on open communication.
A seventh theme is the relationship between light and other physical phenomena. Mach repeatedly draws analogies between light and sound, light and water waves, and light and the behavior of fluids. Grimaldi compared the periodic distribution of diffracted light to water flowing around obstacles or a ship’s wash. Young drew an analogy between light and sound diverging in concentric surfaces. These analogies served both as explanatory devices and as sources of theoretical insight, suggesting that the understanding of light requires understanding its relationships to other natural phenomena.
The Author’s Conclusions
Mach’s conclusions emerge from his historical and philosophical analysis of optical knowledge. He concludes that the development of optics demonstrates the progressive modification and restriction of propositions as new experiences are accumulated. The rectilinear propagation of light, first known to the ancients, was gradually incorporated into more comprehensive theoretical frameworks. Huygens showed that rectilinear propagation, reflection, and refraction could be derived from finite propagation velocity; Fresnel completed this project by incorporating periodicity and polarization. This progression exemplifies Mach’s view that scientific theories are comprehensive descriptions enabling prediction, and that progress consists of modifying and restricting propositions as new experiences limit earlier generalizations.
Mach concludes that the distinction between physical and physiological optics is fundamental and that confusing these domains leads to one-sided theories. The ancient belief that sight comes from emanations from the eye arose from overestimating physiological factors and drawing false analogies with the sense of touch. Goethe and Schopenhauer’s errors in optical theory stemmed from allowing subjective observations to improperly influence their physics. The proper separation of physical and physiological considerations is presented as a precondition for genuine scientific progress.
Mach concludes that theoretical preconceptions can blind scientists to empirical facts and that the history of optics is largely the history of the correction of such errors. Newton’s assumption that refraction and dispersion ran parallel made achromatic lenses seem impossible until Dollond and Klingenstjerna demonstrated otherwise. The supposed necessity of a particular color order in prismatic deviation persisted until Christiansen and Kundt’s observations of anomalous dispersion fundamentally reformed dispersion theory. These episodes support Mach’s broader conclusion that scientific theories must remain open to revision in the face of new experience.
Mach concludes that discoveries are prepared by long chains of prior work and that the attribution of discoveries to individual genius often obscures the collective nature of scientific progress. The telescope could not have been invented without the prior existence of lenses; three hundred years of spectacle use contributed to its chance invention. Similar observations were probably made previously but kept secret, reflecting the strong tendency among early investigators to conceal their results.
Mach’s final conclusions concern the nature of scientific knowledge itself. He disclaims being a forerunner of relativity theory, despite publications suggesting this, and calls himself “merely an unprejudiced rambler.” He withholds from atomistic belief as from relativism, and his objections rest on the physiology of the senses, theoretical ideas, and experimental conceptions. He questions whether relativity will prove more than “a transitory inspiration.” These remarks reveal Mach’s characteristic epistemological modesty: scientific theories are comprehensive descriptions enabling prediction, but they are always subject to revision and never capture reality in any final or absolute sense. The book concludes with the further explanation of light’s behavior through wave theory, tracing how the concepts of rectilinear rays, reflection, and the periodic properties of light were incorporated into increasingly comprehensive theoretical frameworks.
Significance
The significance of Mach’s Principles of Physical Optics lies in its unique combination of historical scholarship, philosophical analysis, and scientific exposition. The book represents a distinctive approach to the understanding of physical science, one that insists on the importance of historical and psychological factors in the development of scientific concepts. Mach’s treatment of optics exemplifies his broader philosophical position that scientific theories are comprehensive descriptions enabling prediction, grounded in experience and subject to revision as new experiences are accumulated.
The book is significant as a document of the transition from classical to modern physics. Written during a period of profound transformation in physical theory, it captures the state of optical knowledge at a moment when the wave theory of light was being challenged by new developments.
Inside the Book
Translators’ Note, Dedication, and Preface
The English edition opens with a note from the translators, John S. Anderson and Ames V. Young, who explain that they have adhered closely to the plan of the German original, preserving its frequent italics and altering few symbols so as not to disturb the lettering of the illustrations. They corrected a number of errors, chiefly algebraic, and added one figure for the sake of clarity. Their aim, they state, was to serve both students of physics and those readers drawn to Mach’s philosophical outlook. The note is dated November 1925, from Teddington.
Mach’s own dedication is addressed to Dr. Paul Carus and Messrs. Hegeler, recalling a correspondence that began when early issues of the Open Court failed to reach him. He credits Carus’s firm with educational work of lasting value and with masterly translations that allowed him to reach a wide public, and he mentions Edward C. Hegeler alongside Carus. The dedication is dated Munich, July 1913.
In his Preface, Mach explains that this book continues the task begun in his works on Mechanics and on Heat, and that its method most closely resembles the latter. He hopes to have revealed the origins of general optical concepts and the historical threads of their development, “extricated from metaphysical ballast,” while acknowledging that some chapters are only cursorily surveyed where other writers have already treated them exhaustively. His approach is critical and psychological: he shows how ideas about the nature of light were shaped by prominent individual workers and transformed by new facts and associated views. The work grew out of lectures beginning with geometrical optics, delivered when he was a Privatdozent in Prague, and he prioritizes salient facts over strict chronology. Owing to age and illness, he yielded to his publisher’s pressure to print this portion, with radiation, the decline of the emission theory, Maxwell’s theory, and relativity deferred to a later volume. His son, a long-time colleague, collaborated on the researches underlying these chapters. Mach states plainly that he must cancel his contemplation of relativity theory, and he disclaims being a forerunner of it, despite publications suggesting otherwise. Philosophers and physicists have crusaded against him, he notes, and he calls himself “merely an unprejudiced rambler,” withholding assent from atomistic belief as from relativism. His objections rest on the physiology of the senses, on theoretical ideas, and on experimental conceptions, all to be treated in the sequel; he questions whether relativity will prove more than “a transitory inspiration.” He thanks the late Professor E. Reusch of Tübingen, the Viennese optician Karl Fritsch-Prokesch, and Dr. Rudolf Steinheil of Munich, and acknowledges Professor J. Plassmann of Münster and J. G. Hagen, S.J., of the Specola Vaticana for discovering F. Grimaldi’s portrait in the possession of Professor M. Rajna at Bologna University Observatory; Professor Majocchi photographed it, and P. Christian Kiehl, S.J., and P. Kofler constructed the block of Grimaldi’s signature.
Chapter I: Introduction
Mach opens with the observation that sensations—brightness, darkness, light, colour—cannot be described, only named. A copper piece visible in sunlight or near a lamp appears red, yet the same red can arise from a galvanic current through the eye, from pressure on the eye, or from hallucination. Red, he says, is “the terminal link of a chain of interrelated circumstances,” and the differentiating feature is called a light-sensation. Chalk appears white near a bright flame but yellow near a sodium flame; after taking santonin, both appear yellow—showing that the same elements are sometimes physical, sometimes physiological features. Physical optical data are occurrences observable by all people with normal sight, while physiological optical data concern dependence on the individual organ’s condition. Confusing these two led Goethe and Schopenhauer into one-sided ideas.
Mach then summarizes six fundamental properties of light: rectilinear propagation with a definite velocity in homogeneous space; reflection and refraction at boundaries; a refractive index differing per colour; diffraction at edges and slits; periodicity in space and time, subject to disturbances; and periodic properties representable only by distances perpendicular to the ray. He credits Euclid with knowledge of rectilinear propagation, Hero with an elegant formulation of the reflection law, Descartes with the correct refraction law, Grimaldi with the discovery of diffraction, Newton with showing the infinite kinds of light differing in colour and refractive index, Thomas Young with the interference principle, Fresnel with the imperfect regularity of periodicity, Huygens with calcite observations, and Malus with further polarization studies. Theories, Mach explains, are comprehensive descriptions enabling prediction: if a phenomenon responds to reaction A, it will also respond to reaction B. Progress consists of modifying and restricting propositions as new experiences—refraction, diffraction—limit earlier generalizations. Huygens showed that rectilinear propagation, reflection, and refraction are derivable from finite propagation velocity, and Fresnel completed this by incorporating periodicity and polarization, revealing Huygens’ laws as special cases of diffraction.
Chapter II: The Rectilinear Propagation of Light
Mach notes that Ptolemy knew that dark objects are visible only by virtue of self-luminous bodies. Epicurus held that sight comes from emanations issuing from the eye—an error born of overestimating physiological factors and of an analogy with feeling. Descartes compared seeing to a blind man’s sticks. Damianus, son of Heliodor of Larissa, held strange views excused by examples of seeing in darkness and of luminous animal eyes—later shown by Gruithuisen and Brücke to result from external scattered light. Deep-sea fish with luminous organs recall these ancient views. Plato proposed συναύγεια, a meeting of rays from object and eye; Aristotle doubted the eye-emanation theory; Alhazen opposed it definitively. Euclid’s references to eye-rays are geometrical, as in modern perspective usage. For the ancient Greeks, “Optics” meant the science of sight.
Euclid’s Theory of Visual Rays
Euclid assumed rays emanate from the eye, forming a cone with its apex in the eye and its base passing through the visible object’s boundaries. These sight rays propagate rectilinearly, densely filling space but with empty interstices between them. Only objects struck by a ray are seen, so each optical image contains finite units, and objects disappear when too distant or small to fall between neighbouring rays. Mach remarks that this resembles Faraday’s lines of force. From these assumptions it follows that objects seen by higher rays appear higher, and that the same object appears greater when near and smaller when far.
Damianus’s Optical Hypotheses
Damianus, son of Heliodor of Larissa, wrote a small book on optical hypotheses listing fourteen propositions: that emanation from ourselves affects the objects seen; that what emanates is light; that light moves in straight lines; that it appears as a cone; that the cone is right-angled; that the cone is not uniformly filled with light; that things are seen within a right or acute angle; that larger things are seen within a greater angle; that sight works by light surrounding the cone’s axis; that sight works forward; that the cone’s apex lies in the pupil, forming the centre of a sphere; that sight occurs by direct ray, by bending, or by throwing back; that there is an affinity between the eye and the sun; and that the ray forms equal angles when reflected, as do sunlight rays.
The Preconditions for the Telescope and Early Optical Devices
The discovery of the telescope was not a spontaneous act of genius but the culmination of a long historical process, one that required a necessary precondition: the existence of suitable optical objects, such as lenses. This postulate is irrefutable, and historical investigation confirms that such objects were indeed present in classical times. The thread of civilization was strained but never broken by the great migrations of peoples in the 4th to 6th centuries, the devastation of the Black Death in the 14th century, and subsequent economic upheavals. The Crusades broadened Europe’s outlook and stimulated its industries, while Arabian culture penetrated into Spain. In 1453, the fall of Constantinople to the Turks drove Greek scholars westward, carrying with them the literary treasures of antiquity.
This period saw a cluster of transformative inventions and discoveries. Around the end of the 13th century, gunpowder, the compass, and spectacles appeared. By the mid-15th century, printed books were circulating, and voyages of exploration led to the discovery of America and the sea route to India. This new impetus in science and discovery produced a constellation of great minds, including Copernicus, Stevin, Gilbert, Kepler, Galileo, and Descartes. The science of optics received a particular stimulus from the world of art, especially from Leonardo da Vinci, who, observing optical phenomena from an artist’s perspective, advanced physical optics through his work on perspective, illumination, shade, and color.
The passage notes that every science has a period of secrecy and magical character. Optical phenomena appear in Porta’s “Natural Magic,” and it is likely that knowledge of many discoveries was deliberately kept secret. The vague descriptions found in the works of Porta and Roger Bacon (1216-1294) can be explained without assuming fabrication. The author recalls that as a boy of ten, he played with a lens and saw images of trees cast upon a wall, a common experience. The tendency to keep results secret was strong, especially when publication entailed danger, as the fate of Roger Bacon proves. The magic lantern is first described in 1671 by A. Kircher (1601-1680), but the biography of Benvenuto Cellini (1500-1571) mentions a frightful apparition that was likely produced by a similar arrangement. The danger threatening those who possessed knowledge was a real drawback. The first burning for witchcraft took place at Toulouse in 1275, and organized persecution began after the publication of the “Malleus maleficarum” in 1489. The Inquisition had been established as early as 1199, and its trials continued for centuries, with a witchcraft trial as late as 1782 and a trial by Inquisition as late as 1852.
Ancient Lenses and the Invention of Spectacles
The concave mirror was known to the ancients, though Euclid treated it faultily. Burning glasses are referenced in Aristophanes’ “Clouds,” and Pliny’s remarks, while vague, are not wholly fabricated. He describes glass spheres filled with water setting clothes on fire and physicians using them for burning bodies. Pliny’s reference to Nero’s emerald is ambiguous. The author himself examined a lens from a Roman grave in Mayence, measuring 5.5 cm in diameter with a focal length of 18-19 cm, which would have been perfectly usable as a reading or burning glass. Lessing denied that the ancients had magnifying glasses, claiming their vision was keener, but the existence of cameos suggests otherwise. Spherical glass vessels filled with water and plano-convex glass knobs demonstrate magnifying action. The Roman scientific disposition was meager, yet other lenses have been found, including a bi-convex glass in England, a plano-convex one in Pompeii, and a quartz lens in Nimrod. Ptolemy makes no mention of lenses, and Alhazen speaks clumsily of optical magnification, as both authors dealt with optics only for astronomical purposes.
Spectacles are first spoken of towards the end of the 13th century, around 1299. An epitaph in Florence from 1317 suggests Salvino degli Armati as the inventor, while Brother Alexander de Spina (d. 1313) is said to have copied his idea. References to reading with spectacles as a sign of old age appear in mid-14th century literature, and an eye-glass appears in a Giotto school picture from about 1350. Spectacles of smoky quartz were known in China in the 13th century, though Marco Polo’s travels contain no reference to them. The increased use of lenses naturally led to the observation of real images, the camera obscura, and possibly the magic lantern before Kircher. The crucial idea of observing a small real image with a magnifying glass may well have led to the invention of the telescope and microscope.
The First Telescopes and Microscopes
According to Wilde, Zacharias Joannides of Middelburg invented the telescope in 1590. Guilelmus Borelius stated that Hans (Joannes) and his son Zacharias invented the microscope, and he saw one in 1619. These early instruments likely used one convex and one concave lens. Franciscus Fontana claimed to have invented the microscope with two convex lenses in 1618. Galileo constructed his first telescope in 1609, using descriptions that had come from Holland, mounting two lenses in a leaden organ-pipe tube. He later made more perfect instruments, which he described in his “Sidereus nuncius” of 1610. Huygens held that only superhuman genius could have invented the telescope theoretically, but the author suggests that three hundred years of spectacle use contributed to a chance invention. Similar observations were likely made previously but kept secret; indeed, an attempt was made to sell a telescope on condition of secrecy. Huygens, in his “Dioptrica,” describes Galileo’s discoveries: the mountains and valleys of the moon, sun-spots, the solar rotation, Jupiter’s satellites, the phases of Venus, the resolution of the Milky Way, and differences in planetary apparent diameters. Galileo considered Saturn a triple star; it was Huygens who observed Saturn’s ring and one satellite, with Cassini adding two more.
Lens Properties and Ray Construction
The discussion then turns to the theoretical foundations of dioptrics, establishing that lens properties derive from a double application of the laws for a single spherical refracting surface. A lens shares characteristics with a single spherical surface: homocentric ray bundles with small incidence angles remain homocentric after refraction. Rays incident parallel to the axis converge at the second focus, while rays emerging parallel to the axis originate from the first focus. Planes perpendicular to the axis through the foci are designated as the first and second focal planes. Distant luminous points image in the second focal plane, and points in the first focal plane produce parallel beams after refraction.
For a parallel beam incident on a lens, the ray through the first focus emerges parallel to the axis and intersects the second focal plane at the beam’s focus. A beam diverging from a point in the first focal plane becomes parallel after refraction; the ray through that point parallel to the axis passes through the second focus, and all other rays emerge parallel to this direction. This construction method originated with R. Smith. To construct the path of an arbitrary ray, one can treat it as belonging to a parallel beam or as belonging to a point in the focal plane. Both constructions show that a ray through the center of an infinitely thin lens is undeviated, a property that holds approximately for moderate thickness. The image of any point can be constructed using any pair of rays.
Hering’s Theory of Color Vision
The discussion turns to the physiological foundations of color perception, asserting that the optic center must contain as many distinct chemical nerve processes as there are psychologically separable optical sensations. Hering, building upon Johannes Müller’s hypotheses, adopted red, yellow, green, blue, white, and black as the fundamental sensations. His crucial advance, following Plateau, was the stipulation that both the disturbance of the visual organ from equilibrium by a stimulus and its subsequent return to equilibrium constitute sense perception. Thus, both katabolic and anabolic processes—which Hering termed dissimilation and assimilation—are sensory. He arranged these fundamentals into three opposing pairs: black-white, red-green, and yellow-blue. White is excited by the entire spectrum together, while black arises as a reaction from white. Red and green, and yellow and blue, hold contending positions, producing complementary colors; equally powerful excitation of a contending pair annuls both, yielding white.
Color Blindness and Goethe’s Theories
Dalton, who observed he could distinguish fewer colors than others, first investigated this scientifically. Three types of color blindness exist: red-green blindness (seeing only yellow and blue), yellow-blue blindness (seeing red at both extremities and green in the center), and the extremely rare total color blindness where the spectrum appears entirely white. Goethe considered color a quality in the eye resulting from action and reaction; his observations on subjective colors were valuable, but he improperly let these insights influence his physics, treating complicated phenomena with turbid media as fundamental. Schopenhauer similarly held that color sensations utilize only part of retinal activity, with red and green as qualitatively equal halves, orange and blue as two-thirds and one-third, and yellow and violet as three-quarters and one-quarter of full activity—assumptions made naively without confirmation.
Newton’s Critics and Fraunhofer’s Discoveries
Newton’s theory was approved by Pardies, Hooke, and Huygens, though Linus and Rizzetti raised objections based on incorrect experiment repetition. Lucas was among Newton’s cleverest opponents, and his rigorous experiments might have advanced dispersion theory had the differences (due to different prism glass) been properly discussed. Newton held that refraction and dispersion ran parallel, making achromatic lenses impossible. Euler thought the eye achromatic and proposed liquid-filled lenses; Dollond failed initially but later manufactured achromatic prisms from two glasses. Wollaston discovered fixed spectral lines but misinterpreted them; Fraunhofer independently observed them, publishing in the Munich Academy Journal. His interest in improving achromatic telescopes drove his investigation. Seeking a reference line in sunlight, Fraunhofer discovered fixed dark lines: sunlight through a narrow slit fell on a prism at minimum deviation, and the spectrum viewed through a theodolite telescope showed many dark lines. These lines always appeared in the same positions, did not constitute color boundaries, were not diffraction effects, and were intrinsic to the solar spectrum. The lamp’s bright R line coincided with dark line D. Fraunhofer denoted principal lines with alphabet letters, determining minimum deviations in a water-filled prism. The D line can be found by illuminating half the slit with a spirit flame containing common salt, producing a bright yellow line matching D; line A corresponds to a bright potassium flame line.
Dispersion Curves and Prism Constructions
Fraunhofer’s lines enabled precise definition of light kinds—speaking of the refractive index of the D line is more precise than “yellow light.” Measurements showed refraction and dispersion bear no simple relation: for crown glass and oil of cassia, mean refractive indices are nearly equal, but oil of cassia’s dispersion is about three times greater. Dispersion curves have different shapes for different substances. The passage describes Snell’s construction applied twice for refraction through prisms, methods for determining refractive indices, and Fraunhofer’s formula for minimum deviation where incident and emergent rays are symmetrical. Newton missed the fixed lines because he and his followers did not simultaneously fulfill requisite conditions: a sufficiently narrow slit, minimum deviation, and a screen at the correct image distance.
Kirchhoff and Early Dispersion Theories
Kirchhoff used his law to explain Fraunhofer lines: if the sun is a glowing solid or liquid body surrounded by a gaseous atmosphere, the continuous spectrum of the solid center contains dark lines at positions where bright lines would appear in the gaseous atmosphere’s spectrum. Investigating the coincidence between Fraunhofer lines and terrestrial spectral lines enables chemical analysis of the sun’s atmosphere. Newton’s dispersion ideas created prejudiced notions—one being proportionality between refractive index and dispersion, another concerning the fixed order of colors. Cauchy’s theory assumed the separation of the smallest divisions of matter was not vanishingly small compared to light’s wavelength, developing a formula expressing refractive index increase as a function of decreasing wavelengths that agreed well with observation.
Anomalous Dispersion Discoveries
When Leroux noticed greater deviation in red than violet with an iodine-filled prism, the observation was dismissed as illusion. Christiansen and Kundt’s observations fundamentally reformed dispersion theory. Christiansen applied Wollaston’s total reflection method to fuchsine, finding abnormal behavior—no distinct total reflection boundary appeared. Spectrally examining reflected light showed green was totally reflected at perpendicular incidence; increasing angle caused total reflection for blue, violet, purple, rose, then yellow successively. His refractive index measurements for Fraunhofer lines showed the usual order completely destroyed. Kundt examined many substances, finding anomalous dispersion characteristic of substances with high absorption and marked surface colors. He recognized a connection between absorption and anomalous dispersion: approaching an absorption region from the red end, refractive indices increase above normal values; approaching from the violet end, they decrease below normal. Normal dispersion is a special case where absorption lies beyond the violet end. The author worked with G. v. Osnobischin to improve total reflection methods, using fuchsine solution between flint glass prisms, where broad colored bands appeared instead of a narrow blue fringe.
Grimaldi’s Diffraction Experiments
The discussion returns to Grimaldi’s observations on diffraction. He found that a shadow cast by an opaque object was somewhat broader than expected if light propagated rectilinearly. Light underwent the same deflection at the edges of a second slit after entering through a first slit. Grimaldi stated that light propagates not only directly, refracted, and reflected, but also by a certain fourth mode—diffracted. He observed three colored fringes bordering the shadow, each narrower than its neighbor toward the shadow. With bright light, fringes appeared inside the shadow, always an odd number of brighter fringes—light therefore penetrated into the shadow. This periodic distribution reminded Grimaldi of water flowing around obstacles, and he remarked that light seems to be a fluid spreading extremely rapidly and at least sometimes in waves through transparent bodies. Mach notes that Grimaldi’s conclusion that light sometimes renders darker a surface already illuminated from elsewhere was based on an illusion—the phenomenon is physiological, not physical, since brightness sensation depends not only on illumination intensity but on its differential coefficient with respect to area.
Hooke’s Micrographia and Newton’s Thin Plates
Hooke described microscope observations of needle-points, razors, fabrics, ice designs, snowflakes, cork, and insects, noticing colors in thin mica sections and rainbow fringes in air interstices, imitated by pressing glass plates together. All transparent bodies in thin layers showed colors, independent of surrounding substances. Opaque substances never showed colors, so interaction between front and back face reflections was essential. Hooke argued fiery bodies have violent internal motion, assuming with Descartes that velocity increases in denser media, making wave-surfaces oblique to rays. Blue resulted from oblique pulses with the weakest part preceding; red from the strongest part preceding.
Fresnel’s Experimental Refinements
Fresnel’s earliest experiments were remarkably simple: he admitted sunlight through a pinhole pierced in tin foil attached to a window shutter. Following a suggestion from Arago, he soon replaced this with the focal image from a small microscope lens, which produced a cone of light with a greater angle and eliminated the need for a heliostat. He also used a drop of honey set in a hole in a metal plate as a light source. Initially, he caught the interference fringes on ground glass and examined them with a magnifying glass. However, after accidentally looking around the edge of the ground glass, he discovered this was unnecessary—the fringes were bright and distinct enough to be viewed directly. So clear were they that he could even use a star as his light source and measure its fringes.
To eliminate the confusing double images caused by reflection from the back surface of ordinary glass mirrors, Fresnel blackened that surface. Modern practice favors obsidian glass for the same purpose. Arago, for his part, chose platinum mirrors, which yielded exceptionally bright images. These mirrors were fixed to a wooden block with blobs of wax, allowing them to be adjusted by pressure so that they inclined at an angle slightly less than 180 degrees. Running a finger down the adjacent edges of the two mirrors revealed whether one was advanced beyond the other. The interference fringes themselves ran perpendicular to the line joining the two mirror-images, a crucial distinction from diffraction fringes, which run parallel to the mirror edges. Fresnel used a prism for monochromatic illumination and adopted another of Arago’s suggestions: using deep red church window glass, which transmitted nearly monochromatic light.
With monochromatic light, the fringes appeared as simple bands of brightness and darkness. A key observation came from comparing the interference field with the side field produced by one mirror alone. The dark fringes were not merely dark in contrast to the bright ones; they were actually darker than the side field. Covering one mirror made the field uniformly bright, a striking demonstration that superposing more light could, in fact, destroy luminosity.
Geometric Analysis of the Fringes
A diagram of the two Fresnel mirrors shows signs indicating which portions of one mirror lie in front of or behind the corresponding portions of the other. The image in the right-hand mirror appears higher than that in the left. The fringes cross the junction of the mirrors obliquely, always perpendicular to the line joining the two virtual images. They are straight only when they run parallel to the line of contact, which occurs when the mirrors are in exact contact along their entire edges. Otherwise, the fringes curve at their ends, bending asymptotically toward the line of contact. A thread analogy clarifies this: imagine two threads of equal length pulled from the mirror-images through openings representing the mirrors. If the thread ends are moved so that the path differences remain equal, the right-hand thread catches on the line of separation when moved upward, bending its direction of motion toward that line. Moving downward, the left-hand thread bends in a similar fashion.
The mirrors can be turned so that the central line of contact is nearer the observer. In this arrangement, instead of a brighter superposition field, a dark shadow region appears where light arrives only by diffraction. This region shows similar oblique and curved fringes, which differ from Grimaldi fringes only in their greater complexity. Arago also suggested using light from a cylindrical lens as a source, which requires exact parallelism with the mirror intersection line. Light from different points of the cylindrical lens line is incoherent, and the corresponding points form separate fringe sets that obliterate one another. A demonstration device uses an oblique slit with a blackened glass plate having diagonal transparent slots. Optimal results come from sunlight impinging directly on the mirrors after passing through a narrow slit parallel to the mirror intersection, with the fringes projected onto a distant screen. For subjective viewing, daylight through a narrow slit with a magnifying lens works well.
Mechanical Adjustments and Troubleshooting
Fresnel invented a mechanical adjustment device with three screws: one screw varies the mirror inclination, while the other two adjust the coincidence of the edges. When fringes appear on the left, the right mirror edge is advanced, and a specific screw corrects this. The two edge-adjusting screws receive alternate small turns in the same direction, while the third screw controls the breadth of the fringes. A troubleshooting method places a short slit before the magnifying lens, perpendicular to the expected fringes. Viewing through a direct-vision prism then shows a spectrum crossed by dark bands. The orientation of these bands indicates the location of zero path difference, and adjusting the screws shifts the fringes toward the violet until they appear vertical.
With sunlight, a narrow slit produces broad diffraction images because the sun is a small source. The mirror edges form external Grimaldi diffraction bands, while the interference fringes appear between these Grimaldi fringes, inside the slit diffraction fringes. The use of extended sources like a bright sky requires explanation: the slit must be narrow enough that diffraction spreads the light so that both mirrors receive approximately equal coherent light from different regions of the sky.
Summary of Periodicity Knowledge
The chapter concludes by summarizing the state of knowledge regarding periodicity. Grimaldi discovered diffraction, and his successors—Hooke, Newton, Young, and initially Fresnel—provided incomplete explanations. Data from Grimaldi, Hooke, and Newton led to the view of a periodic nature, a view strengthened by Newton, Young, and Fresnel. The same properties recur at equal distances, while opposite properties appear at intermediate distances, with properties varying continuously along the ray. This interval relates to colour just as the refractive index does; in a medium of refractive index n, it becomes L/n. The periodicity is temporal—properties vary periodically with time at a point and spatially at an instant. The periodic time derives from the spatial distance and the velocity of light, and colour depends on this periodic time, not on the medium. The periodicity is regular over long regions but experiences disturbances many millions of times per second. The account claims nothing hypothetical; these results derive directly from experimental observation, though instructive analogies contributed to their discovery.
Large Path Differences and the Limits of Coherence
Newton had already used spectral resolution to establish interference at large path differences. Wrede spectrally investigated the linear region of glitter from mica bent into a cylindrical form, ascribing the channelled spectrum to interference between light reflected from the front and back faces. Poggendorff indicated the importance of this experiment for interference theory. Fizeau and Foucault investigated interference at large path differences to question whether a path difference could be determined where two beams cease to be coherent. They produced large path differences by displacing one Fresnel mirror, forming fringes on a narrow slit, and then spectrally resolving the transmitted light. They also investigated light reflected from thicker glass plates and transmitted by thicker crystalline plates between Nicols. If wavelengths λ and λ′ were contained exactly n and n′ times in the total path difference, then nλ = n′λ′. Counting m dark bands between wavelengths gave n′ = n − m and n = mλ′/(λ′ − λ). They found visible interference at path differences of about 4000 wavelengths; beyond this, the bands became too numerous and faint to count, a limit explained by spectral inhomogeneity rather than incoherence. Stefan later obtained interference indications using Talbot fringes for path differences up to 15,560 wavelengths of the H line.
Fresnel’s Mathematical Framework for Light
Fresnel and Young’s assumption that light consisted of mechanical vibrations akin to pendulum motion was a genuine hypothesis, extending beyond what facts alone dictated. Fresnel posited that the acceleration of an ether particle toward equilibrium, given a displacement u, could be expressed as a series au + bu² + cu³ + …, which for small displacements reduces to its first term. This yields a solution of simple harmonic motion, u = m sin √a·t, where m is the amplitude and the period is determined by the constant a. By introducing a phase shift, Fresnel developed transformations allowing him to represent a vibration as the sum of a sine and cosine term, and conversely, to combine such terms into a single sine wave with a new amplitude and phase. This led to his fundamental law for superposing two sine vibrations of the same period and direction, where the resultant amplitude is the square root of the sum of squares plus a cross-term involving the phase difference. He likened this to combining forces, where amplitudes form the sides of a triangle, and the resultant varies between the sum and difference of the individual amplitudes.
Fresnel’s conception of light intensity was tied to the vis viva (energy) of the waves, which for equal periods is proportional to the square of the amplitude. His crucial decision to treat light as transverse vibrations allowed him to explain the combination of oblique vibrations, a step that had been delayed due to hesitation in accepting transverse waves. He repeatedly asserted that perpendicularly polarized rays behave like perpendicular forces, a fact made evident in circular and elliptical polarization by total reflection, where the total intensity remains unaffected by the phase difference between the polarized components. Geometrically, uniform circular motion resolved along perpendicular diameters produces simple harmonic vibrations of equal period and amplitude with a quarter-wave phase difference. The superposition of two such perpendicular vibrations can yield circular motion (with equal amplitudes and a quarter-wave difference) or elliptical motion (with any other amplitude ratio or phase difference). The most general state of polarization is elliptical, which can be decomposed into two perpendicular linear components, and the combination of two opposite circular vibrations produces an elliptical vibration whose major and minor axes are determined by the sum and difference of the component amplitudes.
Applying these principles to interference, Fresnel showed that a wave traversing an additional distance D results in a resultant amplitude of a√(2 + 2 cos 2πD/λ). Extinction occurs when the path difference D is an odd multiple of half a wavelength, while maximum reinforcement occurs at integral multiples of the full wavelength. For white light divided into two equal beams, the spectral resolution of the resulting fringes shows that a point where the extreme red wavelength is contained exactly 100 times in D will be a maximum, but transitioning to a point where it is contained 200 times (a half-wavelength shift for that color) will show a complex pattern of 99 maxima and 100 extinctions.
Crossed Nicols and the Incoherence of Perpendicular Components
When two Nicol prisms are crossed, the components of light resolved into a single plane acquire an additional phase difference of half a wavelength, which is independent of color and diffraction deviation. This causes the minima of one pair of components to coincide with those of others, meaning the symmetrical fringe system has maxima where the previous case had minima. The horizontal bands in the spectrum for crossed Nicols are intermediate in position between those for parallel Nicols, and these effects appear in succession as the eyepiece Nicol is rotated. Fresnel, who did not use spectral resolution, could not readily distinguish these systems and regarded the effects as a special type of superposition of two fringe systems. The most important result from these experiments is the conclusion that perpendicularly polarized components of unpolarized light are mutually incoherent. This view, once laboriously established, now seems self-evident; with a mechanical conception of light, irregular disturbances caused phase changes, but once periodicity was recognized as transverse and two-dimensional, a varying vibration plane causes independent changes in intensity and phase in the perpendicularly resolved components, making them incoherent.
Chromatic Polarization and the Wedge Experiments
The laws of interference of polarized light provide the key to chromatic polarization. Arago discovered the first facts through Newton’s rings experiments, and Biot made measurements of plate thicknesses, developing a theory of “mobile polarization” modeled on Newton’s “fits.” Young recognized the dependence of color on path difference, and Fresnel removed the inconsistency between parallel conceptions. A wedge of a doubly refracting substance, cut with its crystal axis parallel or perpendicular to its sharp edges, is used in a Norrenberg’s polariscope. A quartz wedge with its axis parallel to its long edge, arranged at 45° to the plane of polarization, produces colored bands parallel to the sharp edge, following the order of Newton’s rings. For parallel and perpendicular Nicol positions, the colors are complementary. Biot’s superposition method involves placing a quartz plate over the wedge; with axes parallel, the colors disappear (summation), and with axes perpendicular, the behavior is as if at zero thickness (subtraction). Spectral analysis shows dark fringes entering at the violet end and passing to the red as thickness increases. When the wedge rotates in its own plane, colors strengthen and fade but the hue remains constant, with maximum saturation at 45° inclination of the quartz axis.
The Fresnel Rhomb and Quarter-Wave Plates
Continuing Malus’ experiments on internal reflection, Fresnel discovered circular polarization and constructed parallelepipeds (Fresnel rhombs). A rhomb placed with its plane of reflection at 45° produces light whose brightness is unchanged when a Nicol rotates, differing from both linearly polarized and unpolarized light. Two rhombs with parallel planes retransform the light to linear polarization, as the rhomb imparts a quarter-period retardation to the component polarized in the plane of incidence. Mica can be split to produce a quarter-wave plate, where the longer edge corresponds to the plane of reflection of the rhomb and the more retarded component. Circularly polarized light is separable into two linear components with a quarter-vibration phase difference. Fresnel called the more general state “elliptical” polarization, which results from variations in inclination or incidence angles. The rhomb imparts nearly identical polarization to each color with white light, whereas a quarter-wave plate cannot produce a quarter-wavelength retardation for all colors simultaneously, leading to yellowish and bluish hues.
Experimental Observations with Quartz and Nicol Prisms
In a series of experiments, shifting to a more refrangible spectral region requires turning the Nicol further clockwise to reach maximum or minimum brightness. With a wedge-shaped quartz plate cut perpendicular to its axis, colored bands traverse the quartz parallel to its thin edge, and turning the Nicol clockwise shifts the bands toward the thick end for right-handed quartz. Fresnel explained these effects by proposing that a linearly polarized ray entering quartz along its axis divides into two oppositely circularly polarized rays with slightly different velocities, which recombine in air to produce linearly polarized light with a rotated plane. He experimentally proved the existence of these two rays. When right-handed circularly polarized light from a Fresnel rhomb is covered by a quartz plate, no colors appear for any Nicol position, since one interfering ray is absent. Imitation experiments with quarter-wave plates can reproduce the behavior of quartz cut perpendicular to its axis, with bands shifting toward the thicker end with clockwise Nicol rotation.
The Construction of Cartesian Ovals
The discussion turns to a geometric method for constructing an interface that will perfectly image a point. Given a luminous point A and its image B, with D the intersection of the line AB with the desired surface, a refractive index of 3/2 is assumed. A point H is set so that DH is 3/2 of DB. The construction proceeds by drawing arcs from A and B with radii that maintain the correct optical path length, yielding points on the meridian curve. These curves, known as Cartesian ovals, were first explored by Descartes in his Géométrie. They can be generated mechanically for a refractive index of 2 using a thread looped around two pins and a pencil, tracing an oval that encloses either both points or just one, depending on the configuration. For higher indices, the thread method becomes impractical, so point-by-point construction is necessary, a process that can be accelerated by a clever shifting of the paper to sum the relevant distances.
Descartes’ Dioptrics and Conic Surfaces
Descartes recognized that a refracting surface shaped as a conic of revolution could focus parallel light to a single point. For a glass ellipsoid with a specific eccentricity, a ray parallel to the major axis is refracted toward the far focus. The proof relies on the normal at the point of incidence bisecting the angle between the incident and refracted rays, a property that leads to the correct focal relationship. While Descartes’ own demonstrations were burdened by excessive construction lines and his less elegant form of the refraction law, he established a similar focusing property for a hyperboloid of revolution. Huygens later showed that these results follow directly from his own principle of equal time, reconstructing Descartes’ reasoning more clearly. By considering the path difference as a point moves along the surface, Huygens derived the same condition that the sum of the optical path lengths must be constant.
Wave-Surfaces and Focal Lines
When rays from a point converge after reflection or refraction, they all traverse paths of equal time. By dividing these paths into equal-time elements, one can construct the wave-surface, to which the rays are normal. However, a homocentric pencil of rays loses its perfect convergence after a single refraction at a spherical surface. The wave-surface then develops a fold, contracting and inverting like a stretchable sheet. The continuous intersections of the rays form a caustic, or focal line, which is the envelope of the ray system. A wave-surface section can be described by unwinding a thread stretched around this focal line.
Fresnel’s Examination of Diffraction
Fresnel’s early work on diffraction, in agreement with Young, treated the fringes as interference between direct light and light deflected from the edges of an obstacle. He soon abandoned this reflection-deviation hypothesis, as a razor blade’s polished back produced no stronger fringes than its edge, and slits of equal width between different types of edges gave identical patterns. The measured fringe distances also did not match the theory. Fresnel concluded that all light, not just rays striking an edge, spreads in all directions, and diffraction arises from the mutual interference of this diffracted light. This was his “corrected Huygens’ principle,” combining Huygens’ elementary waves with the concept of interference. He faced mathematical difficulties in explaining why an unobstructed wave produces the same effect as one broken into elementary waves, resolving this by showing that the contributions from most of the wavefront cancel, leaving only the effect of the central region.
Poisson’s Deduction and Arago’s Confirmation
Fresnel developed integral expressions for the intensity of light at a point, which agreed well with experiments for straight-edged screens. From his theory, Poisson deduced the surprising result that the center of the shadow of a small circular disc should be as bright as if the disc were absent. Arago confirmed this experimentally, and Fresnel derived it independently. This led to the elegant zone theory, where the wavefront is divided into circular zones of equal area. Light from an even number of zones cancels, producing darkness, while an odd number produces brightness. A plate constructed with alternating transparent and opaque zones acts as a lens, a device later explored by Soret and improved by Rayleigh.
Fraunhofer’s and Schwerd’s Contributions
Fraunhofer simplified the experimental setup by using a telescope to observe diffraction patterns from slits, a method that did not require precise parallel alignment. He observed that patterns spread as slit width decreases and that two parallel slits produce interference fringes superimposed on the single-slit pattern. Schwerd later provided the mathematical theory for these observations, deriving the intensity formula for a single slit and the general grating equation for multiple slits. The concept of the “amplitude fan” was introduced as a graphical method for visualizing the summation of wave contributions.
Grating Spectra and Their Properties
The principle of the amplitude fan elegantly explains the formation of grating spectra. At the focus, all rays reinforce to form an image of the source. At a point beside the focus, the fan of amplitudes is incomplete, but when the path difference between adjacent slits is an integral number of wavelengths, the resultants from all slits coincide, producing a sharp principal maximum. White light is thus dispersed into a series of spectra, with the violet end closest to the center. These spectra can overlap for higher orders, a problem Fraunhofer solved by using a prism to disperse the light perpendicularly. Certain spectra may be entirely missing if the slit width is such that it cancels the light for that order. The wavelength of light can be precisely determined by measuring the angle of diffraction for a known grating.
Talbot’s Bands and Newton’s Rings
Talbot observed that placing a thin plate over half of the pupil while viewing a spectrum produces dark bands, which he correctly attributed to the relative retardation of the two halves of the beam. The phenomenon was later explained in detail using a Jamin’s compensator. The discussion then returns to Newton’s rings, first observed with a concave mirror. Young provided a rudimentary theory involving the interference of light reflected from the front and back surfaces. Stokes later attempted to reproduce the rings with two parallel semi-opaque planes but failed, concluding that the light must pass to and fro through the same particles.
The Appendix: A Memoir on the Properties of Reflected Light
The book concludes with an appendix containing a memoir that announces a fundamental discovery: light reflected from the surface of a transparent body acquires new properties, distinguishing it from direct light. The author found that light reflected at a specific angle behaves as if it had undergone double refraction. This modified light can entirely escape reflection at other surfaces, allowing for precise absorption measurements. When this specially prepared ray is reflected by a second mirror, its intensity varies according to a specific law, reaching an absolute minimum at a certain orientation. The author determined that the quantity of reflected light is proportional to the square of the cosine of the angle between two lines perpendicular to the ray, one parallel to each mirror. This behavior is explained by proposing that light consists of molecules with three principal axes, and that reflection acts on these molecules according to their orientation. The author argues that these observations are incompatible with Huygens’ wave hypothesis and support a corpuscular theory of light, where the phenomena are governed by the forces acting on the luminous molecules. The memoir details the laws governing these effects for reflection and double refraction, showing how the orientation of the molecules relative to the reflecting or refracting surface determines the outcome. The author concludes by claiming this theory unifies many previously unrelated facts, without speculating on the ultimate cause of the forces involved.
